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+# UNIVERSAL APPROXIMATION WITH DEEP NARROW NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The classical Universal Approximation Theorem certifies that the universal approximation property holds for the class of neural networks of arbitrary width. Here we consider the natural ‘dual’ theorem for width-bounded networks of arbitrary depth. Precisely, let $n$ be the number of inputs neurons, $m$ be the number of output neurons, and let $\rho$ be any nonaffine continuous function, with a continuous nonzero derivative at some point. Then we show that the class of neural networks of arbitrary depth, width $n + m + 2$ , and activation function $\rho$ , exhibits the universal approximation property with respect to the uniform norm on compact subsets of $\mathbb { R } ^ { n }$ . This covers every activation function possible to use in practice; in particular this includes polynomial activation functions, making this genuinely different to the classical case. We go on to consider extensions of this result. First we show an analogous result for a certain class of nowhere differentiable activation functions. Second we establish an analogous result for noncompact domains, by showing that deep narrow networks with the ReLU activation function exhibit the universal approximation property with respect to the $p$ -norm on $\mathbb { R } ^ { n }$ . Finally we show that width of only $n + m + 1$ suffices for ‘most’ activation functions.
+
+# 1 INTRODUCTION
+
+The Universal Approximation Theorem (Cybenko, 1989; Hornik, 1991; Pinkus, 1999) states that universal approximation holds for the class of neural networks with a single hidden layer of arbitrary width, with any continuous nonpolynomial activation function:
+
+Theorem 1.1. Let $\rho \colon \mathbb { R }  \mathbb { R }$ be any continuous function. Let $\mathcal { N } _ { n } ^ { \rho }$ represent the class of neural networks with activation function $\rho _ { i }$ , with n neurons in the input layer, one neuron in the output layer, and one hidden layer with an arbitrary number of neurons. Let $K \subseteq \mathbb { R } ^ { n }$ be compact. Then $\dot { \mathcal { N } } _ { n } ^ { \rho }$ is dense in $C ( K )$ if and only if ρ is nonpolynomial.
+
+What if arbitrary width is replaced with arbitrary depth? Put more precisely, can networks of bounded width and arbitrary depth provide universal approximation? In some sense this poses a question ‘dual’ to the problem answered by the classical Universal Approximation Theorem. We refer to networks of this type as deep, narrow networks.
+
+Furthermore we might ask how narrow the network may be, and what activation functions may be admitted. We provide a near-complete answer to these various questions.
+
+Universal approximation may be established with respect to more than one topology. Continuous activation functions beget networks representing continuous functions. Thus when working with respect to the uniform norm, it is natural to seek density in $C ( K ; \mathbb { R } ^ { m } )$ for $K \subseteq \mathbb { R } ^ { n }$ . When working with respect to the $p$ -norm, it is natural to seek density in $L ^ { p } ( \mathbb { R } ^ { n } ; \mathbb { R } ^ { m } )$ for $p \in [ 1 , \infty )$ . In this latter case we may hope to generalise to noncompact domains, as functions in $L ^ { p } ( \mathbb { R } ^ { n } ; \mathbb { R } ^ { m } )$ must exhibit some sort of decay.
+
+The primary motivation for this work stems from the work of Lu et al. (2017), who study this question in the special case of the popular ReLU activation function, and who establish density in $L ^ { 1 } ( \mathbb { R } ^ { n } )$ . The other notable result we are aware of is the work of Hanin & Sellke (2017), who show another special case: they also consider the ReLU activation function, and establish density in $C ( K ; \mathbb { R } ^ { m } )$ for $K \subseteq \mathbb { R } ^ { n }$ compact.
+
+This article demonstrates generalisations of these results, in particular to general activation functions, without relying on the strong algebraic and analytic properties of the ReLU activation function. This also improves certain results specific to the ReLU.
+
+The rest of the paper is laid out as follows. Section 2 discusses existing work. Section 3 provides a brief summary of our results; these are then presented in detail in Section 4. Section 5 is the conclusion. Several proofs are deferred to the appendices, due to length and technical content.
+
+# 2 CONTEXT
+
+Some positive results have been established showing that particular classes of networks are dense in certain spaces. Some negative results have also been established, showing that insufficiently wide networks will fail to be dense.
+
+Hanin & Sellke (2017) have shown that deep narrow networks with the ReLU activation function exhibit the universal approximation property in $C ( K ; \mathbb { R } ^ { m } )$ for $K \subseteq \mathbb { R } ^ { n }$ compact.
+
+Lu et al. (2017) have shown that deep narrow networks with the ReLU activation function exhibit the universal approximation property in $L ^ { 1 } ( \mathbb { R } ^ { n } )$ , whilst Lin & Jegelka (2018) have shown that a particular description of residual networks, with the ReLU activation function, also exhibit the universal approximation property in this space. We are not aware of any results for the general case of $L ^ { p } ( \mathbb { R } ^ { n } ; \mathbb { R } ^ { m } )$ for $p \in [ 1 , \infty )$ .
+
+We do not know of any positive results applying to activation functions other than the ReLU.
+
+Regarding widths insufficient for a class of deep narrow networks to exhibit the universal approximation property, consider the case of a network with $n$ input neurons and a single output neuron. For certain activation functions, Johnson (2019) shows that width $n$ is insufficient to give density in $C ( K )$ . For the ReLU activation function, Lu et al. (2017) show that width $n$ is insufficient to give density in $L ^ { 1 } ( \mathbb { R } ^ { n } )$ , and that width $n - 1$ is insufficient in $L ^ { 1 } ( [ - 1 , 1 ] ^ { n } )$ . For the ReLU activation function, Hanin & Sellke (2017) shows that width $n$ is insufficient to give density in $C ( K )$ , and that in fact that this is the greatest possible width not achieving universal approximation in this context.
+
+The precise minimum width for activation functions other than ReLU, or for multiple output neurons, remains unknown.
+
+Everything discussed so far is in the most general case of approximating functions on Euclidean space: in the language of machine learning, they are regression tasks. There has been some related work in the special case of classification tasks, for example Beise et al. (2018); Szymanski & McCane (2012); Rojas (2003); Nguyen et al. (2018). There has also been some related work in the special case of certain finite domains; Le Roux & Bengio (2010) show that networks with sigmoid activation function and width $n$ can approximate any distribution on $\{ 0 , 1 \} ^ { n }$ . See also Sutskever & Hinton (2008). Montufar (2014) considers the analogous scenario for distributions on ´ $\{ 0 , 1 , \ldots , q - 1 \} ^ { n }$ .
+
+# 3 SUMMARY OF RESULTS
+
+Definition 3.1. Let $\rho \colon \mathbb { R }  \mathbb { R }$ and $n , m , k \in \mathbb { N }$ . Then let N ρn,m,k represent the class of functions $\mathbb { R } ^ { n } \to \mathbb { R } ^ { m }$ described by neural networks with $n$ neurons in the input layer, $m$ neurons in the output layer, $k$ neurons in each hidden layer, and an arbitrary number of hidden layers, such that every neuron in every hidden layer has activation function $\rho$ , and every neuron in the output layer has the identity activation function.
+
+Our central result is the following theorem.
+
+Theorem 3.2. Let $\rho \colon \mathbb { R }  \mathbb { R }$ be any continuous function which is continuously differentiable at at least one point, with nonzero derivative at that point. Let $K \subseteq \mathbb { R } ^ { n }$ be compact. Then $\mathcal { N N } _ { n , m , n + m + 2 } ^ { \rho }$ is dense in $C ( K ; \mathbb { R } ^ { m } )$ .
+
+The technical condition is very weak; in particular it is satisfied by every piecewise- $C ^ { 1 }$ function not identically zero. Thus any activation function that one might practically imagine using on a computer must satisfy this property.
+
+Theorem 3.2 is proved by handling particular classes of activation functions as special cases. First we have the result for nonpolynomial activation functions, for which the width can be made slightly smaller.
+
+Theorem 4.4. Let $\rho \colon \mathbb { R }  \mathbb { R }$ be any continuous nonpolynomial function which is continuously differentiable at at least one point, with nonzero derivative at that point. Let $K \subseteq \mathbb { R } ^ { n }$ be compact. Then $\mathcal { N N } _ { n , m , n + m + 1 } ^ { \rho }$ is dense in $C ( K ; \mathbb { R } ^ { m } )$ .
+
+We observe a corollary for noncompact domains, which generalises Lu et al. (2017, Theorem 1) to multiple output neurons, a narrower width, and $L ^ { p }$ for $p \geqslant 1$ instead of just $p = 1$ .
+
+Corollary 4.6. Let $\rho$ be the ReLU activation function. Let $p \in [ 1 , \infty )$ . Then $\mathcal { N N } _ { n , m , n + m + 1 } ^ { \rho }$ dense in $L ^ { p } ( \mathbb { R } ^ { n } ; \mathbb { R } ^ { m } )$ .
+
+Moving on to polynomial activation functions, the smaller width of $n + m + 1$ also suffices for a large class of polynomials.
+
+4.8. and $\rho \colon \mathbb { R }  \mathbb { R }$ polynomial for whibe compact. Then a point is dens $\alpha \in \mathbb { R }$ at. $\rho ^ { \prime } ( \alpha ) = 0$ $\rho ^ { \prime \prime } ( \alpha ) \neq 0$ $K \subseteq \mathbb { R } ^ { n }$ $\mathcal { N N } _ { n , m , n + m + 1 } ^ { \rho }$ $C ( K ; \mathbb { R } ^ { m } )$
+
+The simplest example of such a $\rho$ is $x \mapsto x ^ { 2 }$ . Note that in the classical arbitrary-width case it is both necessary and sufficient that the activation function be nonpolynomial. Here, however, the same restriction does not hold. Polynomial activation functions are a reasonable choice in this context.
+
+The technical restrictions on the polynomial may be lifted by allowing the full $n + m + 2$ neurons per hidden layer.
+
+Theorem 4.10. Let $\rho \colon \mathbb { R }  \mathbb { R }$ be any nonaffine polynomial. Let $K \subseteq \mathbb { R } ^ { n }$ be compact. Then $\mathcal { N N } _ { n , m , n + m + 2 } ^ { \rho }$ is dense in $C ( K ; \mathbb { R } ^ { m } )$ .
+
+It is clear that Theorems 4.4 and 4.10 together imply Theorem 3.2.
+
+Finally we observe that even pathological cases not satisfying the technical condition of Theorem 3.2 may exhibit the universal approximation property.
+
+Proposition 4.13. Let $w \colon \mathbb { R }  \mathbb { R }$ be any bounded continuous nowhere differentiable function. Let $\rho ( x ) = \sin ( x ) + w ( x ) \mathrm { e } ^ { - x }$ , whise in o be nowhere differentiable. Let . $K \subseteq \mathbb { R } ^ { n }$ be compact. $\mathcal { N N } _ { n , m , n + m + 1 } ^ { \rho }$ $C ( K ; \mathbb { R } ^ { m } )$
+
+Whilst not of direct practical application, this result exemplifies that little necessarily needs to be assumed about an activation function to understand the corresponding class of neural networks.
+
+Remark 3.3. Every proof in this article is constructive, and can in principle be traced so as to determine how depth changes with approximation error. We have instead chosen to focus on quantifying the width necessary for universal approximation. In fact there are places in our arguments where we have used a deeper network over a shallower one, when the deeper network is more easily explained.
+
+Remark 3.4. An understanding of universal approximation in deep narrow networks is applicable to an understanding of bottlenecks, when information must be discarded due to space constraints, for example in autoencoders (Bengio et al., 2006). This article demonstrates that certain narrow networks will not constitute a bottleneck; a converse example is Johnson (2019), who demonstrates that networks of insufficient width are forced to maintain certain topological invariants.
+
+# 4 UNIVERSAL APPROXIMATION
+
+# 4.1 PRELIMINARIES
+
+Remark 4.1. A neuron is usually defined as an activation function composed with an affine function. For ease, we shall extend the definition of a neuron to allow it to represent a function of the form $\psi \circ \rho \circ \phi$ , where $\psi$ and $\phi$ are affine functions, and $\rho$ is the activation function. This does not increase the representational power of the network, as the new affine functions may be absorbed into the affine parts of the next layer, but it will make the neural representation of many functions easier to present. We refer to these as enhanced neurons. It is similarly allowable to take affine combinations of multiple enhanced neurons; we will use this fact as well.
+
+One of the key ideas behind our constructions is that most reasonable activation functions can be taken to approximate the identity function. Indeed, this is essentially the notion that differentiability captures: that a function is locally affine. This makes it possible to treat neurons as ‘registers’, in which information may be stored and preserved through the layers. This allows for preserving the input values between layers, which is crucial to performing computations in a memory-bounded regime. Thus our constructions have strong overtones of space-limited algorithm design in traditional computer science settings.
+
+Lemma 4.2. Let $\rho \colon \mathbb { R }  \mathbb { R }$ be any continuous function which is continuously differentiable at at least one point, with nonzero derivative at that point. Let $L \subseteq \mathbb { R }$ be compact. Then a single enhanced neuron with activation function $\rho$ may uniformly approximate the identity function $\iota \colon \mathbb { R }  \mathbb { R }$ on $L$ , with arbitrarily small error.
+
+Proof. By assumption, as $\rho$ is continuously differentiable, there exists $[ a , b ] \subseteq \mathbb { R }$ with $a \neq b$ , on some neighbourhood of which $\rho$ is differentiable, and $\alpha \in ( a , b )$ at which $\rho ^ { \prime }$ is continuous, and for which $\rho ^ { \prime } ( \alpha )$ is nonzero.
+
+For $h \in \mathbb { R } \setminus \{ 0 \}$ , let $\phi _ { h } ( x ) = h x + \alpha .$ , and let
+
+$$
+\psi _ { h } ( x ) = \frac { x - \rho ( \alpha ) } { h \rho ^ { \prime } ( \alpha ) } .
+$$
+
+Then
+
+$$
+\iota _ { h } = \psi _ { h } \circ \rho \circ \phi _ { h }
+$$
+
+is of the form that an enhanced neuron can represent. Then for all $u \in [ a , b ]$ , by the Mean Value Theorem there exists $\xi _ { u }$ between $u$ and $\alpha$ such that
+
+$$
+\rho ( u ) = \rho ( \alpha ) + ( u - \alpha ) \rho ^ { \prime } ( \xi _ { u } ) ,
+$$
+
+and hence
+
+$$
+\begin{array} { l } { \iota _ { h } ( x ) = ( \psi _ { h } \circ \rho \circ \phi _ { h } ) ( x ) } \\ { \qquad = \psi _ { h } \left( \rho ( \alpha ) + h x \rho ^ { \prime } ( \xi _ { h x + \alpha } ) \right) } \\ { \qquad = \frac { x \rho ^ { \prime } ( \xi _ { h x + \alpha } ) } { \rho ^ { \prime } ( \alpha ) } } \end{array}
+$$
+
+for $h$ sufficiently small that $\phi _ { h } ( L ) \subseteq [ a , b ]$ .
+
+Now let $\rho ^ { \prime }$ have modulus of continuity $\omega$ on $[ a , b ]$ . Let $\iota \colon \mathbb { R }  \mathbb { R }$ represent the identity function. Then for all $x \in L$ ,
+
+$$
+\begin{array} { l } { | \iota _ { h } ( x ) - \iota ( x ) | = | x | \left| \frac { \rho ^ { \prime } ( \xi _ { h x + \alpha } ) - \rho ^ { \prime } ( \alpha ) } { \rho ^ { \prime } ( \alpha ) } \right| } \\ { \displaystyle \leqslant \frac { | x | } { | \rho ^ { \prime } ( \alpha ) | } \omega ( h x ) , } \end{array}
+$$
+
+and so $\iota _ { h }  \iota$ uniformly over $L$ .
+
+Notation. Throughout the rest of this paper $\iota _ { h }$ will be used to denote such an approximation to the identity function, where $\iota _ { h }  \iota$ uniformly as $h  0$ .
+
+An enhnaced neuron may be described as performing (for example) the computation $x \mapsto \iota _ { h } ( 4 x { + } 3 )$ . This is possible as the affine transformation $x \mapsto 4 x + 3$ and the affine transformation $\phi _ { h }$ (from the description of $\iota _ { h }$ ) may be combined together into a single affine transformation.
+
+# 4.2 NONPOLYNOMIAL ACTIVATION FUNCTIONS
+
+We consider the ‘Register Model’, which represents a simplification of a neural network.
+
+$\mathcal { T } _ { n , m , n + m \pm 1 } ^ { \rho }$ 4.3 repyer, Model). Let class of neurneurons in e $\rho \colon \mathbb { R }  \mathbb { R }$ be any continuous nonpolynomial function. Let with n neurons in the input layer, m neurons in layer, an arbitrary number of hidden layers, and $n + m + 1$   
+for which neuron in $n + m$ of the neurons in each hidden layeidden layer has activation function have. Let tity activation funcbe compact. Then $\rho$ $K \subseteq \mathbb { R } ^ { n }$ $\mathcal { I } _ { n , m , n + m + 1 } ^ { \rho }$   
+is dense in $C ( K ; \mathbb { R } ^ { m } )$ .
+
+![](images/540adab59ada4efbc4c2aa54a6e857e3295f505448124700139b6997dfa485f5.jpg)  
+Figure 1: A simple example of how to prove the Register Model. The values $x _ { 1 } , \ldots , x _ { n }$ are inputs to the network, and the value $\textstyle \sum _ { j = 1 } ^ { M } \sigma _ { j }$ is the output. Each cell represents one neuron. Each $\sigma _ { i }$ is of the form $\psi _ { i } \circ \rho \circ \phi _ { i }$ , where $\psi _ { i }$ and $\phi _ { i }$ are affine functions and $\rho$ is the activation function.
+
+See Appendix A for the proof.
+
+A simplified depiction of the proof of the Register Model is shown in Figure 1, for the special case of $m = 1$ . It uses $n$ neurons in each layer as registers to preserve the input values. A single neuron in each layer performs a computation based off of the input values, which were preserved in the previous layer. The remaining neuron in each layer also acts a register, gradually summing up the results of the computation neurons. The computation neurons may be shown to exist by the classical Universal Approximation Theorem.
+
+The Register Model is similar to Hanin & Sellke (2017), who have a related construction specific to the ReLU. The idea of the Register Model may also be thought of as thematically similar to residual networks, as in Lin & Jegelka (2018): in both cases the network is almost applying the identity transformation at each layer, with only a small amount of nonlinearity.
+
+Theorem 4.4. Let $\rho \colon \mathbb { R }  \mathbb { R }$ be any continuous nonpolynomial function which is continuously differentiable at at least one point, with nonzero derivative at that point. Let $K \subseteq \mathbb { R } ^ { n }$ be compact. Then $\mathcal { N N } _ { n , m , n + m + 1 } ^ { \rho }$ is dense in $C ( K ; \mathbb { R } ^ { m } )$ .
+
+Proof. Let $f \in C ( K ; \mathbb { R } ^ { m } )$ and $\varepsilon > 0$ . Set up a neural network as in the Register Model (Proposition 4.3), approximating $f$ to within $\varepsilon / 2$ . Every neuron requiring an identity activation function in the Register Model will instead approximate the identity, in the manner of Lemma 4.2.
+
+Uniform continuity preserves uniform convergence, compactness is preserved by continuous functions, and a composition of two uniformly convergent sequences of functions with uniformly continuous limits is again uniformly convergent. So as a neural network is a layer-by-layer composition of functions then the new model can be taken within $\varepsilon / 2$ of the Register Model, with respect to $\| \cdot \| _ { \infty }$ in $K$ , by taking $h$ sufficiently small.
+
+Remark 4.5. This of course implies approximation in $L ^ { p } ( K , \mathbb { R } ^ { m } )$ for $p \in [ 1 , \infty )$ . However, when $\rho$ is the ReLU activation function, then the next corollary shows that in fact the result may be generalised to unbounded domains.
+
+Corollary 4.6. Let $\rho$ be the ReLU activation function. Let $p \in [ 1 , \infty )$ . Then $\mathcal { N N } _ { n , m , n + m + 1 } ^ { \rho }$ dense in $L ^ { p } ( \mathbb { R } ^ { n } ; \mathbb { R } ^ { m } )$ .
+
+See Appendix B for the proof.
+
+Given some $f \in L ^ { p } ( \mathbb { R } ^ { n } ; \mathbb { R } ^ { m } )$ , the essential idea of the proof is to choose a compact set $K \subseteq \mathbb { R } ^ { n }$ on which $f$ places most of its mass, and find a neural approximation to $f$ on $K$ in the manner of Theorem 4.4. Once this is done, a cut-off function is applied outside the set, so that the network takes the value zero in $\mathbb { R } ^ { n } \setminus K$ . The interesting bit is finding a neural representation of such cut-off behaviour.
+
+In particular the ‘obvious’ thing to do – multiply by a cut-off function – does not appear to have a suitable neural representation, as merely approximating the multiplication operation is not necessarily enough on an unbounded domain. Instead the strategy is to take a maximum and a minimum with suitable cut-off functions.
+
+For the classical Universal Approximation Theorem, it was necessary that the activation function be nonpolynomial. However that turns out to be unnecessary here; deep narrow networks are different to shallow wide networks, and polynomial activations functions are reasonable choices.
+
+We begin with the simplest possible nonaffine polynomial, namely $\rho ( x ) = x ^ { 2 }$ .
+
+Proposi uare Model). Let $\rho ( x ) = x ^ { 2 }$ . Let $K \subseteq \mathbb { R } ^ { n }$ be compact. Then $\mathcal { N N } _ { n , m , n + m + 1 } ^ { \rho }$ $C ( K ; \mathbb { R } ^ { m } )$
+
+See Appendix C for the proof. As might be expected, density is established with the help of the Stone–Weierstrass theorem, reducing the problem to the approximation of arbitrary polynomials.
+
+We remark that it is actually straightforward to find a construction showing that N n,m,n+m+2 is dense in $C ( K ; \mathbb { R } ^ { m } )$ when $\rho ( x ) = x ^ { 2 }$ , note the increased width. This is because the square activation function can be used to perform multiplication, via $x y = ( ( x + y ) ^ { 2 } - ( x - y ) ^ { 2 } ) / 4$ , and this makes it easy to construct arbitrary polynomials. In fact this is what is done in the proof of Proposition 4.7 for finding $m - 1$ of the $m$ outputs, when there is still a ‘spare’ neuron in each layer. It is computing the final output that actually requires the bulk of the work. The key to this argument is a width-efficient approximation to division.
+
+It is a consequence of Proposition 4.7 that any (polynomial) activation function which can approximate the square activation function, in a suitable manner, is also capable of universal approximation.
+
+4.8. and $\rho \colon \mathbb { R }  \mathbb { R }$ polynomial for whibe compact. Then a point is dens $\alpha \in \mathbb { R }$ at. $\rho ^ { \prime } ( \alpha ) = 0$ $\rho ^ { \prime \prime } ( \alpha ) \neq 0$ $K \subseteq \mathbb { R } ^ { n }$ $\mathcal { N N } _ { n , m , n + m + 1 } ^ { \rho }$ $C ( K ; \mathbb { R } ^ { m } )$
+
+Proof. Let $h \in \mathbb { R } \setminus \{ 0 \}$ . Define $\rho _ { h } \colon \mathbb { R } \to \mathbb { R }$ by
+
+$$
+\rho _ { h } ( x ) = \frac { \rho ( \alpha + h x ) - \rho ( \alpha ) } { h ^ { 2 } \rho ^ { \prime \prime } ( \alpha ) / 2 } .
+$$
+
+Then, taking a Taylor explansion around $\alpha$
+
+$$
+\begin{array} { l c r } { \rho _ { h } ( x ) = \displaystyle \frac { \rho ( \alpha ) + h x \rho ^ { \prime } ( \alpha ) + h ^ { 2 } x ^ { 2 } \rho ^ { \prime \prime } ( \alpha ) / 2 + \mathcal { O } ( h ^ { 3 } x ^ { 3 } ) - \rho ( \alpha ) } { h ^ { 2 } \rho ^ { \prime \prime } ( \alpha ) / 2 } } \\ { = x ^ { 2 } + \mathcal { O } ( h x ^ { 3 } ) . } \end{array}
+$$
+
+Let $s ( x ) = x ^ { 2 }$ . Then $\rho _ { h } \to s$ uniformly over any compact set as $h  0$ .
+
+Now set up a network as in the Square Model (Proposition 4.7), with every neuron using the square activation function. Call this network $N$ . Create a network $N _ { h }$ by copying $N$ and giving every neuron in the network the activation function $\rho _ { h }$ instead.
+
+Uniform continuity preserves uniform convergence, compactness is preserved by continuous functions, and a composition of two uniformly convergent sequences of functions with uniformly continuous limits is again uniformly convergent. So as a neural network is a layer-by-layer composition of functions, then the difference between $N$ and $N _ { h }$ , with respect to $\| \cdot \| _ { \infty }$ on $K$ , may be taken arbitrarily small by taking $h$ arbitrarily small.
+
+Furthermore note that $\rho _ { h }$ is just $\rho$ pre- and post-composed with affine functions. (Note that there is only one term in the definition of $\rho _ { h } ( x )$ which depends on $x$ .) This means that any network which may be represented with activation function $\rho _ { h }$ may be precisely represented with activation function $\rho$ , by combining the affine transformations involved. □
+
+Remark 4.9. That $\rho$ is polynomial is never really used in the proof of Theorem 4.8. Only a certain amount of differentiability is required, and all such nonpolynomial functions are already covered by Theorem 4.4, as a nonzero second derivative at $\alpha$ implies a nonzero first derivative somewhere close to $\alpha$ . Nonetheless in principle this provides another possible construction by which certain networks may be shown to exhibit universal approximation.
+
+Note that the converse strategy (applying nonpolynomial techniques to the polynomial case) fails. This is because the Register Model requires nonpolynomial activation functions due to its dependence on the classical Universal Approximation Theorem.
+
+![](images/77a1f1c40712be63d7bf1a1e9d1de9dcc7f38e3a02e4a29d54050d0305512ccb.jpg)  
+Figure 2: A layer with square activation functions is equivalent to multiple layers with only a single square activation function in each layer. The other neurons use the identity activation function, denoted $\iota$ .
+
+Theorem 4.10. Let $\rho \colon \mathbb { R }  \mathbb { R }$ be any nonaffine polynomial. Let $K \subseteq \mathbb { R } ^ { n }$ be compact. Then $\mathcal { N N } _ { n , m , n + m + 2 } ^ { \rho }$ is dense in $C ( K ; \mathbb { R } ^ { m } )$ .
+
+Proof. Fix $\alpha \in \mathbb { R }$ such that $\rho ^ { \prime \prime } ( \alpha ) \neq 0$ , which exists as $\rho$ is nonaffine. Now let $h \in ( 0 , \infty )$ . Define $\sigma _ { h } \colon  { \mathbb { R } } \to  { \mathbb { R } }$ by
+
+$$
+\sigma _ { h } ( x ) = { \frac { \rho ( \alpha + h x ) - 2 \rho ( \alpha ) + \rho ( \alpha - h x ) } { h ^ { 2 } \rho ^ { \prime \prime } ( \alpha ) } } .
+$$
+
+Then Taylor expanding $\rho ( \alpha + h x )$ and $\rho ( \alpha - h x )$ around $\alpha$ ,
+
+$$
+\begin{array} { c c l } { \displaystyle \sigma _ { h } ( x ) = \frac { \rho ( \alpha ) + h x \rho ^ { \prime } ( \alpha ) + h ^ { 2 } x ^ { 2 } \rho ^ { \prime \prime } ( \alpha ) / 2 + \mathcal { O } ( h ^ { 3 } x ^ { 3 } ) } { h ^ { 2 } \rho ^ { \prime \prime } ( \alpha ) } - \frac { 2 \rho ( \alpha ) } { h ^ { 2 } \rho ^ { \prime \prime } ( \alpha ) } + } \\ { \displaystyle \frac { \rho ( \alpha ) - h x \rho ^ { \prime } ( \alpha ) + h ^ { 2 } x ^ { 2 } \rho ^ { \prime \prime } ( \alpha ) / 2 + \mathcal { O } ( h ^ { 3 } x ^ { 3 } ) } { h ^ { 2 } \rho ^ { \prime \prime } ( \alpha ) } } \\ { = x ^ { 2 } + \mathcal { O } ( h x ^ { 3 } ) . } \end{array}
+$$
+
+Observe that $\sigma _ { h }$ needs precisely two operations of $\rho$ on (affine transformations of) $x$ , and so may be computed by two enhanced neurons with activation function $\rho$ . Thus the operation of a single enhanced neuron with square activation function may be approximated by two enhanced neurons with activation function $\rho$ .
+
+Let $N$ be a network as in the Square Model (Proposition 4.7) with every neuron using the square activation function. Let $\ell$ be any hidden layer of $N$ ; it contains $n + m + 1$ neurons. Let $\eta$ be a vector of the values of the neurons of the previous layer. Let $\phi _ { i }$ be the affine part of the ith neuron of $\ell$ , so that $\ell$ computes $\phi _ { 1 } ( \eta ) ^ { 2 } , \dots , \phi _ { n + m + 1 } \overline { { { ( \eta ) } } } ^ { 2 }$ . Then this may equivalently be calculated with $n + m + 1$ layers of $n + m + 1$ neurons each, with $n + m$ of the neurons in each of these new layers using the identity function, and one neuron using the square activation function. The first of these new layers applies the $\phi _ { i }$ , and the $i$ th layer squares the value of the $i$ th neuron. See Figure 2.
+
+Apply this procedure to every layer of $N$ ; call the resulting network $\widetilde N$ . It will compute exactly the same function as $N$ , and will have $n + m + 1$ times as many layers, but will use only a single squaring operation in each layer.
+
+Create a copy of $\widetilde N$ , call it $\widetilde { N } _ { h }$ . Replace its identity activation functions with approximations in the manner of Lemma 4.2, using activation function $\rho$ . Replace its square activation functions (one in each layer) by approximations in the manner described above with $\sigma _ { h }$ ; this requires an extra neuron in each hidden layer, so that the network is now of width $n + m + 2$ . Thus $\widetilde { N } _ { h }$ uses the activation function $\rho$ throughout.
+
+Uniform continuity preserves uniform convergence, compactness is preserved by continuous functions, and a composition of two uniformly convergent sequences of functions with uniformly continuous limits is again uniformly convergent. So as a neural network is a layer-by-layer composition of functions, then the difference between $\widetilde { N } _ { h }$ and $\widetilde { N }$ , with respect to $\| \cdot \| _ { \infty }$ on $K$ , may be taken arbitrarily small by taking $h$ arbitrarily small.
+
+Remark 4.11. It is possible to construct shallower networks analogous to $\widetilde N$ . The proof of Proposition 4.7 in Appendix C, uses most of the network’s neurons to approximate the identity anyway; only a few in each layer are used to square a valued that is desired to be squared. These are the only neurons that actually require the procedure used in Figure 2 and the proof of Theorem 4.10.
+
+# 4.4 NONDIFFERENTIABLE ACTIVATION FUNCTIONS
+
+Although not of direct practical application, results for nondifferentiable activation functions demonstrate how certain pathological cases are still capable of being handled.
+
+Lemma 4.12. Let $w \colon \mathbb { R }  \mathbb { R }$ be any bounded continuous nowhere differentiable function. Let $\rho ( x ) = \sin ( x ) + w ( x ) \mathrm { e } ^ { - x }$ . Let $L \subseteq \mathbb { R }$ be compact. Then a single enhanced neuron with activation function $\rho$ may uniformly approximate the identity function $\iota \colon \mathbb { R }  \mathbb { R }$ on $L$ , with arbitrarily small error.
+
+Proof. For $h \in \mathbb { R } \setminus \{ 0 \}$ and $A \in 2 \pi \mathbb { N }$ , let $\phi _ { h , A } ( x ) = h x + A$ , and let $\psi ( x ) = x / h$ . Let
+
+$$
+\iota _ { h , A } = \psi _ { h } \circ \rho \circ \phi _ { h , A } ,
+$$
+
+which is of the form that an enhanced neuron can represent. Then jointly taking $h$ small enough and $A$ large enough it is clear that $^ { l } h , A$ may be taken uniformly close to $\iota$ on $L$ . □
+
+Proposition 4.13. Let $w \colon \mathbb { R }  \mathbb { R }$ be any bounded continuous nowhere differentiable function. Let $\rho ( x ) = \sin ( x ) + w ( x ) \mathrm { e } ^ { - x }$ , which will also be nowhere differentiable. Let $K \subseteq \mathbb { R } ^ { n }$ be compact. Then $\mathcal { N N } _ { n , m , n + m + 1 } ^ { \rho }$ is dense in $C ( K ; \mathbb { R } ^ { m } )$ .
+
+Proof. As the proof of Theorem 4.4, except substituting Lemma 4.12 for Lemma 4.2.
+
+This manner of proof may be extended to other nondifferentiable activation functions as well.
+
+# 5 CONCLUSION
+
+There is a large literature on theoretical properties of neural networks, but much of it deals only with the ReLU.1 However how to select an activation function remains a poorly understood topic, and many other options have been proposed: leaky ReLU, PReLU, RRelu, ELU, SELU and other more exotic activation functions as well.2
+
+Our central contribution is to provide results for universal approximation using general activation functions (Theorems 3.2, 4.4, 4.8 and 4.10). In contrast to previous work, these results do not rely on the nice properties of the ReLU, and in particular do not rely on its explicit description. The techniques we use are straightforward, and robust enough to handle even the pathological case of nondifferentiable activation functions (Proposition 4.13).
+
+We also consider approximation in $L ^ { p }$ norm (Remark 4.5), and generalise previous work to smaller widths, multiple output neurons, and $p \geqslant 1$ in place of $p = 1$ (Corollary 4.6).
+
+In contrast to much previous work, every result we show also handles the general case of multiple output neurons.
+
+# ACKNOWLEDGEMENTS
+
+(Redacted from anonymised submission)
+
+REFERENCES   
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+
+# A PROOF OF THE REGISTER MODEL (PROPOSITION 4.3)
+
+First, we recall the classical Universal Approximation Theorem (Pinkus, 1999):
+
+Theorem 1.1. Let $\rho \colon \mathbb { R }  \mathbb { R }$ be any continuous function. Let $\mathcal { N } _ { n } ^ { \rho }$ represent the class of neural networks with activation function $\rho ,$ , with n neurons in the input layer, one neuron in the output layer, and one hidden layer with an arbitrary number of neurons. Let $K \subseteq \mathbb { R } ^ { n }$ be compact. Then $\mathcal { N } _ { n } ^ { \rho }$ is dense in $C ( K )$ if and only if ρ is nonpolynomial.
+
+The Register Model is created by suitably reorganising the neurons from a collection of such shallow networks.
+
+Proposition 4.3 (Register Model). Let $\mathcal { T } _ { n , m , n + m \pm 1 } ^ { \rho }$ repyer, class of neural networks with n neurons in the input layer, m neurons inneurons in each hidden layer, an arbitrary number of hidden layers, and $\rho \colon \mathbb { R }  \mathbb { R }$ be any continuous nonpolynomial function. Let $n + m + 1$   
+for which neuron in $n + m$ of the neurons in each hidden layeidden layer has activation function have. Let tity activation funcbe compact. Then $\rho$ $K \subseteq \mathbb { R } ^ { n }$ $\mathcal { I } _ { n , m , n + m + 1 } ^ { \rho }$   
+is dense in $C ( K ; \mathbb { R } ^ { m } )$ .
+
+Proof. Fix $f \in C ( K ; \mathbb { R } ^ { m } )$ . Let $f = ( f _ { 1 } , \ldots , f _ { m } )$ . Fix $\varepsilon > 0$ . By Theorem 1.1, there exist singlehidden-layer neural networks $g _ { 1 } , \ldots , g _ { m } \in \mathcal { N } _ { n } ^ { \rho }$ with activation function $\rho$ approximating $f _ { 1 } , \ldots , f _ { m }$ respectively. Each approximation is to within error $\varepsilon$ with respect to $\| \cdot \| _ { \infty }$ on $K$ . Let each $g _ { i }$ have $\beta _ { i }$ hidden neurons. Let $\sigma _ { i , j }$ represent the operation of its $j$ th hidden neuron, for $j \in \{ 1 , \dots , \beta _ { i } \}$ . In keeping with the idea of enhanced neurons, let each $\sigma _ { i , j }$ include the affine function that comes after it in the output layer of $g _ { i }$ , so that $\begin{array} { r } { g _ { i } = \sum _ { j = 1 } ^ { \beta _ { i } } \sigma _ { i , j } } \end{array}$ . Let $\begin{array} { r } { M = \sum _ { i = 1 } ^ { m } \beta _ { i } } \end{array}$ .
+
+We seek to construct a neural network $N \in \mathbb { Z } _ { n , m , n + m + 1 } ^ { \rho }$ . Given input $( x _ { 1 } , \ldots , x _ { n } ) \in \mathbb { R } ^ { n }$ , it will output $( G _ { 1 } , \dots , G _ { m } ) \in \mathbb { R } ^ { m }$ , such that $G _ { i } = g _ { i } ( x _ { 1 } , \ldots , x _ { n } )$ for each $i$ . That is, it will compute all of the shallow networks $g _ { 1 } , \ldots , g _ { m }$ . Thus it will approximate $f$ to within error $\varepsilon$ with respect to $\| \cdot \| _ { \infty }$ on $K$ .
+
+The construction of $N$ is mostly easily expressed pictorially; see Figure 3. In each cell, representing a neuron, we define its value as a function of the values of the neurons in the previous layer. In every layer, all but one of the neurons uses the identity activation function $\iota \colon \mathbb { R }  \mathbb { R }$ , whilst one neuron in each layer performs a computation of the form $\sigma _ { i , j }$ .
+
+The construction can be summed up as follows.
+
+Each layer has $n + m + 1$ neurons, arranged into a group of $n$ neurons, a group of a single neuron, and a group of $m$ neurons.
+
+The first $n$ neurons in each layer simply record the input $( x _ { 1 } , \ldots , x _ { n } )$ , by applying an identity activation function. We refer to these as the ‘in-register neurons’.
+
+Next we consider $g _ { 1 } , \ldots , g _ { m }$ , which are all shallow networks. The neurons in the hidden layers of $g _ { 1 } , \ldots , g _ { m }$ are arranged ‘vertically’ in our deep network, one in each layer. This is the neuron in each layer that uses the activation function $\rho$ . We refer to these as the ‘computation neurons’. Each computation neuron performs its computation based off of the inputs preserved in the in-register neurons.
+
+The final group of $m$ neurons also use the identity activation function; their affine parts gradually sum up the results of the computation neurons. We refer to these as the ‘out-register neurons’. The $i$ th out-register neuron in each layer will sum up the results of the computation neurons computing $\sigma _ { i , j }$ for all $j \in \{ 1 , \dots , \beta _ { i } \}$ .
+
+Finally, the neurons in the output layer of the network are connected to the out-register neurons of the final hidden layer. As each of the neurons in the output layer has, as usual, the identity activation function, they will now have computed the desired results. □
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>(1-W1+I-WuS)=Wu5</td><td rowspan=1 colspan=1>（-1+-5）+=I-wu5</td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=2>0=+Tgu5：…</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>0=z+Igu5</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>0=Ig‘u5</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>0=bu5</td><td rowspan=1 colspan=1>0=8u5</td><td rowspan=1 colspan=1>0=5</td><td rowspan=1 colspan=1>0=Iu5</td></tr><tr><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=2>：</td><td rowspan=1 colspan=1>：…</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>…</td><td rowspan=1 colspan=1>：…</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：：</td><td rowspan=1 colspan=1>：…</td><td rowspan=1 colspan=1>：：</td><td rowspan=1 colspan=1>：：</td></tr><tr><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>=W5(1-W25)1</td><td rowspan=1 colspan=1>=1-W5（-5）</td><td rowspan=7 colspan=1>·</td><td rowspan=7 colspan=1>…</td><td rowspan=1 colspan=1>1-++1-=g+g5</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>(2+11++21=8+1g&#x27;5</td><td rowspan=1 colspan=1>(+1g)1= +g&#x27;5</td><td rowspan=1 colspan=1>0=I+1g‘）</td><td rowspan=1 colspan=1>0=1g5</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>0=v5</td><td rowspan=1 colspan=1>0=85</td><td rowspan=1 colspan=1>0=5</td><td rowspan=1 colspan=1>0=15</td></tr><tr><td rowspan=1 colspan=1>W&#x27;S=5</td><td rowspan=1 colspan=1>=w15(1-W&#x27;15)1</td><td rowspan=1 colspan=1>=I-Wt5(-W15)</td><td rowspan=1 colspan=1>(-g+1g15）=g+1g5</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>=8+1g5（+1g15）7</td><td rowspan=1 colspan=1>=+tg&#x27;5(t+1g15）1</td><td rowspan=1 colspan=1>(11+15）=I+Ig5</td><td rowspan=1 colspan=1>(-11+1-1g15)=1g15</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>1+5151= v15</td><td rowspan=1 colspan=1>(+1+=85</td><td rowspan=1 colspan=1>（)=T5</td><td rowspan=1 colspan=1>0=115</td></tr><tr><td></td><td rowspan=1 colspan=1>(I-W‘uLc...I-W&#x27;I)ug‘uo=W⊥</td><td rowspan=1 colspan=1>(z-wu/....-w))-ugu)=I-W</td><td rowspan=1 colspan=1>（-y+1.-+））=g+g</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>(z+g/-.)+)80=s</td><td rowspan=1 colspan=1>(1g&#x27;us.*1+)0= z+⊥</td><td rowspan=1 colspan=1>(g‘u....gu)o= I+g⊥</td><td rowspan=1 colspan=1>(1-1gu..1-1g)1=1</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>(εu...ε/))10=v)</td><td rowspan=1 colspan=1>2/.../)1=)）</td><td rowspan=1 colspan=1>（/..)1=)</td><td rowspan=1 colspan=1>(ux..·x)1&#x27;1o=I</td></tr><tr><td></td><td rowspan=1 colspan=1>0=Wu</td><td rowspan=1 colspan=1>=I-Wu(-Wu)</td><td rowspan=1 colspan=1>（-g+1g）=g+Igu</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>=8+Igu（z+Ig&#x27;uL)</td><td rowspan=1 colspan=1>=+Igu(1+Ig‘u)</td><td rowspan=1 colspan=1>=I+Igu(gu)</td><td rowspan=1 colspan=1>(1-g‘u)=Igu</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>=u(ε²u)</td><td rowspan=1 colspan=1>=8u(u)</td><td rowspan=1 colspan=1>=u(t&#x27;u）</td><td rowspan=1 colspan=1>=u(ux）</td></tr><tr><td></td><td rowspan=1 colspan=1>：：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=2>：：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=2>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=2>：  ：</td></tr><tr><td></td><td rowspan=1 colspan=1>0=W</td><td rowspan=1 colspan=1>=-（-）</td><td rowspan=1 colspan=1>（1-+）=g+g</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>=8+g（+g）</td><td rowspan=1 colspan=1>=+g（+g）</td><td rowspan=1 colspan=1>=+g（g)</td><td rowspan=1 colspan=1>(-g)=g</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>=(8）</td><td rowspan=1 colspan=1>=8(）</td><td rowspan=1 colspan=1>=(1)</td><td rowspan=1 colspan=1>=(x）7</td></tr><tr><td></td><td rowspan=1 colspan=1>0=W</td><td rowspan=1 colspan=1>=-W(-)1</td><td rowspan=1 colspan=1>（1-+=g+g</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>=8+g（+g)</td><td rowspan=1 colspan=1>=+g(1+Ig)</td><td rowspan=1 colspan=1>=I+Ig()</td><td rowspan=1 colspan=1>(1-1g)=g</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>=(1)</td><td rowspan=1 colspan=1>=8()</td><td rowspan=1 colspan=1>=(1)</td><td rowspan=1 colspan=1>=It(1x)7</td></tr></table>
+
+# B PROOF OF COROLLARY 4.6
+
+Lemma B.1. Let $a , b , c , d \in \mathbb { R }$ be such that $a < b < c < d .$ . Let $U _ { a , b , c , d } \colon \mathbb { R } \to \mathbb { R }$ be the unique continuous piecewise affine function which is one on $[ b , c ]$ and zero on $( - \infty , a ] \cup [ d , \infty )$ . Then two layers of two enhanced neurons each, with ReLU activation function, may exactly represent the function $U _ { a , b , c , d }$ .
+
+Proof. Let $x \in \mathbb { R }$ be the input. Let $m _ { 1 } = 1 / ( b - a )$ . Let $m _ { 2 } = 1 / ( d - c )$ . Let $\eta _ { 1 } , \eta _ { 2 }$ represent the first neuron in each layer, and $\zeta _ { 1 } , \zeta _ { 2 }$ represent the second neuron in each layer. We assign them values as follows.
+
+$$
+\begin{array} { l l } { \eta _ { 1 } = \operatorname* { m a x } \{ 0 , m _ { 1 } ( x - a ) \} , } & { \qquad \zeta _ { 1 } = \operatorname* { m a x } \{ 0 , m _ { 2 } ( x - c ) \} , } \\ { \eta _ { 2 } = \operatorname* { m a x } \{ 0 , 1 - \eta _ { 1 } \} , } & { \qquad \zeta _ { 2 } = \operatorname* { m a x } \{ 0 , 1 - \zeta _ { 1 } \} . } \end{array}
+$$
+
+Then $U _ { a , b , c , d } ( x ) = \zeta _ { 2 } - \eta _ { 2 }$ . (This final affine transformation is allowed, in keeping with the notion of enhanced neurons.)
+
+Lemma B.2. One layer of two enhanced neurons, with ReLU activation function, may exactly represent the function $( x , y ) \mapsto \operatorname* { m i n } \{ x , y \}$ on $[ 0 , \infty ) ^ { 2 }$ .
+
+Proof. Let the first neuron compute $\eta = \operatorname* { m a x } \{ 0 , x - y \}$ . Let the second neuron compute $\zeta =$ $\operatorname* { m a x } \{ 0 , x \}$ . Then $\operatorname* { m i n } \{ x , y \} = \bar { \zeta } - \eta$ . $\boxed { \begin{array} { r l } \end{array} }$
+
+Corollary 4.6. Let $\rho$ be the ReLU activation function. Let $p \in [ 1 , \infty )$ . Then $\mathcal { N N } _ { n , m , n + m + 1 } ^ { \rho }$ dense in $L ^ { p } ( \mathbb { R } ^ { n } ; \mathbb { R } ^ { m } )$ .
+
+Proof. Let $f \in L ^ { p } ( \mathbb { R } ^ { n } ; \mathbb { R } ^ { m } )$ and $\varepsilon > 0$ . For simplicity assume that $\mathbb { R } ^ { m }$ is endowed with the $\| \cdot \| _ { \infty }$ norm; other norms are of course equivalent. Let $\widehat { f } = ( \widehat { f } _ { 1 } , \ldots , \widehat { f } _ { m } ) \in C _ { c } ( \mathbb { R } ^ { n } ; \mathbb { R } ^ { m } )$ be such that
+
+$$
+\left\| f - { \widehat { f } } \right\| _ { p } < \varepsilon / 3 .
+$$
+
+Let
+
+$$
+C = \operatorname* { s u p } _ { x \in \mathbb { R } ^ { n } } \operatorname* { m a x } _ { i } { \widehat { f } } _ { i } ( x ) + 1
+$$
+
+and
+
+$$
+c = \operatorname* { i n f } _ { x \in \mathbb { R } ^ { n } } \operatorname* { m i n } _ { i } { \widehat { f } } _ { i } ( x ) - 1
+$$
+
+Pick $a _ { 1 } , b _ { 1 } , \ldots , a _ { n } , b _ { n } \in \mathbb { R }$ such that $J$ defined by
+
+$$
+J = [ a _ { 1 } , b _ { 1 } ] \times \cdot \cdot \cdot \times [ a _ { n } , b _ { n } ]
+$$
+
+is such that $\operatorname { s u p p } { \widehat { f } } \subseteq J$ . Furthermore, for $\delta > 0$ that we shall fix in a moment, let
+
+$$
+\begin{array} { r } { A _ { i } = a _ { i } - \delta , } \\ { B _ { i } = b _ { i } + \delta , } \end{array}
+$$
+
+and let $K$ be defined by
+
+$$
+{ \cal K } = [ A _ { 1 } , B _ { 1 } ] \times \cdots \times [ A _ { n } , B _ { n } ] .
+$$
+
+Fix $\delta$ small enough that
+
+$$
+| K \setminus J | ^ { 1 / p } \cdot \operatorname* { m a x } { \{ | C | , | c | \} } < \frac { \varepsilon } { 6 } .
+$$
+
+$g = ( g _ { 1 } , \ldots , g _ { m } ) \in \mathcal { N } _ { n , m , n + m + 1 } ^ { \rho }$ be such that
+
+$$
+\operatorname* { s u p } _ { x \in K } \left| { \widehat { f } } ( x ) - g ( x ) \right| < \operatorname* { m i n } \left\{ { \frac { \varepsilon } { 3 { \big | } J { \big | } ^ { 1 / p } } } , 1 \right\} ,
+$$
+
+which exists by Theorem 4.4. Note that $g$ is defined on all of $\mathbb { R } ^ { n }$ ; it simply happens to be close to $\widehat { f }$ on $K$ . In particular it will takes values close to zero on $K \backslash J$ , and may take arbitrary values in $\mathbb { R } ^ { n } \setminus K$ . By equations (B.2), (B.3), (B.5), it is the case that
+
+$$
+\begin{array} { r } { C \geqslant \underset { x \in K } { \operatorname* { s u p } } \underset { i } { \operatorname* { m a x } } g _ { i } ( x ) , } \\ { c \leqslant \underset { x \in K } { \operatorname* { i n f } } \underset { i } { \operatorname* { m i n } } g _ { i } ( x ) . } \end{array}
+$$
+
+Now consider the network describing $g$ ; it will be modified slightly. The goal is to create a network which takes value $g$ on $J$ , zero in $\mathbb { R } ^ { n } \setminus K$ , and moves between these values in the interface region $K \backslash J$ . Such a network will provide a suitable approximation to $\widehat { f } .$ . This is done by first constructing a function which is approximately the indicator function for $J$ , with support in $K$ ; call such a function $U$ . The idea then is to construct a neural representation of $G _ { i }$ defined by
+
+$$
+G _ { i } = \operatorname* { m i n } \{ \operatorname* { m a x } \{ g _ { i } , c U \} , C U \} .
+$$
+
+Provided $| K \setminus J |$ is small enough then $G = ( G _ { 1 } , \dots , G _ { m } )$ will be the desired approximation; this is proved this below.
+
+We move on to presenting the neural representation of this construction.
+
+First we observe that because the activation function is the ReLU, then the identity approximations used in the proof of Theorem 4.4 may in fact exactly represent the identity function on some compact set: $x \mapsto \operatorname* { m a x } \{ 0 , x + N \} - N$ is exactly the identity function, for suitably large $N$ , and is of the form that an enhanced neuron may represent. This observation isn’t strictly necessary for the proof, but it does simplify the presentation somewhat, as the values preserved in the in-register neurons of $g$ are now exactly the inputs $x = ( x _ { 1 } , \ldots , x _ { n } ) $ for $x \in K$ . For sufficiently negative $x _ { i }$ , outside of $K$ , they will take the value $- N$ instead, but by insisting that is $N$ sufficiently large that
+
+$$
+- N < A _ { i }
+$$
+
+for all $i$ , then this will not be an issue for the proof.
+
+So take the network representing $g$ , and remove the output layer. (If the output layer is performing any affine transformations then treat them as being part of the final hidden layer, in the manner of enhanced neurons. Thus the output layer that is being removed is just applying the identity function to the out-register neurons.) Some more hidden layers will be placed on top, and then a new output layer will be placed on top. In the following description, all neurons not otherwise specified will be performing the identity function, so as to preserve the values of the corresponding neurons in the preceding layer. As all functions involved are continuous and $K$ is compact, and compactness is preserved by continuous functions, and continuous functions are bounded on compact sets, then this is possible for all $x \in K$ by taking $N$ large enough.
+
+The first task is to modify the value stored in the in-register neurons corresponding to $x _ { 1 }$ . At present it stores the value $x _ { 1 }$ ; by using this in-register neuron and the computation neuron in two extra layers, its value may be replaced with $U _ { A _ { 1 } , a _ { 1 } , b _ { 1 } , B _ { 1 } } ( x _ { 1 } )$ , via Lemma B.1. Place another two layers on top, and use them to replace the value of $x _ { 2 }$ in the second in-register neuron with $U _ { A _ { 2 } , a _ { 2 } , b _ { 2 } , B _ { 2 } } ( x _ { 2 } )$ , and so on. The in-register neurons now store the values $( U _ { A _ { 1 } , a _ { 1 } , b _ { 1 } , B _ { 1 } } ( x _ { 1 } ) , \dots , U _ { A _ { n } , a _ { n } , b _ { n } , B _ { n } } ( x _ { n } ) )$ .
+
+Once this is complete, place another layer on top and use the first two in-register neurons to compute the minimum of their values, in the manner of Lemma B.2, thus computing $\operatorname* { m i n } \{ U _ { A _ { 1 } , a _ { 1 } , b _ { 1 } , B _ { 1 } } ^ { - } ( x _ { 1 } ) , U _ { A _ { 2 } , a _ { 2 } , b _ { 2 } , B _ { 2 } } ( x _ { 2 } ) \}$ . Place another layer on top and use another two inregister neurons to compute the minimum of this value and the value presently stored in the third in-register neuron, that is $U _ { A _ { 3 } , a _ { 3 } , b _ { 3 } , B _ { 3 } } ( x _ { 3 } )$ , so that
+
+$$
+\operatorname* { m i n } \{ U _ { A _ { 1 } , a _ { 1 } , b _ { 1 } , B _ { 1 } } ( x _ { 1 } ) , U _ { A _ { 2 } , a _ { 2 } , b _ { 2 } , B _ { 2 } } ( x _ { 2 } ) , U _ { A _ { 3 } , a _ { 3 } , b _ { 3 } , B _ { 3 } } ( x _ { 3 } ) \} _ { [ \mathrm { f } ] }
+$$
+
+has now been computed. Continue to repeat this process until the in-register neurons have computed.3
+
+$$
+U = \operatorname* { m i n } _ { i \in \{ 1 , . . . , n \} } U _ { A _ { i } , a _ { i } , b _ { i } , B _ { i } } ( x _ { i } ) .
+$$
+
+Observe how $U$ represents an approximation to the indicator function for $J$ , with support in $K$ , evaluated at $( x _ { 1 } , \ldots , x _ { n } )$ .
+
+This is a highly destructive set of operations: the network no longer remembers the values of its inputs. Thankfully, it no longer needs them. Note how the small foible regarding how an in-register neuron would only record $- N$ instead of $x _ { i }$ , for $x _ { i } < - N$ , is not an issue. This is because of equation (B.7), which implies that $U _ { A _ { i } , a _ { i } , b _ { i } , B _ { i } } ( x _ { i } ) = 0 = U _ { A _ { i } , a _ { i } , b _ { i } , B _ { i } } ( - N )$ , thus leaving the value of $U$ unaffected.
+
+The out-register neurons presently store the values $g _ { 1 } , \ldots , g _ { m }$ , where $g _ { i } = g _ { i } ( x _ { 1 } , \ldots , x _ { n } )$ . Now add another layer. Let the value of its out-register neurons be $\theta _ { 1 } , \ldots , \theta _ { m }$ , where
+
+$$
+\theta _ { i } = \operatorname* { m a x } \{ 0 , g _ { i } - c U \} .
+$$
+
+Add one more hidden layer. Let the value of its out-register neurons be $\lambda _ { 1 } , \ldots , \lambda _ { m }$ , where
+
+$$
+\lambda _ { i } = \operatorname* { m a x } \{ 0 , - \theta _ { i } + ( C - c ) U \} .
+$$
+
+Finally place the output layer on top. Let the value of its neurons be $G _ { 1 } , \ldots , G _ { m }$ , where
+
+$$
+G _ { i } = - \lambda _ { i } + C U .
+$$
+
+Then in fact
+
+$$
+G _ { i } = \operatorname* { m i n } \{ \operatorname* { m a x } \{ g _ { i } , c U \} , C U \}
+$$
+
+as desired.
+
+All that remains to show is that $G = ( G _ { 1 } , \dots , G _ { m } )$ of this form is indeed a suitable approximation. First, as $G$ and $g$ coincide in $J$ , and by equation (B.5),
+
+$$
+\begin{array} { l } { \displaystyle \left( \displaystyle \int _ { J } \left| \widehat { f } ( x ) - G ( x ) \right| ^ { p } \mathrm { d } x \right) ^ { 1 / p } \leqslant \left| J \right| ^ { 1 / p } \operatorname* { s u p } _ { x \in J } \left| \widehat { f } ( x ) - G ( x ) \right| } \\ { \displaystyle \qquad = | J | ^ { 1 / p } \operatorname* { s u p } _ { x \in J } \left| \widehat { f } ( x ) - g ( x ) \right| } \\ { \displaystyle \qquad < \frac { \varepsilon } { 3 } . } \end{array}
+$$
+
+Secondly, by equations (B.2), (B.3), (B.6), (B.8) and then equation (B.4),
+
+$$
+\begin{array} { r l } {  { ( \int _ { K \setminus J } \Big | \widehat { f } ( x ) - G ( x ) \Big | ^ { p } \mathrm { d } x ) ^ { 1 / p } \leqslant \vert K \setminus J \vert ^ { 1 / p } \operatorname* { s u p } _ { x \in K \setminus J } \Big \vert \widehat { f } ( x ) - G ( x ) \Big \vert } } \\ & { \leqslant \vert K \setminus J \vert ^ { 1 / p } \cdot 2 \operatorname* { m a x } \big \{ \vert C \vert , \vert c \vert \big \} } \\ & { < \frac { \varepsilon } { 3 } . } \end{array}
+$$
+
+Thirdly,
+
+$$
+\left( \int _ { \mathbb { R } ^ { n } \setminus K } \Big | \widehat { f } ( x ) - G ( x ) \Big | ^ { p } \mathrm { d } x \right) ^ { 1 / p } = 0 ,
+$$
+
+as both $\widehat { f }$ and $G$ have support in $K$ .
+
+So by equations (B.1), (B.9), (B.10) and (B.11),
+
+$$
+\begin{array} { l } { \displaystyle \| f - G \| _ { p } \leqslant \left\| f - \widehat { f } \right\| _ { p } + \left( \int _ { \mathbb R ^ { n } } \left| \widehat { f } ( x ) - G ( x ) \right| ^ { p } \mathrm { d } x \right) ^ { 1 / p } } \\ { \displaystyle < \frac \varepsilon 3 + \frac \varepsilon 3 + \frac \varepsilon 3 } \\ { = \varepsilon . } \end{array}
+$$
+
+# C PROOF OF THE SQUARE MODEL (PROPOSITION 4.7)
+
+Lemma C.1. One layer of two enhanced neurons, with square activation function, may exactly represent the multiplication function $( x , y ) \mapsto x y$ on $\mathbb { R } ^ { 2 }$ .
+
+Proof. Let the first neuron compute $\eta = ( x + y ) ^ { 2 } / 4$ . Let the second neuron compute $\zeta = ( x - y ) ^ { 2 } / 4$ Then $x y = \eta - \zeta$ . □
+
+Lemma C.2. Fix $L \subseteq \mathbb { R } ^ { 2 }$ compact. Three layers of two enhanced neurons each, with square activation function, may uniformly approximate $( x , y )  ( x ^ { 2 } , y ( x + 1 ) )$ arbitrarily well on $L$ .
+
+Proof. Let $h , s \in \mathbb { R } \setminus \{ 0 \}$ . Let $\eta _ { 1 } , \eta _ { 2 } , \eta _ { 3 }$ represent the first neuron in each layer; let $\zeta _ { 1 } , \zeta _ { 2 } , \zeta _ { 3 }$ represent the second neuron in each layer. Let $\iota _ { h }$ represent an approximation to the identity in the manner of Lemma 4.2. Using $\cdot \approx $ as an informal notation to represent ‘equal to up to the use of $\iota _ { h }$ in place $\iota ^ { \prime }$ , just to help keep track of why we are performing these operations, assign values to $\eta _ { 1 } , \eta _ { 2 } , \eta _ { 3 }$ and $\zeta _ { 1 } , \zeta _ { 2 } , \zeta _ { 3 }$ as follows:
+
+$$
+\begin{array} { c c c } { { \eta _ { 1 } = \iota _ { h } ( x ) } } & { { } } & { { \zeta _ { 1 } = ( x + s y + 1 ) ^ { 2 } } } \\ { { } } & { { \approx x , } } & { { = x ^ { 2 } + 2 s x y + s ^ { 2 } y ^ { 2 } + 2 x + 2 s y + 1 , } } \\ { { } } & { { \eta _ { 2 } = ( \eta _ { 1 } ) ^ { 2 } } } & { { \zeta _ { 2 } = \iota _ { h } ( \zeta _ { 1 } - 2 \eta _ { 1 } - 1 ) } } \\ { { } } & { { \approx x ^ { 2 } , } } & { { \approx x ^ { 2 } + 2 s x y + s ^ { 2 } y ^ { 2 } + 2 s y , } } \\ { { } } & { { \eta _ { 3 } = \iota _ { h } ( \eta _ { 2 } ) } } & { { \zeta _ { 3 } = \iota _ { h } ( ( \zeta _ { 2 } - \eta _ { 2 } ) / 2 s ) } } \\ { { } } & { { } } & { { \approx x y + y + s y ^ { 2 } / 2 . } } \end{array}
+$$
+
+And so $\eta _ { 3 }$ may be taken arbitrarily close to $x ^ { 2 }$ and $\zeta _ { 3 }$ may be taken arbitrarily close to $y ( x + 1 )$ , with respect to $\| \cdot \| _ { \infty }$ on $L$ , by first taking $s$ arbitrarily small, and then taking $h$ arbitrarily small. □
+
+Proposition C.3. Fix $L \subseteq ( 0 , 2 )$ compact. Then multiple layers of two enhanced neurons each, with square activation function, may uniformly approximate $x \mapsto 1 / x$ arbitrarily well on $L$ .
+
+(Unlike Lemma C.2, the number of layers necessary will depend on the quality of approximation.)
+
+Proof. First note that
+
+$$
+\prod _ { i = 0 } ^ { n } ( 1 + x ^ { 2 ^ { i } } ) \to { \frac { 1 } { 1 - x } }
+$$
+
+as $n \to \infty$ , uniformly over compact subsets of $( - 1 , 1 )$ . Thus,
+
+$$
+( 2 - x ) \prod _ { i = 1 } ^ { n } ( 1 + ( 1 - x ) ^ { 2 ^ { i } } ) = \prod _ { i = 0 } ^ { n } ( 1 + ( 1 - x ) ^ { 2 ^ { i } } )  { \frac { 1 } { x } }
+$$
+
+uniformly over $L$
+
+This has the following neural approximation: let $\eta _ { 1 } = ( 1 - x ) ^ { 2 }$ and $\zeta _ { 1 } = \iota _ { h } ( 2 - x )$ be the neurons in the first layer, where $\iota _ { h }$ is some approximation of the identity as in Lemma 4.2. Let $\kappa _ { h }$ represent an approximation to $( x , y ) \mapsto ( x ^ { 2 } , { \bar { y ( x + 1 ) } } )$ in the manner of Lemma C.2, with error made arbitrarily small as $h  0$ . Now for $i \in \{ 1 , 4 , 7 , 1 0 , . . . , 3 n - 2 \}$ , recursively define $( \eta _ { i + 3 } , \zeta _ { i + 3 } ) = \kappa _ { h } ( \eta _ { i } , \zeta _ { i } )$ , where we increase the index by three to represent the fact that three layers are used to perform this operation. So up to approximation, $\eta _ { i + 3 } \approx ( \eta _ { i } ) ^ { 2 }$ , and $\zeta _ { i + 3 } \approx \zeta _ { i } ( \eta _ { i } + 1 )$ .
+
+So $\begin{array} { r } { \zeta _ { 3 n + 1 } \to ( 2 - x ) \prod _ { i = 1 } ^ { n } ( 1 + ( 1 - x ) ^ { 2 ^ { i } } ) } \end{array}$ uniformly over $L$ as $h  0$ . Thus the result is obtained by taking first $n$ large enough and then $h$ small enough. □
+
+Remark C.4. Our approach to Proposition C.3 is to find a suitable polynomial approximation of the reciprocal function, and then represent that with a network of multiple layers of two neurons each. It is fortunate, then, that this polynomial happens to be of a form that may be represented by such a network, as it is not clear that this should necessarily be the case for all polynomials. Even if Proposition 4.7 were already known, it requires a network of width three to represent arbitrary polynomials $\mathbb { R } \to \mathbb { R }$ , whereas Proposition C.3 uses a network of only width two. It remains unclear whether an arbitrary-depth network of width two, with square activation function, is capable of universal approximation in $C ( K )$ .
+
+Proposition 4.7 (Square Model). Let $\rho ( x ) = x ^ { 2 }$ . Let $K \subseteq \mathbb { R } ^ { n }$ be compact. Then $\mathcal { N N } _ { n , m , n + m + 1 } ^ { \rho }$ is dense in $C ( K ; \mathbb { R } ^ { m } )$ .
+
+Proof. Fix $f \in C ( K ; \mathbb { R } ^ { m } )$ . Let $f = ( f _ { 1 } , \dots , f _ { m } )$ . Fix $\varepsilon > 0$ . By precomposing with an affine function – which may be absorbed into the first layer of the network – assume without loss of generality that
+
+$$
+K \subseteq ( 1 , 2 ) ^ { n } .
+$$
+
+By the Stone–Weierstrass theorem there exist polynomials $g _ { 1 } , \ldots , g _ { m }$ in $x _ { 1 } , \ldots . x _ { n }$ approximating $f _ { 1 } , \ldots , f _ { m }$ to within $\varepsilon / 3$ with respect to $\| \cdot \| _ { \infty }$ .
+
+We will construct a network that evaluates arbitrarily good approximations to $g _ { 1 } , \ldots , g _ { n }$ . There are a total of $n + m + 1$ neurons in each layer; group them as in the proof of the Register Model (Proposition 4.3), see Appendix A, so that in each layer there is a group of $n$ neurons that we refer to as ‘in-register neurons’, a single neuron that we refer to as the ‘computation neuron’, and a group of $m$ neurons that we refer to as the ‘out-register neurons’.
+
+As before, every in-register neuron will simply apply an approximate identity function to the corresponding in-register neuron in the previous layer, so that they preserve the inputs to the network, up to an arbitrarily good approximation of the identity. (For now, at least – later, when constructing the approximation to $g _ { 1 }$ , which will be the final approximation that is handled, then these neurons will be repurposed to perform that computation.) For the sake of sanity of notation, we shall suppress this detail in our notation, and refer to our neurons in later layers as having e.g. $\cdot _ { x _ { 1 } } ,$ as an input to them to them; in practice this means some arbitrarily good approximation to $x _ { 1 }$ . The out-register neurons will eventually store the desired outputs of the network; thus there is an out-register neuron in each layer ‘corresponding’ to each of the $g _ { i }$ .
+
+Now suppose $m > 1$ ; if $m = 1$ then this paragraph and the next three paragraphs may simply be skipped. It is easy to build a network approximating $g _ { 2 } , \ldots g _ { m }$ , as there is at least one ‘extra’ neuron per layer thafollows. Let o use: the ou, where each egister neuron corresponding to is a monomial. Using the comp $g _ { 1 }$ . The strategy is asation neuron and the $\begin{array} { r } { g _ { 2 } = \sum _ { j = 1 } ^ { N } \delta _ { j } } \end{array}$ $\delta _ { j }$   
+C.1 to compute the value of $\delta _ { 1 }$ . For example, if $\delta _ { 1 } = x _ { 1 } ^ { 2 } x _ { 2 } x _ { 3 }$ , then this chain of multiplications is $x _ { 1 } ( x _ { 1 } ( x _ { 2 } x _ { 3 } ) )$ . The computation neuron and the ‘extra’ out-register neuron in the first layer compute the multiplication $\alpha = x _ { 2 } x _ { 3 }$ , these neurons in the second layer compute the multiplication $\beta = x _ { 1 } \alpha$ , and these neurons in the third layer compute $x _ { 1 } \beta$ . This value is then stored in the out-register neuron corresponding to $g _ { 2 } - \mathsf { b y }$ using the affine part of the operation of this neuron – and kept through the layers via approximate identity functions, as per Lemma 4.2.
+
+This process is then repeated for $\delta _ { 2 }$ . The result is then added on – via the affine part of the operation of the out-register neuron corresponding to $g _ { 2 }$ – to the value stored in the out-register neuron corresponding to $g _ { 2 }$ . Repeat for all $j$ until all of the $\delta _ { j }$ have been computed and the out-register neuron stores an approximation to $g _ { 2 }$ . This is only an approximation in that it requires the use of approximate identity functions; other than that it is exact. As such, by taking sufficiently good approximations of the identity function, this will be a uniform approximation to $g _ { 2 }$ over $K$ .
+
+Now repeat this whole process for $g _ { 3 } , \ldots g _ { m }$
+
+For the rest of the layers of the network, the out-register neurons corresponding to $g _ { 2 } , \ldots , g _ { m }$ will now simply apply approximate identity functions to maintain their values: these will eventually form the outputs of the network. Let these computed values be denoted $\widehat { g } _ { 2 } , \ldots , \widehat { g } _ { m }$ . (With the ‘hat’ notation becuse of the fact that these are not the values $g _ { 2 } , \ldots , g _ { m }$ b b, due to the approximate identity functions in between.)
+
+The difficult bit is computing an approximation to $g _ { 1 }$ , as it must be done without the ‘extra’ neuron in each layer. In total, then, in each layer, there are $n + 2$ neurons available: the $n$ in-register neurons (which have so far been storing the inputs $x _ { 1 } , \ldots x _ { n } )$ ), the computation neuron, and the out-register neuron corresponding to $g _ { 1 }$ .
+
+Written in terms of monomials, let $\begin{array} { r } { g _ { 1 } = \sum _ { j = 1 } ^ { M } \gamma _ { j } } \end{array}$ . Then $g _ { 1 }$ may be written as
+
+$$
+g _ { 1 } = \gamma _ { 1 } \left( 1 + \frac { \gamma _ { 2 } } { \gamma _ { 1 } } \left( 1 + \frac { \gamma _ { 3 } } { \gamma _ { 2 } } \left( \cdots \left( 1 + \frac { \gamma _ { M - 1 } } { \gamma _ { M - 2 } } \left( 1 + \frac { \gamma _ { M } } { \gamma _ { M - 1 } } \right) \right) \cdots \right) \right) \right) .
+$$
+
+Note that this description is defined over $K$ , as $K$ is bounded away from the origin.
+
+In particular, let $\begin{array} { r } { \gamma _ { j } = \prod _ { k = 1 } ^ { n } x _ { k } ^ { \theta _ { j , k } } } \end{array}$ xθj,kk , for θj,k ∈ N0. Substituting this in,
+
+$$
+g _ { 1 } = [ \prod _ { k = 1 } ^ { n } x _ { k } ^ { \theta _ { 1 , k } } ] ( 1 + \prod _ { k = 1 } ^ { n } x _ { k } ^ { \theta _ { 2 , k } } ( 1 + \prod _ { k = 1 } ^ { n } x _ { k } ^ { \theta _ { 3 , k } } ( \cdot \cdot \cdot     \\    \prod _ { k = 1 } ^ { n } x _ { k } ^ { \theta _ { 1 , k } } ( 1 + \prod _ { k = 1 } ^ { n } x _ { k } ^ { \theta _ { 2 , k } } ( 1 + \frac { \prod _ { k = 1 } ^ { n } x _ { k } ^ { \theta _ { M , k } } } { \prod _ { k = 1 } ^ { n } x _ { k } ^ { \theta _ { M - 1 , k } } } ) ) \cdot \cdot \cdot ) ) ) .
+$$
+
+Now let $\operatorname* { s u p } K$ be defined by
+
+$$
+\operatorname* { s u p } K = \operatorname* { s u p } \{ x _ { i } \mid ( x _ { 1 } , \ldots , x _ { n } ) \in K \} ,
+$$
+
+so that $1 < \operatorname* { s u p } K < 2$ . Let $r$ be an approximation to $x \mapsto 1 / x$ in the manner of Proposition C.3, with the $L$ of that proposition given by
+
+$$
+L = [ ( \operatorname* { s u p } K ) ^ { - 1 } - \alpha , \operatorname* { s u p } K + \alpha ] \subseteq ( 0 , 2 ) ,
+$$
+
+where $\alpha > 0$ is taken small enough that the inclusion holds.
+
+Let $r ^ { a }$ denote $r$ composed $a$ times. By taking $r$ to be a suitably good approximation, we may ensure that $\widetilde { g } _ { 1 }$ defined by
+
+$$
+\begin{array} { r l } { \widetilde { \eta } _ { 1 } - \displaystyle \prod _ { i = 1 } ^ { n } \gamma ^ { 2 M - 2 } ( z _ { i \ell } ) ^ { \theta _ { i , i } } \Bigg | } & { \Bigg ( 1 + \displaystyle \prod _ { i = 1 } ^ { n } \gamma ^ { 2 M - 3 } ( z _ { i \ell } ) ^ { \theta _ { i , i } } ) ^ { \theta _ { i , i } } \Bigg | \left[ \displaystyle \prod _ { i = 1 } ^ { n } \gamma ^ { 2 M - 4 } ( z _ { i \ell } ) ^ { \theta _ { i , i } } \right] } \\ & { \Bigg ( 1 + \displaystyle \prod _ { i = 1 } ^ { n } \gamma ^ { 2 M - 3 } ( z _ { i \ell } ) ^ { \theta _ { i , i } } \Bigg ) \displaystyle \prod _ { i = 1 } ^ { n } \gamma ^ { 3 \ell - 3 } ( z _ { i \ell } ) ^ { \theta _ { i , i } } \Bigg ] } \\ & { \Bigg ( \displaystyle \prod _ { i = 1 } ^ { n } \bigg ( \displaystyle \prod _ { i = 1 } ^ { n } \gamma ^ { 3 } ( z _ { i \ell } ) ^ { \theta _ { i , i } } \bigg ) ^ { \theta _ { i } } \Bigg ) } \\ & { \Bigg ( 1 + \displaystyle \prod _ { i = 1 } ^ { n } \gamma ^ { 3 } ( z _ { i \ell } ) ^ { \theta _ { i , i } } \Bigg ) ^ { \theta _ { i } } \left[ \displaystyle \prod _ { i = 1 } ^ { n } \gamma ^ { 2 } ( z _ { i \ell } ) ^ { \theta _ { i , i } } \cdots \right] \Bigg | } \\ & { \Bigg ( 1 + \displaystyle \prod _ { i = 1 } ^ { n } \gamma ^ { 3 } ( z _ { i \ell } ) ^ { \theta _ { i , i } } \cdots \bigg ) \Bigg [ \displaystyle \prod _ { i = 1 } ^ { n } \alpha _ { i } ^ { \theta _ { i , i } } \cdots \bigg ] \Bigg ) \Bigg ) ^ { \theta _ { i } } } \\ & { \quad \cdots \Bigg ) \Bigg ) \Bigg ) } \end{array}
+$$
+
+is an approximation to $g _ { 1 }$ in $K$ , to within $\varepsilon / 3$ , with respect to $\| \cdot \| _ { \infty }$ . This is possible by equations (C.1) and (C.2); in particular the approximation should be sufficiently precise that
+
+$$
+r ^ { 2 M - 2 } ( [ ( \operatorname { s u p } K ) ^ { - 1 } , \operatorname { s u p } K ] ) \subseteq L ,
+$$
+
+which is why $\alpha > 0$ is needed: note how $r ^ { 2 }$ , and thus $r ^ { 4 } , r ^ { 6 } , . . . , r ^ { 2 M - 2 }$ , are approximately the identity function on $L$ .
+
+This description of $\widetilde { g } _ { 1 }$ is now amenable to representation with a neural network. The key fact about this description of $\widetilde { g } _ { 1 }$ is that, working from the most nested set of brackets outwards, the value of $\widetilde { g } _ { 1 }$ e emay be computed by performing a single chain of multiplications and additions, and occasionally taking the reciprocal of all of the input values.
+
+So let the computation neuron and the out-register neuron corresponding to $g _ { 1 }$ perform the multiplications, layer-by-layer, to compute Qnk=1 xk θM,k in the manner of Lemma C.1. Store this value in the out-register neuron.
+
+Now use the computation neurons and the in-register neurons corresponding to $x _ { 1 }$ (across multiple layers) to compute $r ( x _ { 1 } )$ , in the manner of Proposition C.3: eventually the in-register neuron is now storing $r ( x _ { 1 } ) \approx 1 / x _ { 1 }$ . Repeat for the other in-register neurons, so that they are collectively storing $r ( x _ { 1 } ) , \ldots , r ( x _ { n } )$ .
+
+Now the computation neuron and the out-register neuron may start multipying $r ( x _ { 1 } ) , \ldots , r ( x _ { n } )$ on to Qnk=1 xk θM,k (which is the value presently stored in the out-register neuron) the appropriate number to times to compute $\left[ \prod _ { k = 1 } ^ { n } r ( x _ { k } ) ^ { \theta _ { M - 1 , k } } \right] \left[ \prod _ { k = 1 } ^ { n } x _ { k } ^ { \theta _ { M , k } } \right]$ , by Lemma C.1. Store this value in the out-register neuron. Then add one (using the affine part of a layer). The out-register neuron has now computed the expression in the innermost bracket in the description of $\widetilde { g } _ { 1 }$ .
+
+The general pattern is now clear: apply $r$ to all of the in-register neurons again to compute $r ^ { 2 } ( x _ { i } )$ , multiply them on to the value in the out-register neuron, and so on. Eventually the out-register neuron corresponding to $g _ { 1 }$ will have computed the value $\widetilde { g } _ { 1 }$ . Actually, it will have computed an approximation $\widehat { g } _ { 1 }$ eto this value, because of the identity approximations involved.
+
+By taking all of the (many) identity approximations throughout the network to be suitably precise, the values of $\widetilde { g } _ { 1 }$ and $\widehat { g } _ { 1 }$ may be taken within $\varepsilon / 3$ of each other, and the values of $\widehat { g } _ { 2 } , \ldots , \widehat { g } _ { m }$ and $g _ { 2 } , \ldots , g _ { m }$ e b may be taken within $2 \varepsilon / 3$ of each other, in each case with respect to $\| \cdot \| _ { \infty }$ on $K$ .
+
+Thus $( \widehat { g } _ { 1 } , \ldots , \widehat { g } _ { m } )$ approximates $f$ with total error no more than $\varepsilon$ , and the proof is complete.

parse/train/HyQJ-mclg/HyQJ-mclg.md

@@ -0,0 +1,239 @@
+# INCREMENTAL NETWORK QUANTIZATION: TOWARDS LOSSLESS CNNS WITH LOW-PRECISION WEIGHTS
+
+Aojun Zhou∗, Anbang Yao, Yiwen Guo, Lin Xu, and Yurong Chen   
+Intel Labs China   
+{aojun.zhou, anbang.yao, yiwen.guo, lin.x.xu, yurong.chen}@intel.com
+
+# ABSTRACT
+
+This paper presents incremental network quantization (INQ), a novel method, targeting to efficiently convert any pre-trained full-precision convolutional neural network (CNN) model into a low-precision version whose weights are constrained to be either powers of two or zero. Unlike existing methods which are struggled in noticeable accuracy loss, our INQ has the potential to resolve this issue, as benefiting from two innovations. On one hand, we introduce three interdependent operations, namely weight partition, group-wise quantization and re-training. A well-proven measure is employed to divide the weights in each layer of a pretrained CNN model into two disjoint groups. The weights in the first group are responsible to form a low-precision base, thus they are quantized by a variablelength encoding method. The weights in the other group are responsible to compensate for the accuracy loss from the quantization, thus they are the ones to be re-trained. On the other hand, these three operations are repeated on the latest re-trained group in an iterative manner until all the weights are converted into low-precision ones, acting as an incremental network quantization and accuracy enhancement procedure. Extensive experiments on the ImageNet classification task using almost all known deep CNN architectures including AlexNet, VGG-16, GoogleNet and ResNets well testify the efficacy of the proposed method. Specifically, at 5-bit quantization (a variable-length encoding: 1 bit for representing zero value, and the remaining 4 bits represent at most 16 different values for the powers of two) 1, our models have improved accuracy than the 32-bit floating-point references. Taking ResNet-18 as an example, we further show that our quantized models with 4-bit, 3-bit and 2-bit ternary weights have improved or very similar accuracy against its 32-bit floating-point baseline. Besides, impressive results with the combination of network pruning and INQ are also reported. We believe that our method sheds new insights on how to make deep CNNs to be applicable on mobile or embedded devices. The code will be made publicly available.
+
+# 1 INTRODUCTION
+
+Deep convolutional neural networks (CNNs) have demonstrated record breaking results on a variety of computer vision tasks such as image classification (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015), face recognition (Taigman et al., 2014; Sun et al., 2014), semantic segmentation (Long et al., 2015; Chen et al., 2015a) and object detection (Girshick, 2015; Ren et al., 2015). Regardless of the availability of significantly improved training resources such as abundant annotated data, powerful computational platforms and diverse training frameworks, the promising results of deep CNNs are mainly attributed to the large number of learnable parameters, ranging from tens of millions to even hundreds of millions. Recent progress further shows clear evidence that CNNs could easily enjoy the accuracy gain from the increased network depth and width (He et al., 2016; Szegedy et al., 2015; 2016). However, this in turn lays heavy burdens on the memory and other computational resources. For instance, ResNet-152, a specific instance of the latest residual network architecture wining ImageNet classification challenge in 2015, has a model size of about 230 MB and needs to perform about 11.3 billion FLOPs to classify a $2 2 4 \times 2 2 4$ image crop. Therefore, it is very challenging to deploy deep CNNs on the devices with limited computation and power budgets.
+
+Substantial efforts have been made to the speed-up and compression on CNNs during training, feedforward test or both of them. Among existing methods, the category of network quantization methods attracts great attention from researches and developers. Some network quantization works try to compress pre-trained full-precision CNN models directly. Gong et al. (2014) address the storage problem of AlexNet (Krizhevsky et al., 2012) with vector quantization techniques. By replacing the weights in each of the three fully connected layers with respective floating-point centroid values obtained from the clustering, they can get over $2 0 \times$ model compression at about $1 \%$ loss in top-5 recognition rate. HashedNet (Chen et al., 2015b) uses a hash function to randomly map pre-trained weights into hash buckets, and all the weights in the same hash bucket are constrained to share a single floating-point value. In HashedNet, only the fully connected layers of several shallow CNN models are considered. For better compression, Han et al. (2016) present deep compression method which combines the pruning (Han et al., 2015), vector quantization and Huffman coding, and reduce the model storage by $3 5 \times$ on AlexNet and $4 9 \times$ on VGG-16 (Simonyan & Zisserman, 2015). Vanhoucke et al. (2011) use an SSE 8-bit fixed-point implementation to improve the computation of neural networks on the modern Intel $\mathbf { \boldsymbol { x } } 8 6$ CPUs in feed-forward test, yielding $3 \times$ speed-up over an optimized floating-point baseline. Training CNNs by substituting the 32-bit floating-point representation with the 16-bit fixed-point representation has also been explored in Gupta et al. (2015). Other seminal works attempt to restrict CNNs into low-precision versions during training phase. Soudry et al. (2014) propose expectation backpropagation (EBP) to estimate the posterior distribution of deterministic network weights. With EBP, the network weights can be constrained to $+ 1$ and $^ { - 1 }$ during feed-forward test in a probabilistic way. BinaryConnect (Courbariaux et al., 2015) further extends the idea behind EBP to binarize network weights during training phase directly. It has two versions of network weights: floating-point and binary. The floating-point version is used as the reference for weight binarization. BinaryConnect achieves state-of-the-art accuracy using shallow CNNs for small datasets such as MNIST (LeCun et al., 1998) and CIFAR-10. Later on, a series of efforts have been invested to train CNNs with low-precision weights, low-precision activations and even low-precision gradients, including but not limited to BinaryNet (Courbariaux et al., 2016), XNOR-Net (Rastegari et al., 2016), ternary weight network (TWN) (Li & Liu, 2016), DoReFa-Net (Zhou et al., 2016) and quantized neural network (QNN) (Hubara et al., 2016).
+
+Despite these tremendous advances, CNN quantization still remains an open problem due to two critical issues which have not been well resolved yet, especially under scenarios of using low-precision weights for quantization. The first issue is the non-negligible accuracy loss for CNN quantization methods, and the other issue is the increased number of training iterations for ensuring convergence. In this paper, we attempt to address these two issues by presenting a novel incremental network quantization (INQ) method.
+
+In our INQ, there is no assumption on the CNN architecture, and its basic goal is to efficiently convert any pre-trained full-precision (i.e., 32-bit floating-point) CNN model into a low-precision version whose weights are constrained to be either powers of two or zero. The advantage of such kind of low-precision models is that the original floating-point multiplication operations can be replaced by cheaper binary bit shift operations on dedicated hardware like FPGA. We noticed that most existing network quantization methods adopt a global strategy in which all the weights are simultaneously converted to low-precision ones (that are usually in the floating-point types). That is, they have not considered the different importance of network weights, leaving the room to retain network accuracy limited. In sharp contrast to existing methods, our INQ makes a very careful handling for the model accuracy drop from network quantization. To be more specific, it incorporates three interdependent operations: weight partition, group-wise quantization and re-training. Weight partition uses a pruning-inspired measure (Han et al., 2015; Guo et al., 2016) to divide the weights in each layer of a pre-trained full-precision CNN model into two disjoint groups which play complementary roles in our INQ. The weights in the first group are quantized to be either powers of two or zero by a variable-length encoding method, forming a low-precision base for the original model. The weights in the other group are re-trained while keeping the quantized weights fixed, compensating for the accuracy loss resulted from the quantization. Furthermore, these three operations are repeated on the latest re-trained weight group in an iterative manner until all the weights are quantized, acting as an incremental network quantization and accuracy enhancement procedure (as illustrated in Figure 1).
+
+![](images/450f6613f97d9601a580c5c41c8086953fe819a0c5d0d4e38305e967fdbf1359.jpg)  
+Figure 1: An overview of our incremental network quantization method. (a) Pre-trained fullprecision model used as a reference. (b) Model update with three proposed operations: weight partition, group-wise quantization (green connections) and re-training (blue connections). (c) Final low-precision model with all the weights constrained to be either powers of two or zero. In the figure, operation (1) represents a single run of (b), and operation (2) denotes the procedure of repeating operation (1) on the latest re-trained weight group until all the non-zero weights are quantized. Our method does not lead to accuracy loss when using 5-bit, 4-bit and even 3-bit approximations in network quantization. For better visualization, here we just use a 3-layer fully connected network as an illustrative example, and the newly re-trained weights are divided into two disjoint groups of the same size at each run of operation (1) except the last run which only performs quantization on the re-trained floating-point weights occupying $12 . 5 \%$ of the model weights.
+
+The main insight of our INQ is that a compact combination of the proposed weight partition, groupwise quantization and re-training operations has the potential to get a lossless low-precision CNN model from any full-precision reference. We conduct extensive experiments on the ImageNet large scale classification task using almost all known deep CNN architectures to validate the effectiveness of our method. We show that: (1) For AlexNet, VGG-16, GoogleNet and ResNets with 5-bit quantization, INQ achieves improved accuracy in comparison with their respective full-precision baselines. The absolute top-1 accuracy gain ranges from ${ \bf 0 . 1 3 \% }$ to $2 . 2 8 \%$ , and the absolute top-5 accuracy gain is in the range of $0 . 2 3 \%$ to $1 . 6 5 \%$ . (2) INQ has the property of easy convergence in training. In general, re-training with less than 8 epochs could consistently generate a lossless model with 5-bit weights in the experiments. (3) Taking ResNet-18 as an example, our quantized models with 4-bit, 3-bit and 2-bit ternary weights also have improved or very similar accuracy compared with its 32-bit floating-point baseline. (4) Taking AlexNet as an example, the combination of our network pruning and INQ outperforms deep compression method (Han et al., 2016) with significant margins.
+
+# 2 INCREMENTAL NETWORK QUANTIZATION
+
+In this section, we clarify the insight of our INQ, describe its key components, and detail its implementation.
+
+# 2.1 WEIGHT QUANTIZATION WITH VARIABLE-LENGTH ENCODING
+
+Suppose a pre-trained full-precision (i.e., 32-bit floating-point) CNN model can be represented by $\{ \mathbf { W } _ { l } : 1 \le l \le L \}$ , where $\mathbf { W } _ { l }$ denotes the weight set of the ${ \bf \chi } _ { l ^ { t h } }$ layer, and $L$ denotes the number of learnable layers in the model. To simplify the explanation, we only consider convolutional layers and fully connected layers. For CNN models like AlexNet, VGG-16, GoogleNet and ResNets as tested in this paper, $\mathbf { W } _ { l }$ can be a $4 D$ tensor for the convolutional layer, or a $2 D$ matrix for the fully connected layer. For simplicity, here the dimension difference is not considered in the expression. Given a pre-trained full-precision CNN model, the main goal of our INQ is to convert all 32-bit floating-point weights to be either powers of two or zero without loss of model accuracy. Besides, we also attempt to explore the limit of the expected bit-width under the premise of guaranteeing lossless network quantization. Here, we start with our basic network quantization method on how to convert $\mathbf { W } _ { l }$ to be a low-precision version $\widehat { \mathbf { W } } _ { l }$ , and each of its entries is chosen from
+
+$$
+\mathbf { P } _ { l } = \{ \pm 2 ^ { n _ { 1 } } , \cdot \cdot \cdot , \pm 2 ^ { n _ { 2 } } , 0 \} ,
+$$
+
+where $n _ { 1 }$ and $n _ { 2 }$ are two integer numbers, and they satisfy $n _ { 2 } \leq n _ { 1 }$ . Mathematically, $n _ { 1 }$ and $n _ { 2 }$ help to bound ${ \bf P } _ { l }$ in the sense that its non-zero elements are constrained to be in the range of either $[ - 2 ^ { n _ { 1 } } , - 2 ^ { n _ { 2 } } ]$ or $[ 2 ^ { n _ { 2 } } , 2 ^ { n _ { 1 } } ]$ . That is, network weights with absolute values smaller than $2 ^ { n _ { 2 } }$ will be pruned away (i.e., set to zero) in the final low-precision model. Obviously, the problem is how to determine $n _ { 1 }$ and $n _ { 2 }$ . In our INQ, the expected bit-width $b$ for storing the indices in ${ \bf P } _ { l }$ is set beforehand, thus the only hyper-parameter shall be determined is $n _ { 1 }$ because $n _ { 2 }$ can be naturally computed once $b$ and $n _ { 1 }$ are available. Here, $n _ { 1 }$ is calculated by using a tricky yet practically effective formula as
+
+$$
+n _ { 1 } = \mathrm { { f l o o r } ( l o g _ { 2 } ( 4 { s / 3 } ) ) , }
+$$
+
+where $\operatorname { f l o o r } ( \cdot )$ indicates the round down operation and $s$ is calculated by using
+
+$$
+s = \operatorname* { m a x } ( \operatorname { a b s } ( \mathbf { W } _ { l } ) ) ,
+$$
+
+where $\mathrm { a b s } ( \cdot )$ is an element-wise operation and $\operatorname* { m a x } ( { \mathord { \cdot } } )$ outputs the largest element of its input. In fact, Equation (2) helps to match the rounding power of 2 for $s$ , and it could be easily implemented in practical programming. After $n _ { 1 }$ is obtained, $n _ { 2 }$ can be naturally determined as $n _ { 2 } = n _ { 1 } + 1 -$ $2 ^ { ( b - 1 ) } / 2$ . For instance, if $b = 3$ and $n _ { 1 } = - 1$ , it is easy to get $n _ { 2 } = - 2$ .
+
+Once $\mathbf { P } _ { l }$ is determined, we further use the ladder of powers to convert every entry of $\mathbf { W } _ { l }$ into a low-precision one by using
+
+$$
+\widehat { \mathbf { W } } _ { l } ( i , j ) = \left\{ \begin{array} { l l } { \beta \mathrm { s g n } ( \mathbf { W } _ { l } ( i , j ) ) } & { \mathrm { i f } ( \alpha + \beta ) / 2 \leq \mathrm { a b s } ( \mathbf { W } _ { l } ( i , j ) ) < 3 \beta / 2 } \\ { 0 } & { \mathrm { o t h e r w i s e } , } \end{array} \right.
+$$
+
+where $\alpha$ and $\beta$ are two adjacent elements in the sorted ${ \bf P } _ { l }$ , making the above equation as a numerical rounding to the quantum values. It should be emphasized that factor $4 / 3$ in Equation (2) is set to make sure that all the elements in ${ \bf P } _ { l }$ correspond with the quantization rule defined in Equation (4). In other words, factor $4 / 3$ in Equation (2) highly correlates with factor $3 / 2$ in Equation (4).
+
+Here, an important thing we want to clarify is the definition of the expected bit-width $b$ . Taking 5-bit quantization as an example, since zero value cannot be written as the power of two, we use 1 bit to represent zero value, and the remaining 4 bits to represent at most 16 different values for the powers of two. That is, the number of candidate quantum values is at most $2 ^ { b - 1 } + 1$ , so our quantization method actually adopts a variable-length encoding scheme. It is clear that the quantization described above is performed in a linear scale. An alternative solution is to perform the quantization in the log scale. Although it may also be effective, it should be a little bit more difficult in implementation and may cause some extra computational overhead in comparison to our method.
+
+# 2.2 INCREMENTAL QUANTIZATION STRATEGY
+
+We can naturally use the above described method to quantize any pre-trained full-precision CNN model. However, noticeable accuracy loss appeared in the experiments when using small bit-width values (e.g., 5-bit, 4-bit, 3-bit and 2-bit).
+
+In the literature, there are many existing network quantization works such as HashedNet (Chen et al., 2015b), vector quantization (Gong et al., 2014), fixed-point representation (Vanhoucke et al., 2011; Gupta et al., 2015), BinaryConnect (Courbariaux et al., 2015), BinaryNet (Courbariaux et al., 2016), XNOR-Net (Rastegari et al., 2016), TWN (Li & Liu, 2016), DoReFa-Net (Zhou et al., 2016) and QNN (Hubara et al., 2016). Similar to our basic network quantization method, they also suffer from non-negligible accuracy loss on deep CNNs, especially when being applied on the ImageNet large scale classification dataset. For all these methods, a common fact is that they adopt a global strategy in which all the weights are simultaneously converted into low-precision ones, which in turn causes accuracy loss. Compared with the methods focusing on the pre-trained models, accuracy loss becomes worse for the methods such as XNOR-Net, TWN, DoReFa-Net and QNN which intend to train low-precision CNNs from scratch.
+
+Recall that our main goal is to achieve lossless low-precision quantization for any pre-trained fullprecision CNN model with no assumption on its architecture. To this end, our INQ makes a special handling of the strategy for suppressing resulting quantization loss in model accuracy. We are partially inspired by the latest progress in network pruning (Han et al., 2015; Guo et al., 2016). In these methods, the accuracy loss from removing less important network weights of a pre-trained neural network model could be well compensated by following re-training steps. Therefore, we conjecture that the nature of changing network weight importance is critical to achieve lossless network quantization.
+
+![](images/2dc0441050f3b20ced1e13c4a46fed6509dbb2251f6016d5491c6f7ceef84095.jpg)  
+Figure 2: Result illustrations. First row: results from the $1 ^ { s t }$ iteration of the proposed three operations. The top left cube illustrates weight partition operation generating two disjoint groups, the middle image illustrates the quantization operation on the first weight group (green cells), and the top right cube illustrates the re-training operation on the second weight group (light blue cells). Second row: results from the $2 ^ { n d }$ , $3 ^ { r d }$ and $\dot { 4 } ^ { t h }$ iterations of the INQ. In the figure, the accumulated portion of the weights which have been quantized undergoes from $5 0 \% {  } 7 5 \% {  } 8 7 . 5 \% {  } 1 0 0 \%$ .
+
+Base on this assumption, we present INQ which incorporates three interdependent operations: weight partition, group-wise quantization and re-training. Weight partition is to divide the weights in each layer of a pre-trained full-precision CNN model into two disjoint groups which play complementary roles in our INQ. The weights in the first group are responsible for forming a low-precision base for the original model, thus they are quantized by using Equation (4). The weights in the second group adapt to compensate for the loss in model accuracy, thus they are the ones to be re-trained. Once the first run of the quantization and re-training operations is finished, all the three operations are further conducted on the second weight group in an iterative manner, until all the weights are converted to be either powers of two or zero, acting as an incremental network quantization and accuracy enhancement procedure. As a result, accuracy loss under low-precision CNN quantization can be well suppressed by our INQ. Illustrative results at iterative steps of our INQ are provided in Figure 2.
+
+For the $l ^ { t h }$ layer, weight partition can be defined as
+
+$$
+\mathbf { A } _ { l } ^ { ( 1 ) } \cup \mathbf { A } _ { l } ^ { ( 2 ) } = \{ \mathbf { W } _ { l } ( i , j ) \} , \mathrm { a n d } \mathbf { A } _ { l } ^ { ( 1 ) } \cap \mathbf { A } _ { l } ^ { ( 2 ) } = \varnothing ,
+$$
+
+where ${ \bf A } _ { l } ^ { ( 1 ) }$ denotes the first weight group that needs to be quantized, and ${ \bf A } _ { 2 }$ denotes the other weight group that needs to be re-trained. We leave the strategies for group partition to be chosen in the experiment section. Here, we define a binary matrix $\mathbf { T } _ { l }$ to help distinguish above two categories of weights. That is, $\mathbf T _ { l } ( i , j ) = 0$ means $\mathbf { W } _ { l } ( i , j ) \in \mathbf { A } _ { l } ^ { ( 1 ) }$ , and $\mathbf T _ { l } ( i , j ) = 1$ means $\mathbf { W } _ { l } ( i , j ) \in \mathbf { A } _ { l } ^ { ( 2 ) }$ .
+
+# 2.3 INCREMENTAL NETWORK QUANTIZATION ALGORITHM
+
+Now, we come to the training method. Taking the $l ^ { t h }$ layer as an example, the basic optimization problem of making its weights to be either powers of two or zero can be expressed as
+
+$$
+\begin{array} { r l } { \underset { \mathbf { W } _ { l } } { \operatorname* { m i n } } } & { E ( \mathbf { W } _ { l } ) = L ( \mathbf { W } _ { l } ) + \lambda R ( \mathbf { W } _ { l } ) } \\ { \mathrm { s . t . } } & { \mathbf { W } _ { l } ( i , j ) \in \mathbf { P } _ { l } , 1 \leq l \leq L , } \end{array}
+$$
+
+where $L ( \mathbf { W } _ { l } )$ is the network loss, $R ( { \mathbf W } _ { l } )$ is the regularization term, $\lambda$ is a positive coefficient, and the constraint term indicates each weight entry ${ \bf W } _ { l } ( i , j )$ should be chosen from the set $\mathbf { P } _ { l }$ consisting of a fixed number of the values of powers of two plus zero. Direct solving above optimization problem in training from scratch is challenging since it is very easy to undergo convergence problem.
+
+By performing weight partition and group-wise quantization operations beforehand, the optimization problem defined in (6) can be reshaped into a easier version. That is, we only need to optimize the following objective function
+
+$$
+\begin{array} { r l } { \underset { \mathbf { W } _ { l } } { \mathop { \operatorname* { m i n } } } } & { E ( \mathbf { W } _ { l } ) = L ( \mathbf { W } _ { l } ) + \lambda R ( \mathbf { W } _ { l } ) } \\ { \mathrm { s . t . } } & { \mathbf { W } _ { l } ( i , j ) \in \mathbf { P } _ { l } , \mathrm { ~ i f ~ } \mathbf { T } _ { l } ( i , j ) = 0 , 1 \leq l \leq L , } \end{array}
+$$
+
+where $\mathbf { P } _ { l }$ is determined at group-wise quantization operation, and the binary matrix $\mathbf { T } _ { l }$ acts as a mask which is determined by weight partition operation. Since $\mathbf { P } _ { l }$ and $\mathbf { T } _ { l }$ are known, the optimization problem (7) can be solved using popular stochastic gradient decent (SGD) method. That is, in INQ, we can get the update scheme for the re-training as
+
+$$
+\mathbf { W } _ { l } ( i , j )  \mathbf { W } _ { l } ( i , j ) - \gamma \frac { \partial E } { \partial ( \mathbf { W } _ { l } ( i , j ) ) } \mathbf { T } _ { l } ( i , j ) ,
+$$
+
+where $\gamma$ is a positive learning rate. Note that the binary matrix $\mathbf { T } _ { l }$ forces zero update to the weights that have been quantized. That is, only the weights still keep with floating-point values are updated, akin to the latest pruning methods (Han et al., 2015; Guo et al., 2016) in which only the weights that are not currently removed are re-trained to enhance network accuracy. The whole procedure of our INQ is summarized as Algorithm 1.
+
+We would like to highlight that the merits of our INQ are in three aspects: (1) Weight partition introduces the importance-aware weight quantization. (2) Group-wise weight quantization introduces much less accuracy loss than simultaneously quantizing all the network weights, thus making retraining have larger room to recover model accuracy. (3) By integrating the operations of weight partition, group-wise quantization and re-training into a nested loop, our INQ has the potential to obtain lossless low-precision CNN model from the pre-trained full-precision reference.
+
+g   
+Input: $X$ : the training data, $\{ \mathbf { W } _ { l } : 1 \le l \le L \}$ : the pre-trained full-precision CNN model, $\{ \sigma _ { 1 } , \sigma _ { 2 } , \cdots , \sigma _ { N } \}$ : the accumulated portions of weights quantized at iterative steps   
+Output: $\{ \widehat { \mathbf { W } } _ { l } : 1 \leq l \leq L \}$ : the final low-precision model with the weights constrained to be either powers of two or zero 1: Initialize $\mathbf { A } _ { l } ^ { ( 1 ) }  \varnothing , \mathbf { A } _ { l } ^ { ( 2 ) }  \{ \mathbf { W } _ { l } ( i , j ) \} , \mathbf { T } _ { l }  \mathbf { 1 } , \operatorname { f o r } 1 \leq l \leq L$ 2: for $n = 1 , 2 , \ldots , N$ do 3: Reset the base learning rate and the learning policy 4: According to $\sigma _ { n }$ , perform layer-wise weight partition and update $\mathbf { A } _ { l } ^ { ( 1 ) } , \mathbf { A } _ { l } ^ { ( 2 ) }$ and $\mathbf { T } _ { l }$ 5: Based on ${ \bf A } _ { l } ^ { ( 1 ) }$ , determine $\mathbf { P } _ { l }$ layer-wisely 6: Quantize the weights in ${ \bf A } _ { l } ^ { ( 1 ) }$ by Equation (4) layer-wisely 7: Calculate feed-forward loss, and update weights in $\{ \mathbf { A } _ { l } ^ { ( 2 ) } : 1 \le l \le L \}$ by Equation (8) 8: end for
+
+# 3 EXPERIMENTAL RESULTS
+
+To analyze the performance of our INQ, we perform extensive experiments on the ImageNet large scale classification task, which is known as the most challenging image classification benchmark so far. ImageNet dataset has about 1.2 million training images and 50 thousand validation images. Each image is annotated as one of 1000 object classes. We apply our INQ to AlexNet, VGG-16, GoogleNet, ResNet-18 and ResNet-50, covering almost all known deep CNN architectures. Using the center crops of validation images, we report the results with two standard measures: top-1 error rate and top-5 error rate. For fair comparison, all pre-trained full-precision (i.e., 32-bit floatingpoint) CNN models except ResNet-18 are taken from the Caffe model ${ \mathrm { z o o } } ^ { 2 }$ . Note that He et al. (2016) do not release their pre-trained ResNet-18 model to the public, so we use a publicly available re-implementation by Facebook3. Since our method is implemented with Caffe, we make use of an open source tool4 to convert the pre-trained ResNet-18 model from Torch to Caffe.
+
+# 3.1 RESULTS ON IMAGENET
+
+Table 1: Our INQ well converts diverse full-precision deep CNN models (including AlexNet, VGG16, GoogleNet, ResNet-18 and ResNet-50) to 5-bit low-precision versions with consistently improved model accuracy.   
+
+<table><tr><td>Network</td><td>Bit-width</td><td>Top-1 error</td><td>Top-5 error</td><td>Decrease in top-1/top-5 error</td></tr><tr><td>AlexNet ref</td><td>32</td><td>42.76%</td><td>19.77%</td><td></td></tr><tr><td>AlexNet VGG-16 ref</td><td>5 32</td><td>42.61% 31.46%</td><td>19.54% 11.35%</td><td>0.15%/0.23%</td></tr><tr><td>VGG-16</td><td>5</td><td>29.18%</td><td>9.70%</td><td>2.28%/1.65%</td></tr><tr><td>GoogleNet ref</td><td></td><td></td><td></td><td></td></tr><tr><td>GoogleNet</td><td>32</td><td>31.11%</td><td>10.97%</td><td></td></tr><tr><td>ResNet-18 ref</td><td>5</td><td>30.98%</td><td>10.72%</td><td>0.13%/0.25%</td></tr><tr><td>ResNet-18</td><td>32</td><td>31.73%</td><td>11.31%</td><td></td></tr><tr><td></td><td>5</td><td>31.02%</td><td>10.90%</td><td>0.71%/0.41%</td></tr><tr><td>ResNet-50 ref</td><td>32</td><td>26.78%</td><td>8.76%</td><td></td></tr><tr><td>ResNet-50</td><td>5</td><td>25.19%</td><td>7.55%</td><td>1.59%/1.21%</td></tr></table>
+
+Setting expected bit-width to 5, the first set of experiments is performed to testify the efficacy of our INQ on different CNN architectures. Regarding weight partition, there are several candidate strategies as we tried in our previous work for efficient network pruning (Guo et al., 2016). In Guo et al. (2016), we found random partition and pruning-inspired partition are the two best choices compared with the others. Thus in this paper, we directly compare these two strategies for weight partition. In random strategy, the weights in each layer of any pre-trained full-precision deep CNN model are randomly split into two disjoint groups. In pruning-inspired strategy, the weights are divided into two disjoint groups by comparing their absolute values with layer-wise thresholds which are automatically determined by a given splitting ratio. Here we directly use pruning-inspired strategy and the experimental results in Section 3.2 will show why. After the re-training with no more than 8 epochs over each pre-trained full-precision model, we obtain the results as shown in Table 1. It can be concluded that the 5-bit CNN models generated by our INQ show consistently improved top-1 and top-5 recognition rates compared with respective full-precision references. Parameter settings are described below.
+
+AlexNet: AlexNet has 5 convolutional layers and 3 fully-connected layers. We set the accumulated portions of quantized weights at iterative steps as $\{ 0 . 3 , \dot { 0 } . 6 , 0 . 8 , 1 \}$ , the batch size as 256, the weight decay as 0.0005, and the momentum as 0.9.
+
+VGG-16: Compared with AlexNet, VGG-16 has 13 convolutional layers and more parameters. We set the accumulated portions of quantized weights at iterative steps as $\{ 0 . 5 , 0 . 7 5 , 0 . 8 7 5 , 1 \}$ , the batch size as 32, the weight decay as 0.0005, and the momentum as 0.9.
+
+GoogleNet: Compared with AlexNet and VGG-16, GoogleNet is more difficult to quantize due to a smaller number of parameters and the increased network width. We set the accumulated portions of quantized weights at iterative steps as $\{ 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 \}$ , the batch size as 80, the weight decay as 0.0002, and the momentum as 0.9.
+
+ResNet-18: Different from above three networks, ResNets have batch normalization layers and relief the vanishing gradient problem by using shortcut connections. We first test the 18-layer version for exploratory purpose and test the 50-layer version later on. The network architectures of ResNet18 and ResNet-34 are very similar. The only difference is the number of filters in every convolutional layer. We set the accumulated portions of quantized weights at iterative steps as {0.5, 0.75, 0.875, $1 \}$ , the batch size as 80, the weight decay as 0.0005, and the momentum as 0.9.
+
+ResNet-50: Besides significantly increased network depth, ResNet-50 has a more complex network architecture in comparison to ResNet-18. However, regarding network architecture, ResNet-50 is very similar to ResNet-101 and ResNet-152. The only difference is the number of filters in every convolutional layer. We set the accumulated portions of quantized weights at iterative steps as $\{ 0 . 5$ , 0.75, 0.875, 1}, the batch size as 32, the weight decay as 0.0005, and the momentum as 0.9.
+
+# 3.2 ANALYSIS OF WEIGHT PARTITION STRATEGIES
+
+In our INQ, the first operation is weight partition whose result will directly affect the following group-wise quantization and re-training operations. Therefore, the second set of experiments is conducted to analyze two candidate strategies for weight partition. As mentioned in the previous section, we use pruning-inspired strategy for weight partition. Unlike random strategy in which all the weights have equal probability to fall into the two disjoint groups, pruning-inspired strategy considers that the weights with larger absolute values are more important than the smaller ones to form a low-precision base for the original CNN model. We use ResNet-18 as a test case to compare the performance of these two strategies. In the experiments, the parameter settings are completely the same as described in Section 3.1. We set 4 epochs for weight re-training. Table 2 summarizes the results of our INQ with 5-bit quantization. It can be seen that our INQ achieves top-1 error rate of $3 2 . 1 1 \%$ and top-5 error rate of $1 1 . 7 3 \%$ by using random partition. Comparatively, pruning-inspired partition brings $1 . 0 9 \%$ and $0 . 8 3 \%$ decrease in top-1 and top-5 error rates, respectively. Apparently, pruning-inspired partition is better than random partition, and this is the reason why we use it in this paper. For future works, weight partition based on quantization error could also be an option worth exploring.
+
+Table 2: Comparison of two different strategies for weight partition on ResNet-18.   
+
+<table><tr><td>Strategy</td><td>Bit-width</td><td>Top-1 error</td><td>Top-5 error</td></tr><tr><td>Random partition</td><td>5</td><td>32.11%</td><td>11.73%</td></tr><tr><td>Pruning-inspired partition</td><td>5</td><td>31.02%</td><td>10.90%</td></tr></table>
+
+# 3.3 THE TRADE-OFF BETWEEN EXPECTED BIT-WIDTH AND MODEL ACCURACY
+
+The third set of experiments is performed to explore the limit of the expected bit-width under which our INQ can still achieve lossless network quantization. Similar to the second set of experiments, we also use ResNet-18 as a test case, and the parameter settings for the batch size, the weight decay and the momentum are completely the same. Finally, lower-precision models with 4-bit, 3-bit and even 2-bit ternary weights are generated for comparisons. As the expected bit-width goes down, the number of candidate quantum values will be decreased significantly, thus we shall increase the number of iterative steps accordingly for enhancing the accuracy of final low-precision model. Specifically, we set the accumulated portions of quantized weights at iterative steps as $\{ 0 . 3 , 0 . 5$ , 0.8, 0.9, 0.95, $1 \}$ , $\{ 0 . 2 , 0 . 4 , 0 . 6 , 0 . 7 , \bar { 0 } . 8 , 0 . 9 , 0 . 9 \bar { 5 } , 1 \}$ and $\{ 0 . 2 , 0 . 4 , 0 . 6 , 0 . 7 , 0 . 8 , \bar { 0 } . 8 5 , \bar { 0 } . 9 , 0 . 9 5 _ { \cdotp }$ , $0 . 9 7 5 , 1 \}$ for 4-bit, 3-bit and 2-bit ternary models, respectively. The required number of epochs also increases when the expected bit-width goes down, and it reaches 30 when training our 2-bit ternary model. Although our 4-bit model shows slightly decreased accuracy when compared with the 5-bit model, its accuracy is still better than that of the pre-trained full-precision model. Comparatively, even when the expected bit-width goes down to 3, our low-precision model shows only ${ \bar { 0 . 1 9 \% } }$ and
+
+$0 . 3 3 \%$ losses in top-1 and top-5 recognition rates, respectively. As for our 2-bit ternary model, although it incurs $2 . 2 5 \%$ decrease in top-1 error rate and $1 . 5 6 \%$ decrease in top-5 error rate in comparison to the pre-trained full-precision reference, its accuracy is considerably better than stateof-the-art results reported for binary-weight network (BWN) (Rastegari et al., 2016) and ternary weight network (TWN) (Li & Liu, 2016). Detailed results are summarized in Table 3 and Table 4.
+
+Table 3: Our INQ generates extremely low-precision (4-bit and 3-bit) models with improved or very similar accuracy compared with the full-precision ResNet-18 model.   
+
+<table><tr><td>Model</td><td>Bit-width</td><td>Top-1 error</td><td>Top-5 error</td></tr><tr><td>ResNet-18 ref</td><td>32</td><td>31.73%</td><td>11.31%</td></tr><tr><td>INQ</td><td>5</td><td>31.02%</td><td>10.90%</td></tr><tr><td>INQ</td><td>4</td><td>31.11%</td><td>10.99%</td></tr><tr><td>INQ</td><td>3</td><td>31.92%</td><td>11.64%</td></tr><tr><td>INQ</td><td>2 (ternary)</td><td>33.98%</td><td>12.87%</td></tr></table>
+
+Table 4: Comparison of our 2-bit ternary model and some other binary or ternary models, including the BWN and the TWN approximations of ResNet-18.   
+
+<table><tr><td>Method</td><td>Bit-width</td><td>Top-1 error</td><td>Top-5 error</td></tr><tr><td>BWN(Rastegari et al., 2016)</td><td>1</td><td>39.20%</td><td>17.00%</td></tr><tr><td>TWN(Li &amp; Liu, 2016)</td><td>2 (ternary)</td><td>38.20%</td><td>15.80%</td></tr><tr><td>INQ (ours)</td><td>2 (ternary)</td><td>33.98%</td><td>12.87%</td></tr></table>
+
+# 3.4 LOW-BIT DEEP COMPRESSION
+
+In the literature, recently proposed deep compression method (Han et al., 2016) reports so far best results on network compression without loss of model accuracy. Therefore, the last set of experiments is conducted to explore the potential of our INQ for much better deep compression. Note that Han et al. (2016) is a hybrid network compression solution combining three different techniques, namely network pruning (Han et al., 2015), vector quantization (Gong et al., 2014) and Huffman coding. Taking AlexNet as an example, network pruning gets $9 \times$ compression, however this result is mainly obtained from the fully connected layers. Actually its compression performance on the convolutional layers is less than $3 \times$ (as can be seen in the Table 4 of Han et al. (2016)). Besides, network pruning is realized by separately performing pruning and re-training in an iterative way, which is very time-consuming. It will cost at least several weeks for compressing AlexNet. We solved this problem by our dynamic network surgery (DNS) method (Guo et al., 2016) which achieves about $7 \times$ speed-up in training and improves the performance of network pruning from $9 \times$ to $1 7 . 7 \times$ . In Han et al. (2016), after network pruning, vector quantization further improves compression ratio from $9 \times$ to $2 7 \times$ , and Huffman coding finally boosts compression ratio up to $3 5 \times$ . For fair comparison, we combine our proposed INQ and DNS, and compare the resulting method with Han et al. (2016). Detailed results are summarized in Table 5. When combing our proposed INQ and DNS, we achieve much better compression results compared with Han et al. (2016). Specifically, with 5-bit quantization, we can achieve $5 3 \times$ compression with slightly larger gains both in top-5 and top-1 recognition rates, yielding $5 1 . 4 3 \% / 9 6 . 3 0 \%$ absolute improvement in compression performance compared with full version/fair version (i.e., the combination of network pruning and vector quantization) of Han et al. (2016), respectively. Consistently better results have also obtained for our 4-bit and 3-bit models.
+
+Besides, we also perform a set of experiments on AlexNet to compare the performance of our INQ and vector quantization (Gong et al., 2014). For fair comparison, re-training is also used to enhance the performance of vector quantization, and we set the number of cluster centers for all of 5 convolutional layers and 3 fully connect layers to 32 (i.e., 5-bit quantization). In the experiment, vector quantization incurs over $3 \%$ loss in model accuracy. When we change the number of cluster centers for convolutional layers from 32 to 128, it gets an accuracy loss of $0 . 9 8 \%$ . This is consistent with the results reported in (Gong et al., 2014). Comparatively, vector quantization is mainly proposed to compress the parameters in the fully connected layers of a pre-trained full-precision CNN model, while our INQ addresses all network layers simultaneously and has no accuracy loss for 5-bit and 4-bit quantization. Therefore, it is evident that our INQ is much better than vector quantization. Last but not least, the final weights for vector quantization (Gong et al., 2014), network pruning (Han et al., 2015) and deep compression (Han et al., 2016) are still floating-point values, but the final weights for our INQ are in the form of either powers of two or zero. The direct advantage of our INQ is that the original floating-point multiplication operations can be replaced by cheaper binary bit shift operations on dedicated hardware like FPGA.
+
+Table 5: Comparison of the combination of our INQ and DNS, and deep compression method on AlexNet. Conv: Convolutional layer, FC: Fully connected layer, P: Pruning, Q: Quantization, H: Huffman coding.   
+
+<table><tr><td>Method</td><td>Bit-width(Conv/FC)</td><td>Compression ratio</td><td>Decrease in top-1/top5 error</td></tr><tr><td>Han et al. (2016) (P+Q)</td><td>8/5</td><td>27×</td><td>0.00%/0.03%</td></tr><tr><td>Han et al. (2016) (P+Q+H)</td><td>8/5</td><td>35×</td><td>0.00%/0.03%</td></tr><tr><td>Han et al. (2016) (P+Q+H)</td><td>8/4</td><td>1</td><td>-0.01%/0.00%</td></tr><tr><td>Our method (P+Q)</td><td>5/5</td><td>53×</td><td>0.08%/0.03%</td></tr><tr><td>Han et al. (2016) (P+Q+H)</td><td>4/2</td><td>1</td><td>-1.99%/-2.60%</td></tr><tr><td>Our method (P+Q)</td><td>4/4</td><td>71×</td><td>-0.52%/-0.20%</td></tr><tr><td>Our method (P+Q)</td><td>3/3</td><td>89×</td><td>-1.47 %/-0.96%</td></tr></table>
+
+# 4 CONCLUSIONS
+
+In this paper, we present INQ, a new network quantization method, to address the problem of how to convert any pre-trained full-precision (i.e., 32-bit floating-point) CNN model into a lossless lowprecision version whose weights are constrained to be either powers of two or zero. Unlike existing methods which usually quantize all the network weights simultaneously, INQ is a more compact quantization framework. It incorporates three interdependent operations: weight partition, groupwise quantization and re-training. Weight partition splits the weights in each layer of a pre-trained full-precision CNN model into two disjoint groups which play complementary roles in INQ. The weights in the first group is directly quantized by a variable-length encoding method, forming a low-precision base for the original CNN model. The weights in the other group are re-trained while keeping all the quantized weights fixed, compensating for the accuracy loss from network quantization. More importantly, the operations of weight partition, group-wise quantization and re-training are repeated on the latest re-trained weight group in an iterative manner until all the weights are quantized, acting as an incremental network quantization and accuracy enhancement procedure. On the ImageNet large scale classification task, we conduct extensive experiments and show that our quantized CNN models with 5-bit, 4-bit, 3-bit and even 2-bit ternary weights have improved or at least comparable accuracy against their full-precision baselines, including AlexNet, VGG-16, GoogleNet and ResNets. As for future works, we plan to extend incremental idea behind INQ from low-precision weights to low-precision activations and low-precision gradients (we have actually already made some good progress on it, as shown in our supplementary materials). We will also investigate computation and power efficiency by implementing our low-precision CNN models on hardware platforms.
+
+# REFERENCES
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+
+# A APPENDIX 1: STATISTICAL ANALYSIS OF THE QUANTIZED WEIGHTS
+
+Taking our 5-bit AlexNet model as an example, we analyze the distribution of the quantized weights. Detailed statistical results are summarized in Table 6. We can find: (1) in the $1 ^ { s t }$ and $2 ^ { n d }$ convolutional layers, the values of $\{ - 2 ^ { - 6 } , - 2 ^ { - 5 } , - 2 ^ { - 4 } , 2 ^ { - 6 } , 2 ^ { - 5 } , 2 ^ { - 4 } \}$ and $\{ - 2 ^ { - 8 } , - 2 ^ { - 7 } , - 2 ^ { - 6 } , - 2 ^ { - 5 }$ , $0 , 2 ^ { - 8 } , \dot { 2 } ^ { - 7 } , 2 ^ { - 6 } , 2 ^ { - 5 } \}$ occupy over $60 \%$ and $94 \%$ of all quantized weights, respectively; (2) the distributions of the quantized weights in the $3 ^ { r d }$ , $4 ^ { t h }$ and $5 ^ { t h }$ convolutional layers are similar to that of the $2 ^ { n d }$ convolutional layer, and more weights are quantized into zero in the $2 ^ { n d }$ , $3 ^ { r d }$ , $4 ^ { t h }$ and $5 ^ { t h }$ convolutional layers compared with the $1 ^ { s t }$ convolutional layer; (3) in the $1 ^ { s t }$ fully connected layer, the values of $\left\{ - 2 ^ { - 1 0 } , \ - 2 ^ { - 9 } , - 2 ^ { - 8 } , - 2 ^ { - 7 } , 0 , 2 ^ { - 1 0 } , 2 ^ { - 9 } , 2 ^ { - 8 } , 2 ^ { - 7 } \right\}$ occupy about $98 \%$ of all quantized weights, and similar results can be seen for the $2 ^ { n d }$ fully connected layer; (4) generally, the distributions of the quantized weights in the convolutional layers are usually more scattered compared with the fully connected layers. This may be partially the reason why it is much easier to get good compression performance on fully connected layers in comparison to convolutional layers, when using methods such as network hashing (Chen et al., 2015b) and vector quantization (Gong et al., 2014); (5) for 5-bit AlexNet model, the required bit-width for each layer is actually 4 but not 5.
+
+Table 6: A statistical distribution of the quantized weights in our 5-bit AlexNet model.   
+
+<table><tr><td>Weight</td><td>Conv1</td><td>Conv2</td><td>Conv3</td><td>Conv4</td><td>Conv5</td><td>FC6</td><td>FC7</td><td>FC8</td></tr><tr><td>-2-10</td><td></td><td></td><td>=</td><td>=</td><td></td><td>8.95%</td><td>6.37%</td><td>3.86%</td></tr><tr><td>-2-9</td><td>=</td><td></td><td>=</td><td></td><td></td><td>12.29%</td><td>9.58%</td><td>6.19%</td></tr><tr><td>-2-8</td><td>5.04%</td><td>10.55%</td><td>11.58%</td><td>10.09%</td><td>9.88%</td><td>16.48%</td><td>16.13%</td><td>12.90%</td></tr><tr><td>-2-7</td><td>6.56%</td><td>12.09%</td><td>14.34%</td><td>14.24%</td><td>14.68%</td><td>10.84%</td><td>17.87%</td><td>19.51%</td></tr><tr><td>-2-6</td><td>9.22%</td><td>13.08%</td><td>15.26%</td><td>18.49%</td><td>20.66%</td><td>0.79%</td><td>3.43%</td><td>11.09%</td></tr><tr><td>-2-5</td><td>10.52%</td><td>8.73%</td><td>5.92%</td><td>7.77%</td><td>9.79%</td><td>0.002%</td><td>0.004%</td><td>0.40%</td></tr><tr><td>-2-4</td><td>9.75%</td><td>2.70%</td><td>0.49%</td><td>0.38%</td><td>0.55%</td><td>1</td><td>1</td><td>1</td></tr><tr><td>-2-3</td><td>4.61%</td><td>0.39%</td><td>0.02%</td><td>0.01%</td><td>0.004%</td><td></td><td></td><td></td></tr><tr><td>-2-2</td><td>0.67%</td><td>0.01%</td><td>1e-4%</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>0</td><td>5.51%</td><td>11.30%</td><td>12.24%</td><td>9.70%</td><td>8.97%</td><td>8.86%</td><td>6.17%</td><td>3.62%</td></tr><tr><td>2-10</td><td></td><td></td><td></td><td></td><td></td><td>8.30%</td><td>5.81%</td><td>3.40%</td></tr><tr><td>2-9</td><td>1</td><td></td><td></td><td></td><td></td><td>10.51%</td><td>7.84%</td><td>4.69%</td></tr><tr><td>2-8</td><td>5.20%</td><td>9.70%</td><td>10.44%</td><td>8.60%</td><td>7.69%</td><td>12.91%</td><td>11.30%</td><td>8.08%</td></tr><tr><td>2-7</td><td>6.79%</td><td>11.01%</td><td>11.66%</td><td>10.33%</td><td>8.95%</td><td>8.95%</td><td>11.90%</td><td>10.94%</td></tr><tr><td>2-6</td><td>9.99%</td><td>11.05%</td><td>11.86%</td><td>12.25%</td><td>10.67%</td><td>1.12%</td><td>3.54%</td><td>12.56%</td></tr><tr><td>2-5</td><td>11.15%</td><td>6.57%</td><td>5.22%</td><td>6.81%</td><td>6.37%</td><td>0.01%</td><td>0.06%</td><td>2.75%</td></tr><tr><td>2-4</td><td>10.14%</td><td>2.26%</td><td>0.86%</td><td>1.24%</td><td>1.62%</td><td>1e-5%</td><td>2e-5%</td><td>0.01%</td></tr><tr><td>2-3</td><td>4.26%</td><td>0.53%</td><td>0.09%</td><td>0.08%</td><td>0.16%</td><td>-</td><td>=</td><td></td></tr><tr><td>2-2</td><td>0.60%</td><td>0.05%</td><td>0.01%</td><td>0.003%</td><td>0.01%</td><td></td><td></td><td></td></tr><tr><td>2-1</td><td>1</td><td>3e-4%</td><td>2e-4%</td><td>3e-4%</td><td>1</td><td>1</td><td>=</td><td></td></tr><tr><td>Total</td><td>100%</td><td>100%</td><td>100%</td><td>100%</td><td>100%</td><td>100%</td><td>100%</td><td>1</td></tr><tr><td>Bit-width</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>100% 4</td></tr></table>
+
+# B APPENDIX 2: LOSSLESS CNNS WITH LOW-PRECISION WEIGHTS AND LOW-PRECISION ACTIVATIONS
+
+Table 7: Comparison of our VGG-16 model with 5-bit weights and 4-bit activations, and the pretrained reference with 32-bit floating-point weights and 32-bit float-point activations.   
+
+<table><tr><td>Network</td><td>Bit-width for weight/activation</td><td>Top-1 error</td><td>Top-5 error</td><td>Decrease in top-1/top-5 error</td></tr><tr><td>VGG-16 ref</td><td>32/32</td><td>31.46%</td><td>11.35%</td><td></td></tr><tr><td>VGG-16</td><td>5/4</td><td>29.82%</td><td>10.19%</td><td>1.64%/1.16%</td></tr></table>
+
+Recently, we have made some good progress on developing our INQ for lossless CNNs with both low-precision weights and low-precision activations. According to the results summarized in Table 7, it can be seen that our VGG-16 model with 5-bit weights and 4-bit activations shows improved top-5 and top-1 recognition rates in comparison to the pre-trained reference with 32-bit floating-point weights and 32-bit floating-point activations. To the best of our knowledge, this should be the best results reported on VGG-16 architecture so far.

parse/train/HyQJ-mclg/HyQJ-mclg_content_list.json

@@ -0,0 +1,1254 @@
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+    {
+        "type": "text",
+        "text": "INCREMENTAL NETWORK QUANTIZATION: TOWARDS LOSSLESS CNNS WITH LOW-PRECISION WEIGHTS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Aojun Zhou∗, Anbang Yao, Yiwen Guo, Lin Xu, and Yurong Chen   \nIntel Labs China   \n{aojun.zhou, anbang.yao, yiwen.guo, lin.x.xu, yurong.chen}@intel.com ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            250,
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+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper presents incremental network quantization (INQ), a novel method, targeting to efficiently convert any pre-trained full-precision convolutional neural network (CNN) model into a low-precision version whose weights are constrained to be either powers of two or zero. Unlike existing methods which are struggled in noticeable accuracy loss, our INQ has the potential to resolve this issue, as benefiting from two innovations. On one hand, we introduce three interdependent operations, namely weight partition, group-wise quantization and re-training. A well-proven measure is employed to divide the weights in each layer of a pretrained CNN model into two disjoint groups. The weights in the first group are responsible to form a low-precision base, thus they are quantized by a variablelength encoding method. The weights in the other group are responsible to compensate for the accuracy loss from the quantization, thus they are the ones to be re-trained. On the other hand, these three operations are repeated on the latest re-trained group in an iterative manner until all the weights are converted into low-precision ones, acting as an incremental network quantization and accuracy enhancement procedure. Extensive experiments on the ImageNet classification task using almost all known deep CNN architectures including AlexNet, VGG-16, GoogleNet and ResNets well testify the efficacy of the proposed method. Specifically, at 5-bit quantization (a variable-length encoding: 1 bit for representing zero value, and the remaining 4 bits represent at most 16 different values for the powers of two) 1, our models have improved accuracy than the 32-bit floating-point references. Taking ResNet-18 as an example, we further show that our quantized models with 4-bit, 3-bit and 2-bit ternary weights have improved or very similar accuracy against its 32-bit floating-point baseline. Besides, impressive results with the combination of network pruning and INQ are also reported. We believe that our method sheds new insights on how to make deep CNNs to be applicable on mobile or embedded devices. The code will be made publicly available. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep convolutional neural networks (CNNs) have demonstrated record breaking results on a variety of computer vision tasks such as image classification (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015), face recognition (Taigman et al., 2014; Sun et al., 2014), semantic segmentation (Long et al., 2015; Chen et al., 2015a) and object detection (Girshick, 2015; Ren et al., 2015). Regardless of the availability of significantly improved training resources such as abundant annotated data, powerful computational platforms and diverse training frameworks, the promising results of deep CNNs are mainly attributed to the large number of learnable parameters, ranging from tens of millions to even hundreds of millions. Recent progress further shows clear evidence that CNNs could easily enjoy the accuracy gain from the increased network depth and width (He et al., 2016; Szegedy et al., 2015; 2016). However, this in turn lays heavy burdens on the memory and other computational resources. For instance, ResNet-152, a specific instance of the latest residual network architecture wining ImageNet classification challenge in 2015, has a model size of about 230 MB and needs to perform about 11.3 billion FLOPs to classify a $2 2 4 \\times 2 2 4$ image crop. Therefore, it is very challenging to deploy deep CNNs on the devices with limited computation and power budgets. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Substantial efforts have been made to the speed-up and compression on CNNs during training, feedforward test or both of them. Among existing methods, the category of network quantization methods attracts great attention from researches and developers. Some network quantization works try to compress pre-trained full-precision CNN models directly. Gong et al. (2014) address the storage problem of AlexNet (Krizhevsky et al., 2012) with vector quantization techniques. By replacing the weights in each of the three fully connected layers with respective floating-point centroid values obtained from the clustering, they can get over $2 0 \\times$ model compression at about $1 \\%$ loss in top-5 recognition rate. HashedNet (Chen et al., 2015b) uses a hash function to randomly map pre-trained weights into hash buckets, and all the weights in the same hash bucket are constrained to share a single floating-point value. In HashedNet, only the fully connected layers of several shallow CNN models are considered. For better compression, Han et al. (2016) present deep compression method which combines the pruning (Han et al., 2015), vector quantization and Huffman coding, and reduce the model storage by $3 5 \\times$ on AlexNet and $4 9 \\times$ on VGG-16 (Simonyan & Zisserman, 2015). Vanhoucke et al. (2011) use an SSE 8-bit fixed-point implementation to improve the computation of neural networks on the modern Intel $\\mathbf { \\boldsymbol { x } } 8 6$ CPUs in feed-forward test, yielding $3 \\times$ speed-up over an optimized floating-point baseline. Training CNNs by substituting the 32-bit floating-point representation with the 16-bit fixed-point representation has also been explored in Gupta et al. (2015). Other seminal works attempt to restrict CNNs into low-precision versions during training phase. Soudry et al. (2014) propose expectation backpropagation (EBP) to estimate the posterior distribution of deterministic network weights. With EBP, the network weights can be constrained to $+ 1$ and $^ { - 1 }$ during feed-forward test in a probabilistic way. BinaryConnect (Courbariaux et al., 2015) further extends the idea behind EBP to binarize network weights during training phase directly. It has two versions of network weights: floating-point and binary. The floating-point version is used as the reference for weight binarization. BinaryConnect achieves state-of-the-art accuracy using shallow CNNs for small datasets such as MNIST (LeCun et al., 1998) and CIFAR-10. Later on, a series of efforts have been invested to train CNNs with low-precision weights, low-precision activations and even low-precision gradients, including but not limited to BinaryNet (Courbariaux et al., 2016), XNOR-Net (Rastegari et al., 2016), ternary weight network (TWN) (Li & Liu, 2016), DoReFa-Net (Zhou et al., 2016) and quantized neural network (QNN) (Hubara et al., 2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Despite these tremendous advances, CNN quantization still remains an open problem due to two critical issues which have not been well resolved yet, especially under scenarios of using low-precision weights for quantization. The first issue is the non-negligible accuracy loss for CNN quantization methods, and the other issue is the increased number of training iterations for ensuring convergence. In this paper, we attempt to address these two issues by presenting a novel incremental network quantization (INQ) method. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In our INQ, there is no assumption on the CNN architecture, and its basic goal is to efficiently convert any pre-trained full-precision (i.e., 32-bit floating-point) CNN model into a low-precision version whose weights are constrained to be either powers of two or zero. The advantage of such kind of low-precision models is that the original floating-point multiplication operations can be replaced by cheaper binary bit shift operations on dedicated hardware like FPGA. We noticed that most existing network quantization methods adopt a global strategy in which all the weights are simultaneously converted to low-precision ones (that are usually in the floating-point types). That is, they have not considered the different importance of network weights, leaving the room to retain network accuracy limited. In sharp contrast to existing methods, our INQ makes a very careful handling for the model accuracy drop from network quantization. To be more specific, it incorporates three interdependent operations: weight partition, group-wise quantization and re-training. Weight partition uses a pruning-inspired measure (Han et al., 2015; Guo et al., 2016) to divide the weights in each layer of a pre-trained full-precision CNN model into two disjoint groups which play complementary roles in our INQ. The weights in the first group are quantized to be either powers of two or zero by a variable-length encoding method, forming a low-precision base for the original model. The weights in the other group are re-trained while keeping the quantized weights fixed, compensating for the accuracy loss resulted from the quantization. Furthermore, these three operations are repeated on the latest re-trained weight group in an iterative manner until all the weights are quantized, acting as an incremental network quantization and accuracy enhancement procedure (as illustrated in Figure 1). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/450f6613f97d9601a580c5c41c8086953fe819a0c5d0d4e38305e967fdbf1359.jpg",
+        "image_caption": [
+            "Figure 1: An overview of our incremental network quantization method. (a) Pre-trained fullprecision model used as a reference. (b) Model update with three proposed operations: weight partition, group-wise quantization (green connections) and re-training (blue connections). (c) Final low-precision model with all the weights constrained to be either powers of two or zero. In the figure, operation (1) represents a single run of (b), and operation (2) denotes the procedure of repeating operation (1) on the latest re-trained weight group until all the non-zero weights are quantized. Our method does not lead to accuracy loss when using 5-bit, 4-bit and even 3-bit approximations in network quantization. For better visualization, here we just use a 3-layer fully connected network as an illustrative example, and the newly re-trained weights are divided into two disjoint groups of the same size at each run of operation (1) except the last run which only performs quantization on the re-trained floating-point weights occupying $12 . 5 \\%$ of the model weights. "
+        ],
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+    {
+        "type": "text",
+        "text": "The main insight of our INQ is that a compact combination of the proposed weight partition, groupwise quantization and re-training operations has the potential to get a lossless low-precision CNN model from any full-precision reference. We conduct extensive experiments on the ImageNet large scale classification task using almost all known deep CNN architectures to validate the effectiveness of our method. We show that: (1) For AlexNet, VGG-16, GoogleNet and ResNets with 5-bit quantization, INQ achieves improved accuracy in comparison with their respective full-precision baselines. The absolute top-1 accuracy gain ranges from ${ \\bf 0 . 1 3 \\% }$ to $2 . 2 8 \\%$ , and the absolute top-5 accuracy gain is in the range of $0 . 2 3 \\%$ to $1 . 6 5 \\%$ . (2) INQ has the property of easy convergence in training. In general, re-training with less than 8 epochs could consistently generate a lossless model with 5-bit weights in the experiments. (3) Taking ResNet-18 as an example, our quantized models with 4-bit, 3-bit and 2-bit ternary weights also have improved or very similar accuracy compared with its 32-bit floating-point baseline. (4) Taking AlexNet as an example, the combination of our network pruning and INQ outperforms deep compression method (Han et al., 2016) with significant margins. ",
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+    {
+        "type": "text",
+        "text": "2 INCREMENTAL NETWORK QUANTIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "In this section, we clarify the insight of our INQ, describe its key components, and detail its implementation. ",
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+    {
+        "type": "text",
+        "text": "2.1 WEIGHT QUANTIZATION WITH VARIABLE-LENGTH ENCODING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Suppose a pre-trained full-precision (i.e., 32-bit floating-point) CNN model can be represented by $\\{ \\mathbf { W } _ { l } : 1 \\le l \\le L \\}$ , where $\\mathbf { W } _ { l }$ denotes the weight set of the ${ \\bf \\chi } _ { l ^ { t h } }$ layer, and $L$ denotes the number of learnable layers in the model. To simplify the explanation, we only consider convolutional layers and fully connected layers. For CNN models like AlexNet, VGG-16, GoogleNet and ResNets as tested in this paper, $\\mathbf { W } _ { l }$ can be a $4 D$ tensor for the convolutional layer, or a $2 D$ matrix for the fully connected layer. For simplicity, here the dimension difference is not considered in the expression. Given a pre-trained full-precision CNN model, the main goal of our INQ is to convert all 32-bit floating-point weights to be either powers of two or zero without loss of model accuracy. Besides, we also attempt to explore the limit of the expected bit-width under the premise of guaranteeing lossless network quantization. Here, we start with our basic network quantization method on how to convert $\\mathbf { W } _ { l }$ to be a low-precision version $\\widehat { \\mathbf { W } } _ { l }$ , and each of its entries is chosen from ",
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+    {
+        "type": "text",
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+    {
+        "type": "equation",
+        "img_path": "images/063d9025c44a7c383e718512d7decf01b18a85bd4520b478c1affe7741370588.jpg",
+        "text": "$$\n\\mathbf { P } _ { l } = \\{ \\pm 2 ^ { n _ { 1 } } , \\cdot \\cdot \\cdot , \\pm 2 ^ { n _ { 2 } } , 0 \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $n _ { 1 }$ and $n _ { 2 }$ are two integer numbers, and they satisfy $n _ { 2 } \\leq n _ { 1 }$ . Mathematically, $n _ { 1 }$ and $n _ { 2 }$ help to bound ${ \\bf P } _ { l }$ in the sense that its non-zero elements are constrained to be in the range of either $[ - 2 ^ { n _ { 1 } } , - 2 ^ { n _ { 2 } } ]$ or $[ 2 ^ { n _ { 2 } } , 2 ^ { n _ { 1 } } ]$ . That is, network weights with absolute values smaller than $2 ^ { n _ { 2 } }$ will be pruned away (i.e., set to zero) in the final low-precision model. Obviously, the problem is how to determine $n _ { 1 }$ and $n _ { 2 }$ . In our INQ, the expected bit-width $b$ for storing the indices in ${ \\bf P } _ { l }$ is set beforehand, thus the only hyper-parameter shall be determined is $n _ { 1 }$ because $n _ { 2 }$ can be naturally computed once $b$ and $n _ { 1 }$ are available. Here, $n _ { 1 }$ is calculated by using a tricky yet practically effective formula as ",
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+    {
+        "type": "equation",
+        "img_path": "images/a74c241828b54be419125e3da1b0152e0dcce20d6436b884dad81f09a6d2dd39.jpg",
+        "text": "$$\nn _ { 1 } = \\mathrm { { f l o o r } ( l o g _ { 2 } ( 4 { s / 3 } ) ) , }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\operatorname { f l o o r } ( \\cdot )$ indicates the round down operation and $s$ is calculated by using ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/41afd528d11c405728ff4d873f9b8cf19a4c359ff0b1e365eb31b37bd57d4c6c.jpg",
+        "text": "$$\ns = \\operatorname* { m a x } ( \\operatorname { a b s } ( \\mathbf { W } _ { l } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "where $\\mathrm { a b s } ( \\cdot )$ is an element-wise operation and $\\operatorname* { m a x } ( { \\mathord { \\cdot } } )$ outputs the largest element of its input. In fact, Equation (2) helps to match the rounding power of 2 for $s$ , and it could be easily implemented in practical programming. After $n _ { 1 }$ is obtained, $n _ { 2 }$ can be naturally determined as $n _ { 2 } = n _ { 1 } + 1 -$ $2 ^ { ( b - 1 ) } / 2$ . For instance, if $b = 3$ and $n _ { 1 } = - 1$ , it is easy to get $n _ { 2 } = - 2$ . ",
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+    {
+        "type": "text",
+        "text": "Once $\\mathbf { P } _ { l }$ is determined, we further use the ladder of powers to convert every entry of $\\mathbf { W } _ { l }$ into a low-precision one by using ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ae159406671e2bfaa47936d645ee9bd5fb2d14adf82c95388f8bf528ca105806.jpg",
+        "text": "$$\n\\widehat { \\mathbf { W } } _ { l } ( i , j ) = \\left\\{ \\begin{array} { l l } { \\beta \\mathrm { s g n } ( \\mathbf { W } _ { l } ( i , j ) ) } & { \\mathrm { i f } ( \\alpha + \\beta ) / 2 \\leq \\mathrm { a b s } ( \\mathbf { W } _ { l } ( i , j ) ) < 3 \\beta / 2 } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } , } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\alpha$ and $\\beta$ are two adjacent elements in the sorted ${ \\bf P } _ { l }$ , making the above equation as a numerical rounding to the quantum values. It should be emphasized that factor $4 / 3$ in Equation (2) is set to make sure that all the elements in ${ \\bf P } _ { l }$ correspond with the quantization rule defined in Equation (4). In other words, factor $4 / 3$ in Equation (2) highly correlates with factor $3 / 2$ in Equation (4). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, an important thing we want to clarify is the definition of the expected bit-width $b$ . Taking 5-bit quantization as an example, since zero value cannot be written as the power of two, we use 1 bit to represent zero value, and the remaining 4 bits to represent at most 16 different values for the powers of two. That is, the number of candidate quantum values is at most $2 ^ { b - 1 } + 1$ , so our quantization method actually adopts a variable-length encoding scheme. It is clear that the quantization described above is performed in a linear scale. An alternative solution is to perform the quantization in the log scale. Although it may also be effective, it should be a little bit more difficult in implementation and may cause some extra computational overhead in comparison to our method. ",
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+    {
+        "type": "text",
+        "text": "2.2 INCREMENTAL QUANTIZATION STRATEGY ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We can naturally use the above described method to quantize any pre-trained full-precision CNN model. However, noticeable accuracy loss appeared in the experiments when using small bit-width values (e.g., 5-bit, 4-bit, 3-bit and 2-bit). ",
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+        "type": "text",
+        "text": "In the literature, there are many existing network quantization works such as HashedNet (Chen et al., 2015b), vector quantization (Gong et al., 2014), fixed-point representation (Vanhoucke et al., 2011; Gupta et al., 2015), BinaryConnect (Courbariaux et al., 2015), BinaryNet (Courbariaux et al., 2016), XNOR-Net (Rastegari et al., 2016), TWN (Li & Liu, 2016), DoReFa-Net (Zhou et al., 2016) and QNN (Hubara et al., 2016). Similar to our basic network quantization method, they also suffer from non-negligible accuracy loss on deep CNNs, especially when being applied on the ImageNet large scale classification dataset. For all these methods, a common fact is that they adopt a global strategy in which all the weights are simultaneously converted into low-precision ones, which in turn causes accuracy loss. Compared with the methods focusing on the pre-trained models, accuracy loss becomes worse for the methods such as XNOR-Net, TWN, DoReFa-Net and QNN which intend to train low-precision CNNs from scratch. ",
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+    {
+        "type": "text",
+        "text": "Recall that our main goal is to achieve lossless low-precision quantization for any pre-trained fullprecision CNN model with no assumption on its architecture. To this end, our INQ makes a special handling of the strategy for suppressing resulting quantization loss in model accuracy. We are partially inspired by the latest progress in network pruning (Han et al., 2015; Guo et al., 2016). In these methods, the accuracy loss from removing less important network weights of a pre-trained neural network model could be well compensated by following re-training steps. Therefore, we conjecture that the nature of changing network weight importance is critical to achieve lossless network quantization. ",
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+        "image_caption": [
+            "Figure 2: Result illustrations. First row: results from the $1 ^ { s t }$ iteration of the proposed three operations. The top left cube illustrates weight partition operation generating two disjoint groups, the middle image illustrates the quantization operation on the first weight group (green cells), and the top right cube illustrates the re-training operation on the second weight group (light blue cells). Second row: results from the $2 ^ { n d }$ , $3 ^ { r d }$ and $\\dot { 4 } ^ { t h }$ iterations of the INQ. In the figure, the accumulated portion of the weights which have been quantized undergoes from $5 0 \\% {  } 7 5 \\% {  } 8 7 . 5 \\% {  } 1 0 0 \\%$ . "
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+        "text": "Base on this assumption, we present INQ which incorporates three interdependent operations: weight partition, group-wise quantization and re-training. Weight partition is to divide the weights in each layer of a pre-trained full-precision CNN model into two disjoint groups which play complementary roles in our INQ. The weights in the first group are responsible for forming a low-precision base for the original model, thus they are quantized by using Equation (4). The weights in the second group adapt to compensate for the loss in model accuracy, thus they are the ones to be re-trained. Once the first run of the quantization and re-training operations is finished, all the three operations are further conducted on the second weight group in an iterative manner, until all the weights are converted to be either powers of two or zero, acting as an incremental network quantization and accuracy enhancement procedure. As a result, accuracy loss under low-precision CNN quantization can be well suppressed by our INQ. Illustrative results at iterative steps of our INQ are provided in Figure 2. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "For the $l ^ { t h }$ layer, weight partition can be defined as ",
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+    {
+        "type": "equation",
+        "img_path": "images/c74caf5196b64bc0440043dec6582bc0c8aa183c49d8e991fdf6da9e023f90b1.jpg",
+        "text": "$$\n\\mathbf { A } _ { l } ^ { ( 1 ) } \\cup \\mathbf { A } _ { l } ^ { ( 2 ) } = \\{ \\mathbf { W } _ { l } ( i , j ) \\} , \\mathrm { a n d } \\mathbf { A } _ { l } ^ { ( 1 ) } \\cap \\mathbf { A } _ { l } ^ { ( 2 ) } = \\varnothing ,\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where ${ \\bf A } _ { l } ^ { ( 1 ) }$ denotes the first weight group that needs to be quantized, and ${ \\bf A } _ { 2 }$ denotes the other weight group that needs to be re-trained. We leave the strategies for group partition to be chosen in the experiment section. Here, we define a binary matrix $\\mathbf { T } _ { l }$ to help distinguish above two categories of weights. That is, $\\mathbf T _ { l } ( i , j ) = 0$ means $\\mathbf { W } _ { l } ( i , j ) \\in \\mathbf { A } _ { l } ^ { ( 1 ) }$ , and $\\mathbf T _ { l } ( i , j ) = 1$ means $\\mathbf { W } _ { l } ( i , j ) \\in \\mathbf { A } _ { l } ^ { ( 2 ) }$ . ",
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+    {
+        "type": "text",
+        "text": "2.3 INCREMENTAL NETWORK QUANTIZATION ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Now, we come to the training method. Taking the $l ^ { t h }$ layer as an example, the basic optimization problem of making its weights to be either powers of two or zero can be expressed as ",
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+    {
+        "type": "equation",
+        "img_path": "images/0274fd925a2b82221032a8a49d586cda947850729fec2758cc916f7e48f2917a.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\underset { \\mathbf { W } _ { l } } { \\operatorname* { m i n } } } & { E ( \\mathbf { W } _ { l } ) = L ( \\mathbf { W } _ { l } ) + \\lambda R ( \\mathbf { W } _ { l } ) } \\\\ { \\mathrm { s . t . } } & { \\mathbf { W } _ { l } ( i , j ) \\in \\mathbf { P } _ { l } , 1 \\leq l \\leq L , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $L ( \\mathbf { W } _ { l } )$ is the network loss, $R ( { \\mathbf W } _ { l } )$ is the regularization term, $\\lambda$ is a positive coefficient, and the constraint term indicates each weight entry ${ \\bf W } _ { l } ( i , j )$ should be chosen from the set $\\mathbf { P } _ { l }$ consisting of a fixed number of the values of powers of two plus zero. Direct solving above optimization problem in training from scratch is challenging since it is very easy to undergo convergence problem. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "By performing weight partition and group-wise quantization operations beforehand, the optimization problem defined in (6) can be reshaped into a easier version. That is, we only need to optimize the following objective function ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/53397c396e27483e01a1bc37b65e5df028f23d05cfe614ec51875aebad7e67ab.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\underset { \\mathbf { W } _ { l } } { \\mathop { \\operatorname* { m i n } } } } & { E ( \\mathbf { W } _ { l } ) = L ( \\mathbf { W } _ { l } ) + \\lambda R ( \\mathbf { W } _ { l } ) } \\\\ { \\mathrm { s . t . } } & { \\mathbf { W } _ { l } ( i , j ) \\in \\mathbf { P } _ { l } , \\mathrm { ~ i f ~ } \\mathbf { T } _ { l } ( i , j ) = 0 , 1 \\leq l \\leq L , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { P } _ { l }$ is determined at group-wise quantization operation, and the binary matrix $\\mathbf { T } _ { l }$ acts as a mask which is determined by weight partition operation. Since $\\mathbf { P } _ { l }$ and $\\mathbf { T } _ { l }$ are known, the optimization problem (7) can be solved using popular stochastic gradient decent (SGD) method. That is, in INQ, we can get the update scheme for the re-training as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/595d3c22243faffae94fbc176af0274cad661faeb0d9ee751158637e0f3a4f4f.jpg",
+        "text": "$$\n\\mathbf { W } _ { l } ( i , j )  \\mathbf { W } _ { l } ( i , j ) - \\gamma \\frac { \\partial E } { \\partial ( \\mathbf { W } _ { l } ( i , j ) ) } \\mathbf { T } _ { l } ( i , j ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            513
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\gamma$ is a positive learning rate. Note that the binary matrix $\\mathbf { T } _ { l }$ forces zero update to the weights that have been quantized. That is, only the weights still keep with floating-point values are updated, akin to the latest pruning methods (Han et al., 2015; Guo et al., 2016) in which only the weights that are not currently removed are re-trained to enhance network accuracy. The whole procedure of our INQ is summarized as Algorithm 1. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We would like to highlight that the merits of our INQ are in three aspects: (1) Weight partition introduces the importance-aware weight quantization. (2) Group-wise weight quantization introduces much less accuracy loss than simultaneously quantizing all the network weights, thus making retraining have larger room to recover model accuracy. (3) By integrating the operations of weight partition, group-wise quantization and re-training into a nested loop, our INQ has the potential to obtain lossless low-precision CNN model from the pre-trained full-precision reference. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "g   \nInput: $X$ : the training data, $\\{ \\mathbf { W } _ { l } : 1 \\le l \\le L \\}$ : the pre-trained full-precision CNN model, $\\{ \\sigma _ { 1 } , \\sigma _ { 2 } , \\cdots , \\sigma _ { N } \\}$ : the accumulated portions of weights quantized at iterative steps   \nOutput: $\\{ \\widehat { \\mathbf { W } } _ { l } : 1 \\leq l \\leq L \\}$ : the final low-precision model with the weights constrained to be either powers of two or zero 1: Initialize $\\mathbf { A } _ { l } ^ { ( 1 ) }  \\varnothing , \\mathbf { A } _ { l } ^ { ( 2 ) }  \\{ \\mathbf { W } _ { l } ( i , j ) \\} , \\mathbf { T } _ { l }  \\mathbf { 1 } , \\operatorname { f o r } 1 \\leq l \\leq L$ 2: for $n = 1 , 2 , \\ldots , N$ do 3: Reset the base learning rate and the learning policy 4: According to $\\sigma _ { n }$ , perform layer-wise weight partition and update $\\mathbf { A } _ { l } ^ { ( 1 ) } , \\mathbf { A } _ { l } ^ { ( 2 ) }$ and $\\mathbf { T } _ { l }$ 5: Based on ${ \\bf A } _ { l } ^ { ( 1 ) }$ , determine $\\mathbf { P } _ { l }$ layer-wisely 6: Quantize the weights in ${ \\bf A } _ { l } ^ { ( 1 ) }$ by Equation (4) layer-wisely 7: Calculate feed-forward loss, and update weights in $\\{ \\mathbf { A } _ { l } ^ { ( 2 ) } : 1 \\le l \\le L \\}$ by Equation (8) 8: end for ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To analyze the performance of our INQ, we perform extensive experiments on the ImageNet large scale classification task, which is known as the most challenging image classification benchmark so far. ImageNet dataset has about 1.2 million training images and 50 thousand validation images. Each image is annotated as one of 1000 object classes. We apply our INQ to AlexNet, VGG-16, GoogleNet, ResNet-18 and ResNet-50, covering almost all known deep CNN architectures. Using the center crops of validation images, we report the results with two standard measures: top-1 error rate and top-5 error rate. For fair comparison, all pre-trained full-precision (i.e., 32-bit floatingpoint) CNN models except ResNet-18 are taken from the Caffe model ${ \\mathrm { z o o } } ^ { 2 }$ . Note that He et al. (2016) do not release their pre-trained ResNet-18 model to the public, so we use a publicly available re-implementation by Facebook3. Since our method is implemented with Caffe, we make use of an open source tool4 to convert the pre-trained ResNet-18 model from Torch to Caffe. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.1 RESULTS ON IMAGENET ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f0be6d262bbab5baf18e528321b86286200392a5184ed3b0334f7465cc3f708e.jpg",
+        "table_caption": [
+            "Table 1: Our INQ well converts diverse full-precision deep CNN models (including AlexNet, VGG16, GoogleNet, ResNet-18 and ResNet-50) to 5-bit low-precision versions with consistently improved model accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network</td><td>Bit-width</td><td>Top-1 error</td><td>Top-5 error</td><td>Decrease in top-1/top-5 error</td></tr><tr><td>AlexNet ref</td><td>32</td><td>42.76%</td><td>19.77%</td><td></td></tr><tr><td>AlexNet VGG-16 ref</td><td>5 32</td><td>42.61% 31.46%</td><td>19.54% 11.35%</td><td>0.15%/0.23%</td></tr><tr><td>VGG-16</td><td>5</td><td>29.18%</td><td>9.70%</td><td>2.28%/1.65%</td></tr><tr><td>GoogleNet ref</td><td></td><td></td><td></td><td></td></tr><tr><td>GoogleNet</td><td>32</td><td>31.11%</td><td>10.97%</td><td></td></tr><tr><td>ResNet-18 ref</td><td>5</td><td>30.98%</td><td>10.72%</td><td>0.13%/0.25%</td></tr><tr><td>ResNet-18</td><td>32</td><td>31.73%</td><td>11.31%</td><td></td></tr><tr><td></td><td>5</td><td>31.02%</td><td>10.90%</td><td>0.71%/0.41%</td></tr><tr><td>ResNet-50 ref</td><td>32</td><td>26.78%</td><td>8.76%</td><td></td></tr><tr><td>ResNet-50</td><td>5</td><td>25.19%</td><td>7.55%</td><td>1.59%/1.21%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Setting expected bit-width to 5, the first set of experiments is performed to testify the efficacy of our INQ on different CNN architectures. Regarding weight partition, there are several candidate strategies as we tried in our previous work for efficient network pruning (Guo et al., 2016). In Guo et al. (2016), we found random partition and pruning-inspired partition are the two best choices compared with the others. Thus in this paper, we directly compare these two strategies for weight partition. In random strategy, the weights in each layer of any pre-trained full-precision deep CNN model are randomly split into two disjoint groups. In pruning-inspired strategy, the weights are divided into two disjoint groups by comparing their absolute values with layer-wise thresholds which are automatically determined by a given splitting ratio. Here we directly use pruning-inspired strategy and the experimental results in Section 3.2 will show why. After the re-training with no more than 8 epochs over each pre-trained full-precision model, we obtain the results as shown in Table 1. It can be concluded that the 5-bit CNN models generated by our INQ show consistently improved top-1 and top-5 recognition rates compared with respective full-precision references. Parameter settings are described below. ",
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+    },
+    {
+        "type": "text",
+        "text": "AlexNet: AlexNet has 5 convolutional layers and 3 fully-connected layers. We set the accumulated portions of quantized weights at iterative steps as $\\{ 0 . 3 , \\dot { 0 } . 6 , 0 . 8 , 1 \\}$ , the batch size as 256, the weight decay as 0.0005, and the momentum as 0.9. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "VGG-16: Compared with AlexNet, VGG-16 has 13 convolutional layers and more parameters. We set the accumulated portions of quantized weights at iterative steps as $\\{ 0 . 5 , 0 . 7 5 , 0 . 8 7 5 , 1 \\}$ , the batch size as 32, the weight decay as 0.0005, and the momentum as 0.9. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "GoogleNet: Compared with AlexNet and VGG-16, GoogleNet is more difficult to quantize due to a smaller number of parameters and the increased network width. We set the accumulated portions of quantized weights at iterative steps as $\\{ 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 \\}$ , the batch size as 80, the weight decay as 0.0002, and the momentum as 0.9. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "ResNet-18: Different from above three networks, ResNets have batch normalization layers and relief the vanishing gradient problem by using shortcut connections. We first test the 18-layer version for exploratory purpose and test the 50-layer version later on. The network architectures of ResNet18 and ResNet-34 are very similar. The only difference is the number of filters in every convolutional layer. We set the accumulated portions of quantized weights at iterative steps as {0.5, 0.75, 0.875, $1 \\}$ , the batch size as 80, the weight decay as 0.0005, and the momentum as 0.9. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "ResNet-50: Besides significantly increased network depth, ResNet-50 has a more complex network architecture in comparison to ResNet-18. However, regarding network architecture, ResNet-50 is very similar to ResNet-101 and ResNet-152. The only difference is the number of filters in every convolutional layer. We set the accumulated portions of quantized weights at iterative steps as $\\{ 0 . 5$ , 0.75, 0.875, 1}, the batch size as 32, the weight decay as 0.0005, and the momentum as 0.9. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.2 ANALYSIS OF WEIGHT PARTITION STRATEGIES",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In our INQ, the first operation is weight partition whose result will directly affect the following group-wise quantization and re-training operations. Therefore, the second set of experiments is conducted to analyze two candidate strategies for weight partition. As mentioned in the previous section, we use pruning-inspired strategy for weight partition. Unlike random strategy in which all the weights have equal probability to fall into the two disjoint groups, pruning-inspired strategy considers that the weights with larger absolute values are more important than the smaller ones to form a low-precision base for the original CNN model. We use ResNet-18 as a test case to compare the performance of these two strategies. In the experiments, the parameter settings are completely the same as described in Section 3.1. We set 4 epochs for weight re-training. Table 2 summarizes the results of our INQ with 5-bit quantization. It can be seen that our INQ achieves top-1 error rate of $3 2 . 1 1 \\%$ and top-5 error rate of $1 1 . 7 3 \\%$ by using random partition. Comparatively, pruning-inspired partition brings $1 . 0 9 \\%$ and $0 . 8 3 \\%$ decrease in top-1 and top-5 error rates, respectively. Apparently, pruning-inspired partition is better than random partition, and this is the reason why we use it in this paper. For future works, weight partition based on quantization error could also be an option worth exploring. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/73ecc432bc17a02559e3f0592b3b349f982811d9d8005b8a84ffa0394ac1efa1.jpg",
+        "table_caption": [
+            "Table 2: Comparison of two different strategies for weight partition on ResNet-18. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Strategy</td><td>Bit-width</td><td>Top-1 error</td><td>Top-5 error</td></tr><tr><td>Random partition</td><td>5</td><td>32.11%</td><td>11.73%</td></tr><tr><td>Pruning-inspired partition</td><td>5</td><td>31.02%</td><td>10.90%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.3 THE TRADE-OFF BETWEEN EXPECTED BIT-WIDTH AND MODEL ACCURACY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The third set of experiments is performed to explore the limit of the expected bit-width under which our INQ can still achieve lossless network quantization. Similar to the second set of experiments, we also use ResNet-18 as a test case, and the parameter settings for the batch size, the weight decay and the momentum are completely the same. Finally, lower-precision models with 4-bit, 3-bit and even 2-bit ternary weights are generated for comparisons. As the expected bit-width goes down, the number of candidate quantum values will be decreased significantly, thus we shall increase the number of iterative steps accordingly for enhancing the accuracy of final low-precision model. Specifically, we set the accumulated portions of quantized weights at iterative steps as $\\{ 0 . 3 , 0 . 5$ , 0.8, 0.9, 0.95, $1 \\}$ , $\\{ 0 . 2 , 0 . 4 , 0 . 6 , 0 . 7 , \\bar { 0 } . 8 , 0 . 9 , 0 . 9 \\bar { 5 } , 1 \\}$ and $\\{ 0 . 2 , 0 . 4 , 0 . 6 , 0 . 7 , 0 . 8 , \\bar { 0 } . 8 5 , \\bar { 0 } . 9 , 0 . 9 5 _ { \\cdotp }$ , $0 . 9 7 5 , 1 \\}$ for 4-bit, 3-bit and 2-bit ternary models, respectively. The required number of epochs also increases when the expected bit-width goes down, and it reaches 30 when training our 2-bit ternary model. Although our 4-bit model shows slightly decreased accuracy when compared with the 5-bit model, its accuracy is still better than that of the pre-trained full-precision model. Comparatively, even when the expected bit-width goes down to 3, our low-precision model shows only ${ \\bar { 0 . 1 9 \\% } }$ and ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "$0 . 3 3 \\%$ losses in top-1 and top-5 recognition rates, respectively. As for our 2-bit ternary model, although it incurs $2 . 2 5 \\%$ decrease in top-1 error rate and $1 . 5 6 \\%$ decrease in top-5 error rate in comparison to the pre-trained full-precision reference, its accuracy is considerably better than stateof-the-art results reported for binary-weight network (BWN) (Rastegari et al., 2016) and ternary weight network (TWN) (Li & Liu, 2016). Detailed results are summarized in Table 3 and Table 4. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/709c2174f9fa8ab8d02fdce20bc36d23eb293418d1f13d38326d001991459693.jpg",
+        "table_caption": [
+            "Table 3: Our INQ generates extremely low-precision (4-bit and 3-bit) models with improved or very similar accuracy compared with the full-precision ResNet-18 model. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Bit-width</td><td>Top-1 error</td><td>Top-5 error</td></tr><tr><td>ResNet-18 ref</td><td>32</td><td>31.73%</td><td>11.31%</td></tr><tr><td>INQ</td><td>5</td><td>31.02%</td><td>10.90%</td></tr><tr><td>INQ</td><td>4</td><td>31.11%</td><td>10.99%</td></tr><tr><td>INQ</td><td>3</td><td>31.92%</td><td>11.64%</td></tr><tr><td>INQ</td><td>2 (ternary)</td><td>33.98%</td><td>12.87%</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/e5fd9da4669cecfaa7d6bd62ed8e045e5cf2622c2c9d9e5021727c06dd6ed38f.jpg",
+        "table_caption": [
+            "Table 4: Comparison of our 2-bit ternary model and some other binary or ternary models, including the BWN and the TWN approximations of ResNet-18. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Bit-width</td><td>Top-1 error</td><td>Top-5 error</td></tr><tr><td>BWN(Rastegari et al., 2016)</td><td>1</td><td>39.20%</td><td>17.00%</td></tr><tr><td>TWN(Li &amp; Liu, 2016)</td><td>2 (ternary)</td><td>38.20%</td><td>15.80%</td></tr><tr><td>INQ (ours)</td><td>2 (ternary)</td><td>33.98%</td><td>12.87%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "3.4 LOW-BIT DEEP COMPRESSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the literature, recently proposed deep compression method (Han et al., 2016) reports so far best results on network compression without loss of model accuracy. Therefore, the last set of experiments is conducted to explore the potential of our INQ for much better deep compression. Note that Han et al. (2016) is a hybrid network compression solution combining three different techniques, namely network pruning (Han et al., 2015), vector quantization (Gong et al., 2014) and Huffman coding. Taking AlexNet as an example, network pruning gets $9 \\times$ compression, however this result is mainly obtained from the fully connected layers. Actually its compression performance on the convolutional layers is less than $3 \\times$ (as can be seen in the Table 4 of Han et al. (2016)). Besides, network pruning is realized by separately performing pruning and re-training in an iterative way, which is very time-consuming. It will cost at least several weeks for compressing AlexNet. We solved this problem by our dynamic network surgery (DNS) method (Guo et al., 2016) which achieves about $7 \\times$ speed-up in training and improves the performance of network pruning from $9 \\times$ to $1 7 . 7 \\times$ . In Han et al. (2016), after network pruning, vector quantization further improves compression ratio from $9 \\times$ to $2 7 \\times$ , and Huffman coding finally boosts compression ratio up to $3 5 \\times$ . For fair comparison, we combine our proposed INQ and DNS, and compare the resulting method with Han et al. (2016). Detailed results are summarized in Table 5. When combing our proposed INQ and DNS, we achieve much better compression results compared with Han et al. (2016). Specifically, with 5-bit quantization, we can achieve $5 3 \\times$ compression with slightly larger gains both in top-5 and top-1 recognition rates, yielding $5 1 . 4 3 \\% / 9 6 . 3 0 \\%$ absolute improvement in compression performance compared with full version/fair version (i.e., the combination of network pruning and vector quantization) of Han et al. (2016), respectively. Consistently better results have also obtained for our 4-bit and 3-bit models. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Besides, we also perform a set of experiments on AlexNet to compare the performance of our INQ and vector quantization (Gong et al., 2014). For fair comparison, re-training is also used to enhance the performance of vector quantization, and we set the number of cluster centers for all of 5 convolutional layers and 3 fully connect layers to 32 (i.e., 5-bit quantization). In the experiment, vector quantization incurs over $3 \\%$ loss in model accuracy. When we change the number of cluster centers for convolutional layers from 32 to 128, it gets an accuracy loss of $0 . 9 8 \\%$ . This is consistent with the results reported in (Gong et al., 2014). Comparatively, vector quantization is mainly proposed to compress the parameters in the fully connected layers of a pre-trained full-precision CNN model, while our INQ addresses all network layers simultaneously and has no accuracy loss for 5-bit and 4-bit quantization. Therefore, it is evident that our INQ is much better than vector quantization. Last but not least, the final weights for vector quantization (Gong et al., 2014), network pruning (Han et al., 2015) and deep compression (Han et al., 2016) are still floating-point values, but the final weights for our INQ are in the form of either powers of two or zero. The direct advantage of our INQ is that the original floating-point multiplication operations can be replaced by cheaper binary bit shift operations on dedicated hardware like FPGA. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8d0fa65f7749b8cbbbf674b92d9532548aa470e8745f5e064fea8412d23c52a4.jpg",
+        "table_caption": [
+            "Table 5: Comparison of the combination of our INQ and DNS, and deep compression method on AlexNet. Conv: Convolutional layer, FC: Fully connected layer, P: Pruning, Q: Quantization, H: Huffman coding. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Bit-width(Conv/FC)</td><td>Compression ratio</td><td>Decrease in top-1/top5 error</td></tr><tr><td>Han et al. (2016) (P+Q)</td><td>8/5</td><td>27×</td><td>0.00%/0.03%</td></tr><tr><td>Han et al. (2016) (P+Q+H)</td><td>8/5</td><td>35×</td><td>0.00%/0.03%</td></tr><tr><td>Han et al. (2016) (P+Q+H)</td><td>8/4</td><td>1</td><td>-0.01%/0.00%</td></tr><tr><td>Our method (P+Q)</td><td>5/5</td><td>53×</td><td>0.08%/0.03%</td></tr><tr><td>Han et al. (2016) (P+Q+H)</td><td>4/2</td><td>1</td><td>-1.99%/-2.60%</td></tr><tr><td>Our method (P+Q)</td><td>4/4</td><td>71×</td><td>-0.52%/-0.20%</td></tr><tr><td>Our method (P+Q)</td><td>3/3</td><td>89×</td><td>-1.47 %/-0.96%</td></tr></table>",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present INQ, a new network quantization method, to address the problem of how to convert any pre-trained full-precision (i.e., 32-bit floating-point) CNN model into a lossless lowprecision version whose weights are constrained to be either powers of two or zero. Unlike existing methods which usually quantize all the network weights simultaneously, INQ is a more compact quantization framework. It incorporates three interdependent operations: weight partition, groupwise quantization and re-training. Weight partition splits the weights in each layer of a pre-trained full-precision CNN model into two disjoint groups which play complementary roles in INQ. The weights in the first group is directly quantized by a variable-length encoding method, forming a low-precision base for the original CNN model. The weights in the other group are re-trained while keeping all the quantized weights fixed, compensating for the accuracy loss from network quantization. More importantly, the operations of weight partition, group-wise quantization and re-training are repeated on the latest re-trained weight group in an iterative manner until all the weights are quantized, acting as an incremental network quantization and accuracy enhancement procedure. On the ImageNet large scale classification task, we conduct extensive experiments and show that our quantized CNN models with 5-bit, 4-bit, 3-bit and even 2-bit ternary weights have improved or at least comparable accuracy against their full-precision baselines, including AlexNet, VGG-16, GoogleNet and ResNets. As for future works, we plan to extend incremental idea behind INQ from low-precision weights to low-precision activations and low-precision gradients (we have actually already made some good progress on it, as shown in our supplementary materials). We will also investigate computation and power efficiency by implementing our low-precision CNN models on hardware platforms. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "type": "text",
+        "text": "A APPENDIX 1: STATISTICAL ANALYSIS OF THE QUANTIZED WEIGHTS",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            782,
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+    {
+        "type": "text",
+        "text": "Taking our 5-bit AlexNet model as an example, we analyze the distribution of the quantized weights. Detailed statistical results are summarized in Table 6. We can find: (1) in the $1 ^ { s t }$ and $2 ^ { n d }$ convolutional layers, the values of $\\{ - 2 ^ { - 6 } , - 2 ^ { - 5 } , - 2 ^ { - 4 } , 2 ^ { - 6 } , 2 ^ { - 5 } , 2 ^ { - 4 } \\}$ and $\\{ - 2 ^ { - 8 } , - 2 ^ { - 7 } , - 2 ^ { - 6 } , - 2 ^ { - 5 }$ , $0 , 2 ^ { - 8 } , \\dot { 2 } ^ { - 7 } , 2 ^ { - 6 } , 2 ^ { - 5 } \\}$ occupy over $60 \\%$ and $94 \\%$ of all quantized weights, respectively; (2) the distributions of the quantized weights in the $3 ^ { r d }$ , $4 ^ { t h }$ and $5 ^ { t h }$ convolutional layers are similar to that of the $2 ^ { n d }$ convolutional layer, and more weights are quantized into zero in the $2 ^ { n d }$ , $3 ^ { r d }$ , $4 ^ { t h }$ and $5 ^ { t h }$ convolutional layers compared with the $1 ^ { s t }$ convolutional layer; (3) in the $1 ^ { s t }$ fully connected layer, the values of $\\left\\{ - 2 ^ { - 1 0 } , \\ - 2 ^ { - 9 } , - 2 ^ { - 8 } , - 2 ^ { - 7 } , 0 , 2 ^ { - 1 0 } , 2 ^ { - 9 } , 2 ^ { - 8 } , 2 ^ { - 7 } \\right\\}$ occupy about $98 \\%$ of all quantized weights, and similar results can be seen for the $2 ^ { n d }$ fully connected layer; (4) generally, the distributions of the quantized weights in the convolutional layers are usually more scattered compared with the fully connected layers. This may be partially the reason why it is much easier to get good compression performance on fully connected layers in comparison to convolutional layers, when using methods such as network hashing (Chen et al., 2015b) and vector quantization (Gong et al., 2014); (5) for 5-bit AlexNet model, the required bit-width for each layer is actually 4 but not 5. ",
+        "bbox": [
+            173,
+            133,
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+            340
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/4dce5fa55b39479a05fa2990215f3f0a3bccd931f1c5604e7dd0b85e60cf11fd.jpg",
+        "table_caption": [
+            "Table 6: A statistical distribution of the quantized weights in our 5-bit AlexNet model. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Weight</td><td>Conv1</td><td>Conv2</td><td>Conv3</td><td>Conv4</td><td>Conv5</td><td>FC6</td><td>FC7</td><td>FC8</td></tr><tr><td>-2-10</td><td></td><td></td><td>=</td><td>=</td><td></td><td>8.95%</td><td>6.37%</td><td>3.86%</td></tr><tr><td>-2-9</td><td>=</td><td></td><td>=</td><td></td><td></td><td>12.29%</td><td>9.58%</td><td>6.19%</td></tr><tr><td>-2-8</td><td>5.04%</td><td>10.55%</td><td>11.58%</td><td>10.09%</td><td>9.88%</td><td>16.48%</td><td>16.13%</td><td>12.90%</td></tr><tr><td>-2-7</td><td>6.56%</td><td>12.09%</td><td>14.34%</td><td>14.24%</td><td>14.68%</td><td>10.84%</td><td>17.87%</td><td>19.51%</td></tr><tr><td>-2-6</td><td>9.22%</td><td>13.08%</td><td>15.26%</td><td>18.49%</td><td>20.66%</td><td>0.79%</td><td>3.43%</td><td>11.09%</td></tr><tr><td>-2-5</td><td>10.52%</td><td>8.73%</td><td>5.92%</td><td>7.77%</td><td>9.79%</td><td>0.002%</td><td>0.004%</td><td>0.40%</td></tr><tr><td>-2-4</td><td>9.75%</td><td>2.70%</td><td>0.49%</td><td>0.38%</td><td>0.55%</td><td>1</td><td>1</td><td>1</td></tr><tr><td>-2-3</td><td>4.61%</td><td>0.39%</td><td>0.02%</td><td>0.01%</td><td>0.004%</td><td></td><td></td><td></td></tr><tr><td>-2-2</td><td>0.67%</td><td>0.01%</td><td>1e-4%</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>0</td><td>5.51%</td><td>11.30%</td><td>12.24%</td><td>9.70%</td><td>8.97%</td><td>8.86%</td><td>6.17%</td><td>3.62%</td></tr><tr><td>2-10</td><td></td><td></td><td></td><td></td><td></td><td>8.30%</td><td>5.81%</td><td>3.40%</td></tr><tr><td>2-9</td><td>1</td><td></td><td></td><td></td><td></td><td>10.51%</td><td>7.84%</td><td>4.69%</td></tr><tr><td>2-8</td><td>5.20%</td><td>9.70%</td><td>10.44%</td><td>8.60%</td><td>7.69%</td><td>12.91%</td><td>11.30%</td><td>8.08%</td></tr><tr><td>2-7</td><td>6.79%</td><td>11.01%</td><td>11.66%</td><td>10.33%</td><td>8.95%</td><td>8.95%</td><td>11.90%</td><td>10.94%</td></tr><tr><td>2-6</td><td>9.99%</td><td>11.05%</td><td>11.86%</td><td>12.25%</td><td>10.67%</td><td>1.12%</td><td>3.54%</td><td>12.56%</td></tr><tr><td>2-5</td><td>11.15%</td><td>6.57%</td><td>5.22%</td><td>6.81%</td><td>6.37%</td><td>0.01%</td><td>0.06%</td><td>2.75%</td></tr><tr><td>2-4</td><td>10.14%</td><td>2.26%</td><td>0.86%</td><td>1.24%</td><td>1.62%</td><td>1e-5%</td><td>2e-5%</td><td>0.01%</td></tr><tr><td>2-3</td><td>4.26%</td><td>0.53%</td><td>0.09%</td><td>0.08%</td><td>0.16%</td><td>-</td><td>=</td><td></td></tr><tr><td>2-2</td><td>0.60%</td><td>0.05%</td><td>0.01%</td><td>0.003%</td><td>0.01%</td><td></td><td></td><td></td></tr><tr><td>2-1</td><td>1</td><td>3e-4%</td><td>2e-4%</td><td>3e-4%</td><td>1</td><td>1</td><td>=</td><td></td></tr><tr><td>Total</td><td>100%</td><td>100%</td><td>100%</td><td>100%</td><td>100%</td><td>100%</td><td>100%</td><td>1</td></tr><tr><td>Bit-width</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>100% 4</td></tr></table>",
+        "bbox": [
+            173,
+            385,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B APPENDIX 2: LOSSLESS CNNS WITH LOW-PRECISION WEIGHTS AND LOW-PRECISION ACTIVATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            756,
+            787,
+            790
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/fd2dabc808ee2c829122068119fef895f5c81e0c10026102779a914047f2ca0d.jpg",
+        "table_caption": [
+            "Table 7: Comparison of our VGG-16 model with 5-bit weights and 4-bit activations, and the pretrained reference with 32-bit floating-point weights and 32-bit float-point activations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network</td><td>Bit-width for weight/activation</td><td>Top-1 error</td><td>Top-5 error</td><td>Decrease in top-1/top-5 error</td></tr><tr><td>VGG-16 ref</td><td>32/32</td><td>31.46%</td><td>11.35%</td><td></td></tr><tr><td>VGG-16</td><td>5/4</td><td>29.82%</td><td>10.19%</td><td>1.64%/1.16%</td></tr></table>",
+        "bbox": [
+            222,
+            854,
+            774,
+            915
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Recently, we have made some good progress on developing our INQ for lossless CNNs with both low-precision weights and low-precision activations. According to the results summarized in Table 7, it can be seen that our VGG-16 model with 5-bit weights and 4-bit activations shows improved top-5 and top-1 recognition rates in comparison to the pre-trained reference with 32-bit floating-point weights and 32-bit floating-point activations. To the best of our knowledge, this should be the best results reported on VGG-16 architecture so far. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            186
+        ],
+        "page_idx": 13
+    }
+]

parse/train/SJlpM3RqKQ/SJlpM3RqKQ.md

@@ -0,0 +1,246 @@
+# EXPANDING THE REACH OF FEDERATED LEARNING BY REDUCING CLIENT RESOURCE REQUIREMENTS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Communication on heterogeneous edge networks is a fundamental bottleneck in Federated Learning (FL), restricting both model capacity and user participation. To address this issue, we introduce two novel strategies to reduce communication costs: (1) the use of lossy compression on the global model sent serverto-client; and (2) Federated Dropout, which allows users to efficiently train locally on smaller subsets of the global model and also provides a reduction in both client-to-server communication and local computation. We empirically show that these strategies, combined with existing compression approaches for client-toserver communication, collectively provide up to a $1 4 \times$ reduction in server-toclient communication, a $1 . 7 \times$ reduction in local computation, and a $2 8 \times$ reduction in upload communication, all without degrading the quality of the final model. We thus comprehensively reduce FL’s impact on client device resources, allowing higher capacity models to be trained, and a more diverse set of users to be reached.
+
+# 1 INTRODUCTION
+
+Federated Learning (FL) allows users to reap the benefits of models trained from rich yet sensitive data captured by their mobile devices, without the need to centrally store such data (McMahan et al., 2017; Konecnˇ y et al., 2016a; Smith et al., 2017). Under the FL paradigm, each device performs ´ training on samples available locally and only communicates intermediate model updates.
+
+Network speed and number of nodes are two of the core systems aspects that differentiate FL from traditional distributed learning in data centers, with network bandwidth being potentially orders of magnitude slower and the number of worker nodes orders of magnitude larger. Together, these issues exacerbate the communication bottlenecks usually associated with distributed learning, increasing both the number of stragglers and the probability of devices dropping out altogether. The problem is further aggravated when working with high capacity models with large numbers of parameters.
+
+Insisting on training these large models using existing federated optimization methods can lead to the systematic exclusion of clients with restricted bandwidth or limited network access from the training stage, and thus to a degraded user experience once these models are served. One naive solution involves training low capacity models with smaller communication footprints, at the expense of model accuracy. As a middle ground, we could develop strategies to reduce the communication footprint of larger, high-capacity models. Recent work (Konecnˇ y et al., 2016b) has in fact taken ´ this approach, but only in the context of client-to-server FL communication. Their success with lossy compression strategies is perhaps not surprising, as the clients’ lossy, yet unbiased, updates are eventually averaged over many users. However, server-to-client exchanges do not benefit from such averaging. As such, they remain a main bottleneck in our goal of expanding FL’s reach.
+
+In this work, we propose two novel strategies to mitigate the server-to-client communication footprint, and empirically demonstrate their efficacy and seamless integration with existing client-toserver strategies. The specific contributions of this paper are as follows:
+
+1. We study lossily compressing the models downloaded by the clients, thus addressing the open question as to whether these approaches are amenable in the context of server-to-client exchanges. We also introduce the use of the theoretically motivated Kashin’s representation to reduce the error associated with the lossy compression (Lyubarskii & Vershynin, 2010; Kashin, 1977).
+
+![](images/09f2c19b96d2be5f52ed90bc55800a74b6fd0ad93b78c70fc4916cc96768b8a1.jpg)  
+Figure 1: Combination of our proposed strategies during FL training. We reduce the size of the model to be communicated by (1) constructing a sub-model via Federated Dropout, and by (2) lossily compressing the resulting object. This compressed model is then sent to the client, who (3) decompresses and trains it using local data, and (4) compresses the final update. This update is sent back to the server, where it is (5) decompressed and finally, (6) aggregated into the global model.
+
+2. We introduce Federated Dropout, a technique that builds upon the popular idea of dropout (Srivastava et al., 2014), yet is primarily motivated by systems-related concerns. Our approach enables each device to locally operate on a smaller sub-model (i.e. with smaller weight matrices) while still providing updates that can be applied to the larger global model on the server. It thus reduces communication costs by allowing for these smaller sub-models to be exchanged between server and clients, while also reducing the computational cost of local training.
+
+3. We empirically show that not only are these approaches compatible with one another, but with existing client-to-server compression. Combining these approaches during FL training (see Figure 1) reduces the size of the downloaded models up to $1 4 \times$ , the size of the corresponding updates up to $2 8 \times$ , and the required local computations by up to $1 . 7 \times$ , all without degrading the model’s accuracy and only at the expense of a slightly slower convergence rate (in terms of number of communication rounds).
+
+# 2 RELATED WORK
+
+We review the relevant related work given our objective of reducing the communication footprint in server-to-client exchanges in Federated Learning (FL).
+
+Federated Learning Federated Learning (FL) is a technique that aims to learn a global model over data distributed across multiple edge devices (usually mobile phones) without the data ever leaving the device on which it was generated (McMahan et al., 2017). It brings along a set of statistical (nonIID, unbalanced data) and systems (stragglers, communication bottlenecks, etc.) challenges which differentiate it from traditional distributed learning in the data center, and which have been tackled by several works. For instance, McMahan et al. (2017) propose Federated Averaging (FedAvg), which in its canonical form works by (1) sending the global model to a subset of the available devices, (2) training the model on each device using the available local data, and (3) averaging the local updates to thus end a round of training. In contrast Smith et al. (2017) present a multi-task variant that also models the relationship between clients in order to learn personalized yet related models for each device. Nonetheless, all approaches we are aware of (including the two aforementioned ones) require continued exchanges between a central server and its clients across a potentially slow network.
+
+Communication-efficient distributed learning Distributed learning is known to suffer from communication overheads associated with the frequent gradient updates exchanged among nodes (Wang et al., 2018; Dean et al., 2012; Smith et al., 2018; Reddi et al., 2016). To reduce these bottlenecks, recent studies focus on communicating a sparsified, quantized or randomly subsampled version of the updates. Although these operations introduce noise, they have been shown both empirically and theoretically to maintain the quality of the trained models. We refer the reader to the introduction of Wang et al. (2018) for more details and references.
+
+In the context of FL, Konecnˇ y et al. (2016b) successfully perform lossy compression on the client- ´ to-server exchanges (i.e. the model updates). Of particular interest is their use of the randomized Hadamard transform to reduce the error incurred by the subsequent quantization. This is due to the fact that the Hadamard transform, in expectation, spreads a vector’s information more evenly across its components (Suresh et al., 2017; Konecnˇ y & Richt ´ arik, 2016). ´
+
+We note, however, that neither the work on traditional distributed learning nor the work of Konecnˇ y´ et al. (2016b) considers compressing the server-to-client exchanges. Nevertheless, in FL, downloading a large model can still be a considerable burden for users, particularly for those in regions with network constraints. Furthermore, as FL is expected to deal with a large number of devices, communicating the global model may even become a bottleneck for the server (as it would, ideally, send the model to the clients in parallel).
+
+Model compression Deep models tend to demand significant computational resources both for training and inference. Using them on edge devices is therefore not a straightforward task. Because of this, several recent works have proposed compressing the models before deploying them on-device (Ravi, 2018). Popular alternatives include pruning the least useful connections in a network (Han et al., 2016; 2015), weight quantization (Hubara et al., 2016; Lin et al., 2017; De Sa et al., 2018), and model distillation (Hinton et al., 2015). Many of these approaches, however, are not applicable for the problems addressed in this work, as they are either ingrained in the training procedure (and our server holds no data and performs no actual training) or are mostly optimized for inference. In the context of FL, we need something computationally light that can be efficiently applied in every round and that also allows for subsequent local training. We do note, however, that some of the previously mentioned approaches could potentially be leveraged at inference time in the federated setting, and exploring these directions would be an interesting avenue for further research.
+
+# 3 METHODS
+
+In this section, we present our proposed strategies for reducing Federated Learning’s (FL) serverto-client communication costs, namely lossy compression techniques (Section 3.1) and Federated Dropout (Section 3.2). We introduce the strategies separately, but they are fully compatible with one another (as we show in Section 4.4).
+
+# 3.1 LOSSY COMPRESSION
+
+Our first approach at reducing bandwidth usage consists of using lightweight lossy compression techniques that can be applied to an already trained model and that, when reversed (i.e. after decompression), maintain the model’s quality. The particular set of techniques we propose are inspired by those successfully used by Konecnˇ y et al. (2016b) to compress the client-to-server updates. We ap- ´ ply them, however, to the server-to-client exchanges, meaning we do not get the benefit of averaging the noisy decompressions over many updates.
+
+Our method works as follows: we reshape each to-be-compressed weight matrix in our model into a vector $\textbf { \em w }$ and (1) apply a basis transform to it. We then (2) subsample and (3) quantize the resulting vector and finally send it through the network. Once received, we simply execute the respective inverse transformations to finally obtain a noisy version of $\pmb { w }$ .
+
+Basis transform Previous work (Lyubarskii & Vershynin, 2010; Konecnˇ y et al., 2016b) has ex- ´ plored the idea of using a basis transform to reduce the error that will later be incurred by perturbations such as quantization. In particular, Konecnˇ y et al. (2016b) use the random Hadamard transform ´ to more evenly spread out a vector’s information among its dimensions. We go even further and also apply the classical results of Kashin (1977) to spread a vector’s information as much as possible in every dimension (Lyubarskii $\&$ Vershynin, 2010). Thus, Kashin’s representation mitigates the error incurred by subsequent quantization compared to using the random Hadamard transform. For a more detailed discussion, we refer the reader to Section A.3 in the Appendix.
+
+Subsampling For $s \in [ 0 , 1 )$ , we zero out a $1 - s$ fraction of the elements in each weight matrix, appropriately re-scaling the remaining values. The elements to zero out are picked uniformly at random. Thus, we only communicate the non-zero values and a random seed which allows recovery of the corresponding indices.
+
+(i) Original network, with a , ${ \sf b } _ { 2 ^ { \prime } }$ and ${ \mathsf { C } } _ { 3 }$ marked for dropout (ii) On-device network after Federated Dropout
+
+![](images/509a5f9d97f548d4b7f1844c109cda9e7aa6e67e540756f0f785fd692a5e0cd1.jpg)
+
+![](images/68b77fed3a5f430e2c1b733186023a45d7375fed0a3e875ae3bd3a6d27095687.jpg)  
+Figure 2: Federated Dropout applied to two fully-connected layers. Notices activation vectors $a , b =$ $\sigma ( U a )$ and $c = \sigma ( V b )$ in (I). In this example, we randomly select exactly one activation from each layer to drop, namely $a _ { 1 } , b _ { 2 }$ , and $c _ { 3 }$ , producing a sub-model with $2 \times 2$ dense matrices, as in (II).
+
+Probabilistic quantization For a vector $\pmb { w } = ( w _ { 1 } , \dots , w _ { n } )$ , let us denote $\begin{array} { r } { w _ { \operatorname* { m i n } } = \operatorname* { m i n } _ { j } \{ w _ { j } \} _ { j = 1 } ^ { n } } \end{array}$ and $w _ { \mathrm { m a x } } = \operatorname* { m a x } _ { j } \{ w _ { j } \} _ { j = 1 } ^ { n }$ . Uniform probabilistic 1-bit quantization replaces every element $w _ { i }$ by $w _ { \mathrm { m i n } }$ with probability $\frac { w _ { \mathrm { m a x } } - w _ { i } } { w _ { \mathrm { m a x } } - w _ { \mathrm { m i n } } }$ , and by $w _ { \mathrm { m a x } }$ otherwise. It is straightforward to verify this yields an unbiased estimate of $\pmb { w }$ . Now, for $q$ -bit uniform quantization, we first equally divide $[ w _ { \mathrm { m i n } } , w _ { \mathrm { m a x } } ]$ into $2 ^ { q }$ intervals. If $w _ { i }$ falls in the interval bounded by $w ^ { \prime }$ and $w ^ { \prime \prime }$ , the quantization operates by replacing $w _ { \mathrm { m i n } }$ and $w _ { \mathrm { m a x } }$ in step two of the above algorithm by $w ^ { \prime }$ and $w ^ { \prime \prime }$ , respectively.
+
+# 3.2 FEDERATED DROPOUT
+
+To further reduce communication costs, we propose an algorithm in which each client, instead of locally training an update to the whole global model, trains an update to a smaller sub-model. These sub-models are subsets of the global model and, as such, the computed local updates have a natural interpretation as updates to the larger global model. We call this technique Federated Dropout as it is inspired by the well known idea of dropout (Srivastava et al., 2014), albeit motivated primarily by systems-level concerns rather than as a strategy for regularization.
+
+In traditional dropout, hidden units are multiplied by a random binary mask in order to drop an expected fraction of neurons during each training pass through the network. Because the mask changes in each pass, each pass is effectively computing a gradient with respect to a different submodel. These sub-models can have different sizes (architectures) depending on how many neurons are dropped in each layer. Now, even though some units are dropped, in all implementations we are aware of, activations are still multiplied with the original weight matrices, they just have some useless rows and columns.
+
+To extend this idea to FL and realize communication and computation savings, we instead zero out a fixed number of activations at each fully-connected layer, so all possible sub-models have the same reduced architecture; see Figure 2. The server can map the necessary values into this reduced architecture, meaning only the necessary coefficients are transmitted to the client, re-packed as smaller dense matrices. The client (which may be fully unaware of the original model’s architecture) trains its sub-model and sends its update, which the server then maps back to the global model1. For convolutional layers, zeroing out activations would not realize any space savings, so we instead drop out a fixed percentage of filters.
+
+This technique brings two additional benefits beyond savings in server-to-client communication. First, the size of the client-to-server updates is also reduced. Second, the local training procedure now requires a smaller number of FLOPS per gradient evaluation, either because all matrixmultiplies are now of smaller dimensions (for fully-connected layers) or because less filters have to be applied (for convolutional ones). Thus, we reduce local computational costs.
+
+# 4 EXPERIMENTAL RESULTS
+
+In this section, we first present our experimental setup (Section 4.1) before presenting results for our lossy compression (Section 4.2) and Federated Dropout (Section 4.3) strategies. Finally, we
+
+Table 1: Summary of Datasets used in the experiments.   
+
+<table><tr><td>Dataset</td><td># of users</td><td>ID</td><td>Training samples per user mean</td><td>0</td><td>Test samples per user</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>mean</td><td>0</td></tr><tr><td>MNIST</td><td>100</td><td>Yes</td><td>600</td><td>0</td><td>100</td><td>0</td></tr><tr><td>CIFAR-10</td><td>100</td><td>Yes</td><td>500</td><td>0</td><td>100</td><td>0</td></tr><tr><td>EMNIST</td><td>3550</td><td>No</td><td>181.46</td><td>71.15</td><td>45.37</td><td>17.79</td></tr></table>
+
+show experiments that use both of these strategies in tandem with those proposed in Konecnˇ y et al.´ (2016b) to also compress client-to-server exchanges (Section 4.4).
+
+# 4.1 EXPERIMENTAL SETUP
+
+Optimization Algorithm We focus on testing our strategies against already established FL benchmarks. In particular, we restrict our experiments to the use of Federated Averaging (FedAvg) (McMahan et al., 2017).
+
+Datasets We use three datasets in our experiments: MNIST (LeCun et al., 1998), CIFAR10 (Krizhevsky & Hinton, 2009) and Extended MNIST or EMNIST (Cohen et al., 2017). The first two were used to benchmark the performance of FedAvg and of lossy compression for clientto-server updates (Konecnˇ y et al., 2016b). For these two datasets, we use the artificial IID partition ´ proposed by these previous works. Meanwhile, EMNIST is a dataset that has only recently been introduced as a useful benchmark for FL. Derived from the same source as MNIST, it also includes the identifier of the user that wrote the character (digit, lower or upper case letter), creating a natural and much more realistic partition of the data. Table 1 summarizes the basic dataset properties. Due to space constraints, we relegate the MNIST results to Appendix B, though all conclusions presented here also qualitatively hold for these experiments.
+
+Models For MNIST’s digit recognition task we use the same model as McMahan et al. (2017): a CNN with two $5 \mathrm { x } 5$ convolution layers (the first with 32 channels, the second with 64, each followed by $2 \mathbf { x } 2$ max pooling), a fully connected layer with 512 units and ReLu activation, and a final softmax output layer, for a total of more than $\mathrm { 1 0 ^ { 6 } }$ parameters. For CIFAR-10, we use the all convolutional model taken from what is described as “Model $\mathbf { { C } } ^ { \ast }$ in Springenberg et al. (2015), which also has a total of over $1 0 ^ { 6 }$ parameters. Finally, for EMNIST we use a variant of the MNIST model with 2048 units in the final fully connected layer. While none of these models is the state-of-the-art, they are sufficient for evaluating our methods, as we wish to measure accuracy degradation against a baseline and not to achieve the best possible accuracy on these tasks.
+
+Hyperparameters We do not optimize our experiments for FedAvg’s hyperparameters, always using those that proved to work reasonably well in our baseline setting which involves no compression and no Federated Dropout. For local training at each client we use static learning rates of 0.15 for MNIST, 0.05 for CIFAR-10 and 0.035 for EMNIST. We select 10 random clients per round for MNIST and CIFAR-10, and 35 for EMNIST. Finally, each selected client trains for one epoch per round using a batch size of 10.
+
+# 4.2 LOSSY COMPRESSION
+
+We focus on testing how the compression strategies presented in Section 3.1 impact the global model’s accuracy. Like Konecnˇ y et al. (2016b), we don’t compress all variables of our models. As ´ they mention, compressing smaller variables causes significant accuracy degradation but translates into minuscule communication savings. As such, we don’t compress biases for any of the models2.
+
+In our experiments, we vary three parameters:
+
+1. The type of basis transform applied: no transform or identity (I), randomized Hadamard transform (HD) and Kashin’s representation (K).
+
+![](images/4e198c0b75739883172b561a02f9a11936f1fe7be1f337102e3e6aec37d9a0af.jpg)  
+Figure 3: Effect of varying our lossy compression parameters on CIFAR-10 and EMNIST.
+
+2. The subsampling rate $s$ , which refers to the fraction of weights that are kept (i.e. $1 - s$ of the weights are zeroed out).
+
+3. The number of quantization bits $q$
+
+Figure 3 shows the effect of varying these parameters for CIFAR-10 and EMNIST. We repeat each experiment 10 times and report the mean accuracy among these repetitions. The three main takeaways from these experiments are: (1) for every model, we are able find a setting of compression parameters that at the very least matches our baseline; (2) Kashin’s representation proves to be most useful for aggressive quantization values; and (3) it appears that subsampling is not all that helpful in the server-to-client setting. We proceed to give more details about these highlights.
+
+The first takeaway is that, for every model, we are indeed able find a setting of compression parameters that matches or, in some cases, slightly outperforms our baseline. In particular, we are able to quantize every model to 4 bits, which translates to a reduction in communication of nearly $8 \times$ .
+
+The second takeaway is that Kashin’s representation proves to be most useful for aggressive quantization values, i.e. for low values of $q$ . In our experiments, gains were observed only in regimes where the overall accuracy had already degraded, but we hypothesize that the use of Kashin’s representation may provide clearer benefits in the compression of client-to-server gradient updates, where more aggressive quantization is admissible. We also highlight that using Kashin’s representation may be beneficial for other datasets. Indeed, its computational costs are comparable to that of the random Hadamard transform while also providing better theoretical error rates (see Section A.1). We refer the reader to Section A.3 in the Appendix, where we show preliminary results that demonstrate Kashin’s potential to dominate over the randomized Hadamard transform in compressing fully-trained models, particularly for small values of $q$ .
+
+Finally, it appears that subsampling is not all that helpful in this server-to-client setting. This contrasts with the results presented by Konecnˇ y et al. (2016b) for compressed client-to-server updates, ´ where aggressive values of s were admissible. This trend extends to the other compression parameters: server-to-client compression of global models requires much more conservative settings than client-to-server compression of model updates. For example, for CIFAR-10, Konecnˇ y et al. ´ (2016b) get away with using $s = 0 . 2 5$ and $q = 8$ under a random Hadamard transform representation3. Meanwhile, in Figure 3 we can see that, for the same $q$ and representation, $s = 0 . 5$ already causes an unacceptable degradation of the accuracy. This is not surprising, since it is expected that the updates’ error will cancel out once several of them get aggregated at the server, which is not true for model downloads.
+
+![](images/b8cee57370ffcda99e8a75f325bfa49d39c0015a0e8228e901acbf70aef7b22e.jpg)  
+Figure 4: Results for Federated Dropout, varying the percentage of neurons kept in each layer.
+
+# 4.3 FEDERATED DROPOUT
+
+We focus on testing how the global model’s accuracy deteriorates once we use the strategy proposed in Section 3.2. In these experiments, we vary the percentage of neurons (or filters for the case of convolutional layers) that are kept on each layer of our models (we call this the federated dropout rate). We always keep the totality of the input and logits layers, and never drop the neuron that can be associated to the bias term.
+
+Figure 4 shows how the convergence of our three models behaves under different federated dropout rates. We repeat each experiment 10 times and report the mean among these repetitions. The main takeaway from these experiments is that, for every model, it is possible to find a federated dropout rate less than 1.0 that matches or, in some cases, even improves on the final accuracy of the model.
+
+A federated dropout rate of 0.75 seems to work across the board. This corresponds to dropping $2 5 \%$ of the rows and columns of the weight matrices of fully-connected layers (which translates to $\smash { 1 \sim 4 3 \% }$ reduction in size), and to dropping the same percentage of filters of each convolutional layer. Now, because fully connected layers correspond to most of the parameters of the MNIST and EMNIST models, the $\sim 4 3 \%$ reduction will apply to them both in terms of the amount of data that has to be communicated and of the number of FLOPS required for local training. Meanwhile, because our CIFAR model is fully convolutional, gains will be of $2 5 \%$ .
+
+As a final comment, we note that more aggressive federated dropout rates tend to slow down the convergence rate of the model, even if they sometimes result in a higher accuracy.
+
+# 4.4 REDUCING THE OVERALL COMMUNICATION COST
+
+Our final set of experiments shows how our models behave once we combine our two strategies, lossy compression and Federated Dropout, with existing client-to-server compression schemes (Konecnˇ y et al., 2016b), in order to explore how the different components of this end- ´ to-end, communication efficient framework interact. To do this, we evaluate how our models behave under 3 different compression schemes (aggressive, moderate and conservative) and 4 different federated dropout rates (0.5, 0.625, 0.75 and 0.875). The values for these schemes and rates were picked based on the observed behavior during the previous experiments, being somewhat more conservative as we are now combining different sources of noise. Table 2 describes the settings for each scheme.
+
+Figure 5 shows how our CIFAR-10 and EMNIST models behave under each of the previously mentioned conditions. We repeat each experiment 5 times and report the mean among these repetitions. For all three models, a federated dropout rate of 0.75 resulted in models with no accuracy degradation under all compression schemes except for the most aggressive. For MNIST and EMNIST, this translates into server-to-client communication savings of $1 4 \times$ , client-to-server savings of $2 8 \times$ and a reduction of $1 . 7 \times$ in local computation, all without degrading the accuracy of the final global model (and sometimes even improving it). For CIFAR-10, we provide server-to-client communication savings of $1 0 \times$ , client-to-server savings of $2 1 \times$ and local computation savings of $1 . 3 \times$ .
+
+Based on these results, we also hypothesize that a federated dropout rate of 0.75 combined with a moderate or conservative compression scheme will be a good starting point when setting these parameters in practice.
+
+Table 2: Settings for each of our proposed compression schemes.   
+
+<table><tr><td>Scheme</td><td colspan="3">Client-to-Server</td><td colspan="3">Server-to-Client</td></tr><tr><td></td><td>transf.</td><td>S</td><td>q</td><td>transf.</td><td>S</td><td>q</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Aggressive</td><td>Kashin&#x27;s</td><td>0.4</td><td>2</td><td>Kashin&#x27;s</td><td>1.0</td><td>3</td></tr><tr><td>Moderate</td><td>Kashin&#x27;s</td><td>0.5</td><td>4</td><td>Kashin&#x27;s</td><td>1.0</td><td>5</td></tr><tr><td>Conservative</td><td>Kashin&#x27;s</td><td>1.0</td><td>8</td><td>Kashin&#x27;s</td><td>1.0</td><td>8</td></tr></table>
+
+![](images/6f647972ab35ad2d0bac0fe55163b5e6c42b4362445a0a8fa5dba0fb4a5fdebd.jpg)  
+Figure 5: Effect of using both compression and Federated Dropout on CIFAR-10 and EMNIST.
+
+# 5 CONCLUSIONS AND OPEN QUESTIONS
+
+The ecosystem currently targeted by Federated Learning (FL) is marked by heterogeneous edge networks that can potentially be orders of magnitude slower than the ones in datacenters. At the same time, FL can be quite demanding in terms of bandwidth, particularly when used to train deep models. We are thus at risk of either restricting the type of models we are able to train using this technique, or of excluding large groups of users from federated training. Both issues are problematic, but because access to high-end networks also appears to be correlated to sensitive factors such as income and age (Anzilotti, 2016; Pew Research Center, 2018), the latter may have implications related to fairness, making it particularly sensitive as we continue the adoption of FL systems.
+
+Our work dramatically reduces the communication overheads in FL by (1) using lossy compression techniques on the server-to-client exchanges and by (2) using Federated Dropout, a technique that only communicates subsets of the global model to each client. We empirically show that a combination of our strategies with previous work allows for up to a $1 4 \times$ reduction in server-to-client communication, a $1 . 7 \times$ reduction in local computation and a $2 8 \times$ reduction in client-to-server communication.
+
+In future work, we plan to: explore the efficacy of introducing a server step size in order to account for the use of different sub-models in Federated Dropout; investigate the possibility of using the same sub-models for all the selected clients in one round; and further characterize the benefits of Kashin’s representation in compressing the gradient updates in FL and in traditional model serving. An additional future direction to pursue related to fairness involves studying the effect of adaptively using these strategies (i.e. using more aggressive compression and federated dropout rates for some users) to prevent unfairly biased models. Finally, we note that the success of Federated Dropout suggests an entirely new avenue of research in which smaller, perhaps personalized, sub-models are eventually aggregated into a larger, more complex model that can be managed by the server. Contrary to the classic datacenter setting, the computational overhead associated with first creating and then aggregating the sub-models is justified in FL.
+
+REFERENCES   
+Eillie Anzilotti. Visualizing the state of global internet connectivity, Aug 2016. URL https://www.citylab.com/life/2016/08/ visualizing-the-state-of-global-internet-connectivity/496328/.   
+Emmanuel J Candes, Justin K Romberg, and Terence Tao. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 59(8):1207–1223, 2006.   
+Gregory Cohen, Saeed Afshar, Jonathan Tapson, and Andre van Schaik. EMNIST: an extension of ´ MNIST to handwritten letters. arXiv preprint arXiv:1702.05373, 2017.   
+Christopher De Sa, Megan Leszczynski, Jian Zhang, Alana Marzoev, Christopher R Aberger, Kunle Olukotun, and Christopher Re. High-accuracy low-precision training. ´ arXiv preprint arXiv:1803.03383, 2018.   
+Jeffrey Dean, Greg Corrado, Rajat Monga, Kai Chen, Matthieu Devin, Mark Mao, Andrew Senior, Paul Tucker, Ke Yang, Quoc V Le, et al. Large scale distributed deep networks. In Advances in neural information processing systems, pp. 1223–1231, 2012.   
+Simon Foucart and Holger Rauhut. A mathematical introduction to compressive sensing, volume 1. Birkhauser Basel, 2013. ¨   
+Song Han, Jeff Pool, John Tran, and William J Dally. Learning both weights and connections for efficient neural network. In Advances in neural information processing systems, pp. 1135–1143, 2015.   
+Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. International Conference on Learning Representations, 2016.   
+Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.   
+Itay Hubara, Matthieu Courbariaux, Daniel Soudry, Ran El-Yaniv, and Yoshua Bengio. Binarized neural networks. In Advances in neural information processing systems, pp. 4107–4115, 2016.   
+Boris Sergeevich Kashin. Diameters of some finite-dimensional sets and classes of smooth functions. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 41(2):334–351, 1977.   
+Jakub Konecnˇ y and Peter Richt ´ arik. Randomized distributed mean estimation: Accuracy vs com- ´ munication. arXiv preprint arXiv:1611.07555, 2016.   
+Jakub Konecnˇ y, H Brendan McMahan, Daniel Ramage, and Peter Richt ´ arik. Federated optimization: ´ Distributed machine learning for on-device intelligence. arXiv preprint arXiv:1610.02527, 2016a.   
+Jakub Konecnˇ y, H Brendan McMahan, Felix X Yu, Peter Richt ´ arik, Ananda Theertha Suresh, and ´ Dave Bacon. Federated learning: Strategies for improving communication efficiency. arXiv preprint arXiv:1610.05492, 2016b.   
+Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, Citeseer, 2009.   
+Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to ´ document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.   
+Xiaofan Lin, Cong Zhao, and Wei Pan. Towards accurate binary convolutional neural network. In Advances in Neural Information Processing Systems, pp. 345–353, 2017.   
+Yurii Lyubarskii and Roman Vershynin. Uncertainty principles and vector quantization. IEEE Transactions on Information Theory, 56(7):3491–3501, 2010.
+
+H Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Arcas. Communication-efficient learning of deep networks from decentralized data. pp. 1273–1282, 2017.
+
+Pew Research Center. Mobile fact sheet, Feb 2018. URL http://www.pewinternet.org/ fact-sheet/mobile/.
+
+Sujith Ravi. Custom On-Device ML Models with Learn2Compress, May 2018. URL https: //ai.googleblog.com/2018/05/custom-on-device-ml-models.html.
+
+Sashank J Reddi, Jakub Konecnˇ y, Peter Richt ´ arik, Barnab ´ as P ´ ocz ´ os, and Alex Smola. Aide: fast ´ and communication efficient distributed optimization. arXiv preprint arXiv:1608.06879, 2016.
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+Virginia Smith, Chao-Kai Chiang, Maziar Sanjabi, and Ameet S Talwalkar. Federated multi-task learning. In Advances in Neural Information Processing Systems, pp. 4424–4434, 2017.
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+Virginia Smith, Simone Forte, Ma Chenxin, Martin Taka´c, Michael I Jordan, and Martin Jaggi. ˇ Cocoa: A general framework for communication-efficient distributed optimization. Journal of Machine Learning Research, 18:230, 2018.
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+Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research, 15(1):1929–1958, 2014.
+
+Ananda Theertha Suresh, Felix X Yu, Sanjiv Kumar, and H Brendan McMahan. Distributed mean estimation with limited communication. In International Conference on Machine Learning, pp. 3329–3337, 2017.
+
+Hongyi Wang, Scott Sievert, Zachary Charles, Dimitris Papailiopoulos, and Stephen Wright. ATOMO: Communication-efficient Learning via Atomic Sparsification. arXiv preprint arXiv:1806.04090, 2018.
+
+# A KASHIN’S REPRESENTATION
+
+For reasons of space, we have relegated a more detailed discussion of Kashin’s representation (see Section 3) to the Appendix. In this section, we briefly discuss Kashin’s representation both from a theoretical (Section A.1) and practical (Section A.2) standpoints. Finally, we present some preliminary results that argue the potential of Kashin’s representation to dominate over the random Hadamard transform with respect to the size vs. accuracy trade-off (Section A.3).
+
+# A.1 THEORETICAL OVERVIEW
+
+The idea of using the classical results of Kashin (1977) to increase the robustness of coefficients to perturbations was first introduced by Lyubarskii & Vershynin (2010). Their result states that, given a tight frame satisfying a form of uncertainty principle, a weaker notion of the RIP (Candes et al., 2006), it is possible to convert the frame representation of every vector into the more robust Kashin’s representation, whose coefficients will have the smallest possible dynamic range.
+
+Error rates Since the results of Suresh et al. (2017) (who quantified the reduction in quantization error due to the Hadamard transform) rely on exactly this notion of dynamic range, and assuming the subsampled randomized Hadamard transform satisfies the uncertainty principle, Theorem 3.5 of Lyubarskii & Vershynin (2010) can be directly used as a drop-in replacement for Lemma 7 in Suresh et al. (2017), removing the logarithmic dependence on dimension from Theorem 3 therein, matching the lower bounds. We do not provide the complete proof as, beyond drawing this connection, it does not imply any novelty whatsoever. However, an open question remains, as we are not aware of a result showing what are the parameters of the uncertainty principle guaranteed by the subsampled randomized Hadamard transform. They exist however, as the transform is known to satisfy the RIP (Foucart & Rauhut, 2013), which is a stronger notion.
+
+# A.2 PRACTICAL CONSIDERATIONS
+
+In practice, given a tight frame, the algorithm for computing Kashin’s representation is straightforward. It runs for $n$ iterations, and takes parameters $\eta , \delta$ as input. In a single iteration, one first computes the frame coefficients, projects them onto a $L _ { \infty }$ ball, and reconstructs the error in the original domain. Another iteration proceeds starting with the reconstructed error and a smaller ball. We refer the reader to Lyubarskii & Vershynin (2010) for more details regarding $\eta , \delta$ and their relationship with the uncertainty principle.
+
+In our work, we use the randomized Hadamard transform as the initial tight frame (see Section A.1 for details on why this is possible). We also run the algorithm for just $n = 2$ iterations (as very often this provides most of the benefit), fixed $\delta = 1$ , and used a variant of the algorithm which yields an exact representation (omitting the $L _ { \infty }$ projection in the last iteration). Given this, the choice of $\eta$ is irrelevant. The dominant part of the computation is then three applications of the fast Walsh-Hadamard transform, as opposed to a single one in Konecnˇ y et al. (2016b)). ´
+
+As a particular example, say we are to compress an 80-dimensional vector. We first pad the vector with zeros, so that its dimension is 128 (the closest larger power of 2). Then, we multiply the vector by a diagonal matrix with independent Rademacher random variables $( D \in \mathbb { R } ^ { 1 2 8 \times 1 2 8 }$ ), followed by the application of the fast Walsh-Hadamard transform $( H \in \mathbb { R } ^ { 1 2 8 \times 1 2 8 } ,$ ). The first 80 columns of the matrix $H D$ correspond to the tight frame used to find the Kashin’s representation. Nonetheless, we avoid representing this explicitly.
+
+Finally, note that, if the initial dimension was a power of 2, we need to pad zeros to the next power of 2 in order to realize any benefit over just using the Hadamard transform.
+
+# A.3 DOMINANCE OVER HADAMARD
+
+Given the theoretical properties of Kashin’s representation, we hypothesize it should dominate the random Hadamard transform when it comes to the size vs. accuracy trade-off. A preliminary experiment to corroborate this hypothesis is the following:
+
+1. We train an MNIST model until we get an accuracy of around $9 9 . 3 \%$
+
+2. We compress the original model using some linear transform, some subsampling ratio and some number of quantization bits.
+
+3. We decompress the model and evaluate both its new accuracy and its $L _ { 2 }$ distance to the original model.
+
+4. We repeat the previous two steps for different linear transforms (identity, random Hadamard transform and Kashin’s representation), subsampling ratios (0.25, 0.5 and 1.0) and quantization bits (1, 2, 4, 8, 16).
+
+An important detail is that, whenever we use Kashin’s representation, we do a grid search over the best values for $n$ (from 1 to 10) and $\eta$ . However, $\delta$ is kept fixed as 1.
+
+The results of this experiment are shown in Figure 6. In the legend, R corresponds to rotation — I for identity, HD for randomized Hadamard, Kashin for Kashin based on the randomized Hadamard; and SR corresponds to subsampling ratio — the fraction of elements to be kept non-zero. In the top row, the figure shows the relationship of the accuracy of the compressed model vs. the number of bits used for quantization, and vs. the model’s size (in MB). In the bottom row, the $L _ { 2 }$ error incurred is plotted against the same. It is very clear then that Kashin’s representation does dominate the other two representations when it comes to the size vs. accuracy trade-off, making up the Pareto frontier for all combinations of subsampling ratio and quantization bits. Nevertheless, we did optimize over the parameters associated with Kashin’s algorithm, something that does not need to be done for the random Hadamard transform. In Section A.2, we propose a set of values that worked well enough for our experiments, but further exploration on how to easily determine these values is in order.
+
+![](images/abf34396d464b2bb8d7b9fa4e35ae23bba50f9800a6aa216d47354b141f7e471.jpg)  
+Figure 6: Compressing an already trained MNIST model with linear transform $^ +$ subsampling $^ +$ uniform quantization.
+
+# B MNIST EXPERIMENTAL RESULTS
+
+For reasons of space, we have relegated the experimental results using MNIST (see Section 4) (Section A) to the Appendix.
+
+Figure 7 shows the results of using our lossy compression on MNIST under the experimental setup presented in Section 4.2. Meanwhile, Figure 8 shows the results of using Federated Dropout (see Section 4.3 for details). Finally, Figure 9 shows the results of performing both lossy compression for downloads and uploads, as well as Federated Dropout, as described in Section 4.4.
+
+![](images/76b0d59f01e9eb0328dc020065b77b91433f5eec90e30a1e84877121f28377b3.jpg)  
+Figure 7: Effect of varying our lossy compression parameters on the convergence MNIST.
+
+![](images/09ea07f075bbf2194d3e9f2122c5ec2706362418c4a9d51d750b66d8b33eca43.jpg)  
+Figure 8: Effect of varying the percentage of neurons kept in each layer on MNIST.
+
+![](images/76fe8639e916dacc9a7c5c53cddd74600a52a6c614b2572b179f9d9a8004aa1c.jpg)  
+Figure 9: Effect of using both lossy compression and Federated Dropout on MNIST.

parse/train/SJlpM3RqKQ/SJlpM3RqKQ_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "EXPANDING THE REACH OF FEDERATED LEARNING BY REDUCING CLIENT RESOURCE REQUIREMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            172,
+            398,
+            200
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            237,
+            544,
+            252
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Communication on heterogeneous edge networks is a fundamental bottleneck in Federated Learning (FL), restricting both model capacity and user participation. To address this issue, we introduce two novel strategies to reduce communication costs: (1) the use of lossy compression on the global model sent serverto-client; and (2) Federated Dropout, which allows users to efficiently train locally on smaller subsets of the global model and also provides a reduction in both client-to-server communication and local computation. We empirically show that these strategies, combined with existing compression approaches for client-toserver communication, collectively provide up to a $1 4 \\times$ reduction in server-toclient communication, a $1 . 7 \\times$ reduction in local computation, and a $2 8 \\times$ reduction in upload communication, all without degrading the quality of the final model. We thus comprehensively reduce FL’s impact on client device resources, allowing higher capacity models to be trained, and a more diverse set of users to be reached. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            472,
+            336,
+            487
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Federated Learning (FL) allows users to reap the benefits of models trained from rich yet sensitive data captured by their mobile devices, without the need to centrally store such data (McMahan et al., 2017; Konecnˇ y et al., 2016a; Smith et al., 2017). Under the FL paradigm, each device performs ´ training on samples available locally and only communicates intermediate model updates. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Network speed and number of nodes are two of the core systems aspects that differentiate FL from traditional distributed learning in data centers, with network bandwidth being potentially orders of magnitude slower and the number of worker nodes orders of magnitude larger. Together, these issues exacerbate the communication bottlenecks usually associated with distributed learning, increasing both the number of stragglers and the probability of devices dropping out altogether. The problem is further aggravated when working with high capacity models with large numbers of parameters. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Insisting on training these large models using existing federated optimization methods can lead to the systematic exclusion of clients with restricted bandwidth or limited network access from the training stage, and thus to a degraded user experience once these models are served. One naive solution involves training low capacity models with smaller communication footprints, at the expense of model accuracy. As a middle ground, we could develop strategies to reduce the communication footprint of larger, high-capacity models. Recent work (Konecnˇ y et al., 2016b) has in fact taken ´ this approach, but only in the context of client-to-server FL communication. Their success with lossy compression strategies is perhaps not surprising, as the clients’ lossy, yet unbiased, updates are eventually averaged over many users. However, server-to-client exchanges do not benefit from such averaging. As such, they remain a main bottleneck in our goal of expanding FL’s reach. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose two novel strategies to mitigate the server-to-client communication footprint, and empirically demonstrate their efficacy and seamless integration with existing client-toserver strategies. The specific contributions of this paper are as follows: ",
+        "bbox": [
+            174,
+            801,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. We study lossily compressing the models downloaded by the clients, thus addressing the open question as to whether these approaches are amenable in the context of server-to-client exchanges. We also introduce the use of the theoretically motivated Kashin’s representation to reduce the error associated with the lossy compression (Lyubarskii & Vershynin, 2010; Kashin, 1977). ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "image",
+        "img_path": "images/09f2c19b96d2be5f52ed90bc55800a74b6fd0ad93b78c70fc4916cc96768b8a1.jpg",
+        "image_caption": [
+            "Figure 1: Combination of our proposed strategies during FL training. We reduce the size of the model to be communicated by (1) constructing a sub-model via Federated Dropout, and by (2) lossily compressing the resulting object. This compressed model is then sent to the client, who (3) decompresses and trains it using local data, and (4) compresses the final update. This update is sent back to the server, where it is (5) decompressed and finally, (6) aggregated into the global model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            248,
+            104,
+            754,
+            246
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. We introduce Federated Dropout, a technique that builds upon the popular idea of dropout (Srivastava et al., 2014), yet is primarily motivated by systems-related concerns. Our approach enables each device to locally operate on a smaller sub-model (i.e. with smaller weight matrices) while still providing updates that can be applied to the larger global model on the server. It thus reduces communication costs by allowing for these smaller sub-models to be exchanged between server and clients, while also reducing the computational cost of local training. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3. We empirically show that not only are these approaches compatible with one another, but with existing client-to-server compression. Combining these approaches during FL training (see Figure 1) reduces the size of the downloaded models up to $1 4 \\times$ , the size of the corresponding updates up to $2 8 \\times$ , and the required local computations by up to $1 . 7 \\times$ , all without degrading the model’s accuracy and only at the expense of a slightly slower convergence rate (in terms of number of communication rounds). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            570,
+            344,
+            585
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We review the relevant related work given our objective of reducing the communication footprint in server-to-client exchanges in Federated Learning (FL). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Federated Learning Federated Learning (FL) is a technique that aims to learn a global model over data distributed across multiple edge devices (usually mobile phones) without the data ever leaving the device on which it was generated (McMahan et al., 2017). It brings along a set of statistical (nonIID, unbalanced data) and systems (stragglers, communication bottlenecks, etc.) challenges which differentiate it from traditional distributed learning in the data center, and which have been tackled by several works. For instance, McMahan et al. (2017) propose Federated Averaging (FedAvg), which in its canonical form works by (1) sending the global model to a subset of the available devices, (2) training the model on each device using the available local data, and (3) averaging the local updates to thus end a round of training. In contrast Smith et al. (2017) present a multi-task variant that also models the relationship between clients in order to learn personalized yet related models for each device. Nonetheless, all approaches we are aware of (including the two aforementioned ones) require continued exchanges between a central server and its clients across a potentially slow network. ",
+        "bbox": [
+            173,
+            638,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Communication-efficient distributed learning Distributed learning is known to suffer from communication overheads associated with the frequent gradient updates exchanged among nodes (Wang et al., 2018; Dean et al., 2012; Smith et al., 2018; Reddi et al., 2016). To reduce these bottlenecks, recent studies focus on communicating a sparsified, quantized or randomly subsampled version of the updates. Although these operations introduce noise, they have been shown both empirically and theoretically to maintain the quality of the trained models. We refer the reader to the introduction of Wang et al. (2018) for more details and references. ",
+        "bbox": [
+            173,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the context of FL, Konecnˇ y et al. (2016b) successfully perform lossy compression on the client- ´ to-server exchanges (i.e. the model updates). Of particular interest is their use of the randomized Hadamard transform to reduce the error incurred by the subsequent quantization. This is due to the fact that the Hadamard transform, in expectation, spreads a vector’s information more evenly across its components (Suresh et al., 2017; Konecnˇ y & Richt ´ arik, 2016). ´ ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We note, however, that neither the work on traditional distributed learning nor the work of Konecnˇ y´ et al. (2016b) considers compressing the server-to-client exchanges. Nevertheless, in FL, downloading a large model can still be a considerable burden for users, particularly for those in regions with network constraints. Furthermore, as FL is expected to deal with a large number of devices, communicating the global model may even become a bottleneck for the server (as it would, ideally, send the model to the clients in parallel). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Model compression Deep models tend to demand significant computational resources both for training and inference. Using them on edge devices is therefore not a straightforward task. Because of this, several recent works have proposed compressing the models before deploying them on-device (Ravi, 2018). Popular alternatives include pruning the least useful connections in a network (Han et al., 2016; 2015), weight quantization (Hubara et al., 2016; Lin et al., 2017; De Sa et al., 2018), and model distillation (Hinton et al., 2015). Many of these approaches, however, are not applicable for the problems addressed in this work, as they are either ingrained in the training procedure (and our server holds no data and performs no actual training) or are mostly optimized for inference. In the context of FL, we need something computationally light that can be efficiently applied in every round and that also allows for subsequent local training. We do note, however, that some of the previously mentioned approaches could potentially be leveraged at inference time in the federated setting, and exploring these directions would be an interesting avenue for further research. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3 METHODS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            460,
+            290,
+            477
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present our proposed strategies for reducing Federated Learning’s (FL) serverto-client communication costs, namely lossy compression techniques (Section 3.1) and Federated Dropout (Section 3.2). We introduce the strategies separately, but they are fully compatible with one another (as we show in Section 4.4). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 LOSSY COMPRESSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            569,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our first approach at reducing bandwidth usage consists of using lightweight lossy compression techniques that can be applied to an already trained model and that, when reversed (i.e. after decompression), maintain the model’s quality. The particular set of techniques we propose are inspired by those successfully used by Konecnˇ y et al. (2016b) to compress the client-to-server updates. We ap- ´ ply them, however, to the server-to-client exchanges, meaning we do not get the benefit of averaging the noisy decompressions over many updates. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our method works as follows: we reshape each to-be-compressed weight matrix in our model into a vector $\\textbf { \\em w }$ and (1) apply a basis transform to it. We then (2) subsample and (3) quantize the resulting vector and finally send it through the network. Once received, we simply execute the respective inverse transformations to finally obtain a noisy version of $\\pmb { w }$ . ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Basis transform Previous work (Lyubarskii & Vershynin, 2010; Konecnˇ y et al., 2016b) has ex- ´ plored the idea of using a basis transform to reduce the error that will later be incurred by perturbations such as quantization. In particular, Konecnˇ y et al. (2016b) use the random Hadamard transform ´ to more evenly spread out a vector’s information among its dimensions. We go even further and also apply the classical results of Kashin (1977) to spread a vector’s information as much as possible in every dimension (Lyubarskii $\\&$ Vershynin, 2010). Thus, Kashin’s representation mitigates the error incurred by subsequent quantization compared to using the random Hadamard transform. For a more detailed discussion, we refer the reader to Section A.3 in the Appendix. ",
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+    {
+        "type": "text",
+        "text": "Subsampling For $s \\in [ 0 , 1 )$ , we zero out a $1 - s$ fraction of the elements in each weight matrix, appropriately re-scaling the remaining values. The elements to zero out are picked uniformly at random. Thus, we only communicate the non-zero values and a random seed which allows recovery of the corresponding indices. ",
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+    {
+        "type": "text",
+        "text": "(i) Original network, with a , ${ \\sf b } _ { 2 ^ { \\prime } }$ and ${ \\mathsf { C } } _ { 3 }$ marked for dropout (ii) On-device network after Federated Dropout ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "image",
+        "img_path": "images/509a5f9d97f548d4b7f1844c109cda9e7aa6e67e540756f0f785fd692a5e0cd1.jpg",
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+    {
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+        "image_caption": [
+            "Figure 2: Federated Dropout applied to two fully-connected layers. Notices activation vectors $a , b =$ $\\sigma ( U a )$ and $c = \\sigma ( V b )$ in (I). In this example, we randomly select exactly one activation from each layer to drop, namely $a _ { 1 } , b _ { 2 }$ , and $c _ { 3 }$ , producing a sub-model with $2 \\times 2$ dense matrices, as in (II). "
+        ],
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+    {
+        "type": "text",
+        "text": "Probabilistic quantization For a vector $\\pmb { w } = ( w _ { 1 } , \\dots , w _ { n } )$ , let us denote $\\begin{array} { r } { w _ { \\operatorname* { m i n } } = \\operatorname* { m i n } _ { j } \\{ w _ { j } \\} _ { j = 1 } ^ { n } } \\end{array}$ and $w _ { \\mathrm { m a x } } = \\operatorname* { m a x } _ { j } \\{ w _ { j } \\} _ { j = 1 } ^ { n }$ . Uniform probabilistic 1-bit quantization replaces every element $w _ { i }$ by $w _ { \\mathrm { m i n } }$ with probability $\\frac { w _ { \\mathrm { m a x } } - w _ { i } } { w _ { \\mathrm { m a x } } - w _ { \\mathrm { m i n } } }$ , and by $w _ { \\mathrm { m a x } }$ otherwise. It is straightforward to verify this yields an unbiased estimate of $\\pmb { w }$ . Now, for $q$ -bit uniform quantization, we first equally divide $[ w _ { \\mathrm { m i n } } , w _ { \\mathrm { m a x } } ]$ into $2 ^ { q }$ intervals. If $w _ { i }$ falls in the interval bounded by $w ^ { \\prime }$ and $w ^ { \\prime \\prime }$ , the quantization operates by replacing $w _ { \\mathrm { m i n } }$ and $w _ { \\mathrm { m a x } }$ in step two of the above algorithm by $w ^ { \\prime }$ and $w ^ { \\prime \\prime }$ , respectively. ",
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+    {
+        "type": "text",
+        "text": "3.2 FEDERATED DROPOUT ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To further reduce communication costs, we propose an algorithm in which each client, instead of locally training an update to the whole global model, trains an update to a smaller sub-model. These sub-models are subsets of the global model and, as such, the computed local updates have a natural interpretation as updates to the larger global model. We call this technique Federated Dropout as it is inspired by the well known idea of dropout (Srivastava et al., 2014), albeit motivated primarily by systems-level concerns rather than as a strategy for regularization. ",
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+    {
+        "type": "text",
+        "text": "In traditional dropout, hidden units are multiplied by a random binary mask in order to drop an expected fraction of neurons during each training pass through the network. Because the mask changes in each pass, each pass is effectively computing a gradient with respect to a different submodel. These sub-models can have different sizes (architectures) depending on how many neurons are dropped in each layer. Now, even though some units are dropped, in all implementations we are aware of, activations are still multiplied with the original weight matrices, they just have some useless rows and columns. ",
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+    {
+        "type": "text",
+        "text": "To extend this idea to FL and realize communication and computation savings, we instead zero out a fixed number of activations at each fully-connected layer, so all possible sub-models have the same reduced architecture; see Figure 2. The server can map the necessary values into this reduced architecture, meaning only the necessary coefficients are transmitted to the client, re-packed as smaller dense matrices. The client (which may be fully unaware of the original model’s architecture) trains its sub-model and sends its update, which the server then maps back to the global model1. For convolutional layers, zeroing out activations would not realize any space savings, so we instead drop out a fixed percentage of filters. ",
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+    {
+        "type": "text",
+        "text": "This technique brings two additional benefits beyond savings in server-to-client communication. First, the size of the client-to-server updates is also reduced. Second, the local training procedure now requires a smaller number of FLOPS per gradient evaluation, either because all matrixmultiplies are now of smaller dimensions (for fully-connected layers) or because less filters have to be applied (for convolutional ones). Thus, we reduce local computational costs. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we first present our experimental setup (Section 4.1) before presenting results for our lossy compression (Section 4.2) and Federated Dropout (Section 4.3) strategies. Finally, we ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7fab6f6b1b6426028fb4ac2120eba03ea0f7f0869de40e84c17928908754b11a.jpg",
+        "table_caption": [
+            "Table 1: Summary of Datasets used in the experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td># of users</td><td>ID</td><td>Training samples per user mean</td><td>0</td><td>Test samples per user</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>mean</td><td>0</td></tr><tr><td>MNIST</td><td>100</td><td>Yes</td><td>600</td><td>0</td><td>100</td><td>0</td></tr><tr><td>CIFAR-10</td><td>100</td><td>Yes</td><td>500</td><td>0</td><td>100</td><td>0</td></tr><tr><td>EMNIST</td><td>3550</td><td>No</td><td>181.46</td><td>71.15</td><td>45.37</td><td>17.79</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "show experiments that use both of these strategies in tandem with those proposed in Konecnˇ y et al.´ (2016b) to also compress client-to-server exchanges (Section 4.4). ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Optimization Algorithm We focus on testing our strategies against already established FL benchmarks. In particular, we restrict our experiments to the use of Federated Averaging (FedAvg) (McMahan et al., 2017). ",
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+    {
+        "type": "text",
+        "text": "Datasets We use three datasets in our experiments: MNIST (LeCun et al., 1998), CIFAR10 (Krizhevsky & Hinton, 2009) and Extended MNIST or EMNIST (Cohen et al., 2017). The first two were used to benchmark the performance of FedAvg and of lossy compression for clientto-server updates (Konecnˇ y et al., 2016b). For these two datasets, we use the artificial IID partition ´ proposed by these previous works. Meanwhile, EMNIST is a dataset that has only recently been introduced as a useful benchmark for FL. Derived from the same source as MNIST, it also includes the identifier of the user that wrote the character (digit, lower or upper case letter), creating a natural and much more realistic partition of the data. Table 1 summarizes the basic dataset properties. Due to space constraints, we relegate the MNIST results to Appendix B, though all conclusions presented here also qualitatively hold for these experiments. ",
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+    {
+        "type": "text",
+        "text": "Models For MNIST’s digit recognition task we use the same model as McMahan et al. (2017): a CNN with two $5 \\mathrm { x } 5$ convolution layers (the first with 32 channels, the second with 64, each followed by $2 \\mathbf { x } 2$ max pooling), a fully connected layer with 512 units and ReLu activation, and a final softmax output layer, for a total of more than $\\mathrm { 1 0 ^ { 6 } }$ parameters. For CIFAR-10, we use the all convolutional model taken from what is described as “Model $\\mathbf { { C } } ^ { \\ast }$ in Springenberg et al. (2015), which also has a total of over $1 0 ^ { 6 }$ parameters. Finally, for EMNIST we use a variant of the MNIST model with 2048 units in the final fully connected layer. While none of these models is the state-of-the-art, they are sufficient for evaluating our methods, as we wish to measure accuracy degradation against a baseline and not to achieve the best possible accuracy on these tasks. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameters We do not optimize our experiments for FedAvg’s hyperparameters, always using those that proved to work reasonably well in our baseline setting which involves no compression and no Federated Dropout. For local training at each client we use static learning rates of 0.15 for MNIST, 0.05 for CIFAR-10 and 0.035 for EMNIST. We select 10 random clients per round for MNIST and CIFAR-10, and 35 for EMNIST. Finally, each selected client trains for one epoch per round using a batch size of 10. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 LOSSY COMPRESSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We focus on testing how the compression strategies presented in Section 3.1 impact the global model’s accuracy. Like Konecnˇ y et al. (2016b), we don’t compress all variables of our models. As ´ they mention, compressing smaller variables causes significant accuracy degradation but translates into minuscule communication savings. As such, we don’t compress biases for any of the models2. ",
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+    {
+        "type": "text",
+        "text": "In our experiments, we vary three parameters: ",
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+    {
+        "type": "text",
+        "text": "1. The type of basis transform applied: no transform or identity (I), randomized Hadamard transform (HD) and Kashin’s representation (K). ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/4e198c0b75739883172b561a02f9a11936f1fe7be1f337102e3e6aec37d9a0af.jpg",
+        "image_caption": [
+            "Figure 3: Effect of varying our lossy compression parameters on CIFAR-10 and EMNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2. The subsampling rate $s$ , which refers to the fraction of weights that are kept (i.e. $1 - s$ of the weights are zeroed out). ",
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+    {
+        "type": "text",
+        "text": "3. The number of quantization bits $q$ ",
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+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows the effect of varying these parameters for CIFAR-10 and EMNIST. We repeat each experiment 10 times and report the mean accuracy among these repetitions. The three main takeaways from these experiments are: (1) for every model, we are able find a setting of compression parameters that at the very least matches our baseline; (2) Kashin’s representation proves to be most useful for aggressive quantization values; and (3) it appears that subsampling is not all that helpful in the server-to-client setting. We proceed to give more details about these highlights. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The first takeaway is that, for every model, we are indeed able find a setting of compression parameters that matches or, in some cases, slightly outperforms our baseline. In particular, we are able to quantize every model to 4 bits, which translates to a reduction in communication of nearly $8 \\times$ . ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "The second takeaway is that Kashin’s representation proves to be most useful for aggressive quantization values, i.e. for low values of $q$ . In our experiments, gains were observed only in regimes where the overall accuracy had already degraded, but we hypothesize that the use of Kashin’s representation may provide clearer benefits in the compression of client-to-server gradient updates, where more aggressive quantization is admissible. We also highlight that using Kashin’s representation may be beneficial for other datasets. Indeed, its computational costs are comparable to that of the random Hadamard transform while also providing better theoretical error rates (see Section A.1). We refer the reader to Section A.3 in the Appendix, where we show preliminary results that demonstrate Kashin’s potential to dominate over the randomized Hadamard transform in compressing fully-trained models, particularly for small values of $q$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Finally, it appears that subsampling is not all that helpful in this server-to-client setting. This contrasts with the results presented by Konecnˇ y et al. (2016b) for compressed client-to-server updates, ´ where aggressive values of s were admissible. This trend extends to the other compression parameters: server-to-client compression of global models requires much more conservative settings than client-to-server compression of model updates. For example, for CIFAR-10, Konecnˇ y et al. ´ (2016b) get away with using $s = 0 . 2 5$ and $q = 8$ under a random Hadamard transform representation3. Meanwhile, in Figure 3 we can see that, for the same $q$ and representation, $s = 0 . 5$ already causes an unacceptable degradation of the accuracy. This is not surprising, since it is expected that the updates’ error will cancel out once several of them get aggregated at the server, which is not true for model downloads. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/b8cee57370ffcda99e8a75f325bfa49d39c0015a0e8228e901acbf70aef7b22e.jpg",
+        "image_caption": [
+            "Figure 4: Results for Federated Dropout, varying the percentage of neurons kept in each layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 FEDERATED DROPOUT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We focus on testing how the global model’s accuracy deteriorates once we use the strategy proposed in Section 3.2. In these experiments, we vary the percentage of neurons (or filters for the case of convolutional layers) that are kept on each layer of our models (we call this the federated dropout rate). We always keep the totality of the input and logits layers, and never drop the neuron that can be associated to the bias term. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 4 shows how the convergence of our three models behaves under different federated dropout rates. We repeat each experiment 10 times and report the mean among these repetitions. The main takeaway from these experiments is that, for every model, it is possible to find a federated dropout rate less than 1.0 that matches or, in some cases, even improves on the final accuracy of the model. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A federated dropout rate of 0.75 seems to work across the board. This corresponds to dropping $2 5 \\%$ of the rows and columns of the weight matrices of fully-connected layers (which translates to $\\smash { 1 \\sim 4 3 \\% }$ reduction in size), and to dropping the same percentage of filters of each convolutional layer. Now, because fully connected layers correspond to most of the parameters of the MNIST and EMNIST models, the $\\sim 4 3 \\%$ reduction will apply to them both in terms of the amount of data that has to be communicated and of the number of FLOPS required for local training. Meanwhile, because our CIFAR model is fully convolutional, gains will be of $2 5 \\%$ . ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "As a final comment, we note that more aggressive federated dropout rates tend to slow down the convergence rate of the model, even if they sometimes result in a higher accuracy. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "4.4 REDUCING THE OVERALL COMMUNICATION COST ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our final set of experiments shows how our models behave once we combine our two strategies, lossy compression and Federated Dropout, with existing client-to-server compression schemes (Konecnˇ y et al., 2016b), in order to explore how the different components of this end- ´ to-end, communication efficient framework interact. To do this, we evaluate how our models behave under 3 different compression schemes (aggressive, moderate and conservative) and 4 different federated dropout rates (0.5, 0.625, 0.75 and 0.875). The values for these schemes and rates were picked based on the observed behavior during the previous experiments, being somewhat more conservative as we are now combining different sources of noise. Table 2 describes the settings for each scheme. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 5 shows how our CIFAR-10 and EMNIST models behave under each of the previously mentioned conditions. We repeat each experiment 5 times and report the mean among these repetitions. For all three models, a federated dropout rate of 0.75 resulted in models with no accuracy degradation under all compression schemes except for the most aggressive. For MNIST and EMNIST, this translates into server-to-client communication savings of $1 4 \\times$ , client-to-server savings of $2 8 \\times$ and a reduction of $1 . 7 \\times$ in local computation, all without degrading the accuracy of the final global model (and sometimes even improving it). For CIFAR-10, we provide server-to-client communication savings of $1 0 \\times$ , client-to-server savings of $2 1 \\times$ and local computation savings of $1 . 3 \\times$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Based on these results, we also hypothesize that a federated dropout rate of 0.75 combined with a moderate or conservative compression scheme will be a good starting point when setting these parameters in practice. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/3c0006cd346f022b4e611c0a82122cb23d3252ffcbac246b703d442e7e406ef9.jpg",
+        "table_caption": [
+            "Table 2: Settings for each of our proposed compression schemes. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Scheme</td><td colspan=\"3\">Client-to-Server</td><td colspan=\"3\">Server-to-Client</td></tr><tr><td></td><td>transf.</td><td>S</td><td>q</td><td>transf.</td><td>S</td><td>q</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Aggressive</td><td>Kashin&#x27;s</td><td>0.4</td><td>2</td><td>Kashin&#x27;s</td><td>1.0</td><td>3</td></tr><tr><td>Moderate</td><td>Kashin&#x27;s</td><td>0.5</td><td>4</td><td>Kashin&#x27;s</td><td>1.0</td><td>5</td></tr><tr><td>Conservative</td><td>Kashin&#x27;s</td><td>1.0</td><td>8</td><td>Kashin&#x27;s</td><td>1.0</td><td>8</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/6f647972ab35ad2d0bac0fe55163b5e6c42b4362445a0a8fa5dba0fb4a5fdebd.jpg",
+        "image_caption": [
+            "Figure 5: Effect of using both compression and Federated Dropout on CIFAR-10 and EMNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS AND OPEN QUESTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            496,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The ecosystem currently targeted by Federated Learning (FL) is marked by heterogeneous edge networks that can potentially be orders of magnitude slower than the ones in datacenters. At the same time, FL can be quite demanding in terms of bandwidth, particularly when used to train deep models. We are thus at risk of either restricting the type of models we are able to train using this technique, or of excluding large groups of users from federated training. Both issues are problematic, but because access to high-end networks also appears to be correlated to sensitive factors such as income and age (Anzilotti, 2016; Pew Research Center, 2018), the latter may have implications related to fairness, making it particularly sensitive as we continue the adoption of FL systems. ",
+        "bbox": [
+            174,
+            527,
+            825,
+            640
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our work dramatically reduces the communication overheads in FL by (1) using lossy compression techniques on the server-to-client exchanges and by (2) using Federated Dropout, a technique that only communicates subsets of the global model to each client. We empirically show that a combination of our strategies with previous work allows for up to a $1 4 \\times$ reduction in server-to-client communication, a $1 . 7 \\times$ reduction in local computation and a $2 8 \\times$ reduction in client-to-server communication. ",
+        "bbox": [
+            174,
+            647,
+            825,
+            729
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In future work, we plan to: explore the efficacy of introducing a server step size in order to account for the use of different sub-models in Federated Dropout; investigate the possibility of using the same sub-models for all the selected clients in one round; and further characterize the benefits of Kashin’s representation in compressing the gradient updates in FL and in traditional model serving. An additional future direction to pursue related to fairness involves studying the effect of adaptively using these strategies (i.e. using more aggressive compression and federated dropout rates for some users) to prevent unfairly biased models. Finally, we note that the success of Federated Dropout suggests an entirely new avenue of research in which smaller, perhaps personalized, sub-models are eventually aggregated into a larger, more complex model that can be managed by the server. Contrary to the classic datacenter setting, the computational overhead associated with first creating and then aggregating the sub-models is justified in FL. ",
+        "bbox": [
+            174,
+            737,
+            825,
+            890
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES   \nEillie Anzilotti. Visualizing the state of global internet connectivity, Aug 2016. URL https://www.citylab.com/life/2016/08/ visualizing-the-state-of-global-internet-connectivity/496328/.   \nEmmanuel J Candes, Justin K Romberg, and Terence Tao. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 59(8):1207–1223, 2006.   \nGregory Cohen, Saeed Afshar, Jonathan Tapson, and Andre van Schaik. EMNIST: an extension of ´ MNIST to handwritten letters. arXiv preprint arXiv:1702.05373, 2017.   \nChristopher De Sa, Megan Leszczynski, Jian Zhang, Alana Marzoev, Christopher R Aberger, Kunle Olukotun, and Christopher Re. High-accuracy low-precision training. ´ arXiv preprint arXiv:1803.03383, 2018.   \nJeffrey Dean, Greg Corrado, Rajat Monga, Kai Chen, Matthieu Devin, Mark Mao, Andrew Senior, Paul Tucker, Ke Yang, Quoc V Le, et al. Large scale distributed deep networks. In Advances in neural information processing systems, pp. 1223–1231, 2012.   \nSimon Foucart and Holger Rauhut. A mathematical introduction to compressive sensing, volume 1. Birkhauser Basel, 2013. ¨   \nSong Han, Jeff Pool, John Tran, and William J Dally. Learning both weights and connections for efficient neural network. In Advances in neural information processing systems, pp. 1135–1143, 2015.   \nSong Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. International Conference on Learning Representations, 2016.   \nGeoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.   \nItay Hubara, Matthieu Courbariaux, Daniel Soudry, Ran El-Yaniv, and Yoshua Bengio. Binarized neural networks. In Advances in neural information processing systems, pp. 4107–4115, 2016.   \nBoris Sergeevich Kashin. Diameters of some finite-dimensional sets and classes of smooth functions. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 41(2):334–351, 1977.   \nJakub Konecnˇ y and Peter Richt ´ arik. Randomized distributed mean estimation: Accuracy vs com- ´ munication. arXiv preprint arXiv:1611.07555, 2016.   \nJakub Konecnˇ y, H Brendan McMahan, Daniel Ramage, and Peter Richt ´ arik. Federated optimization: ´ Distributed machine learning for on-device intelligence. arXiv preprint arXiv:1610.02527, 2016a.   \nJakub Konecnˇ y, H Brendan McMahan, Felix X Yu, Peter Richt ´ arik, Ananda Theertha Suresh, and ´ Dave Bacon. Federated learning: Strategies for improving communication efficiency. arXiv preprint arXiv:1610.05492, 2016b.   \nAlex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, Citeseer, 2009.   \nYann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to ´ document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.   \nXiaofan Lin, Cong Zhao, and Wei Pan. Towards accurate binary convolutional neural network. In Advances in Neural Information Processing Systems, pp. 345–353, 2017.   \nYurii Lyubarskii and Roman Vershynin. Uncertainty principles and vector quantization. IEEE Transactions on Information Theory, 56(7):3491–3501, 2010. ",
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+        "text": "Virginia Smith, Simone Forte, Ma Chenxin, Martin Taka´c, Michael I Jordan, and Martin Jaggi. ˇ Cocoa: A general framework for communication-efficient distributed optimization. Journal of Machine Learning Research, 18:230, 2018. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Jost Tobias Springenberg, Alexey Dosovitskiy, Thomas Brox, and Martin Riedmiller. Striving for simplicity: The all convolutional net. International Conference on Learning Representations (workshop track), 2015. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research, 15(1):1929–1958, 2014. ",
+        "bbox": [
+            176,
+            409,
+            823,
+            452
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ananda Theertha Suresh, Felix X Yu, Sanjiv Kumar, and H Brendan McMahan. Distributed mean estimation with limited communication. In International Conference on Machine Learning, pp. 3329–3337, 2017. ",
+        "bbox": [
+            171,
+            460,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Hongyi Wang, Scott Sievert, Zachary Charles, Dimitris Papailiopoulos, and Stephen Wright. ATOMO: Communication-efficient Learning via Atomic Sparsification. arXiv preprint arXiv:1806.04090, 2018. ",
+        "bbox": [
+            174,
+            512,
+            825,
+            554
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A KASHIN’S REPRESENTATION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            102,
+            449,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For reasons of space, we have relegated a more detailed discussion of Kashin’s representation (see Section 3) to the Appendix. In this section, we briefly discuss Kashin’s representation both from a theoretical (Section A.1) and practical (Section A.2) standpoints. Finally, we present some preliminary results that argue the potential of Kashin’s representation to dominate over the random Hadamard transform with respect to the size vs. accuracy trade-off (Section A.3). ",
+        "bbox": [
+            174,
+            135,
+            825,
+            204
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 THEORETICAL OVERVIEW ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            222,
+            398,
+            237
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The idea of using the classical results of Kashin (1977) to increase the robustness of coefficients to perturbations was first introduced by Lyubarskii & Vershynin (2010). Their result states that, given a tight frame satisfying a form of uncertainty principle, a weaker notion of the RIP (Candes et al., 2006), it is possible to convert the frame representation of every vector into the more robust Kashin’s representation, whose coefficients will have the smallest possible dynamic range. ",
+        "bbox": [
+            174,
+            248,
+            825,
+            319
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Error rates Since the results of Suresh et al. (2017) (who quantified the reduction in quantization error due to the Hadamard transform) rely on exactly this notion of dynamic range, and assuming the subsampled randomized Hadamard transform satisfies the uncertainty principle, Theorem 3.5 of Lyubarskii & Vershynin (2010) can be directly used as a drop-in replacement for Lemma 7 in Suresh et al. (2017), removing the logarithmic dependence on dimension from Theorem 3 therein, matching the lower bounds. We do not provide the complete proof as, beyond drawing this connection, it does not imply any novelty whatsoever. However, an open question remains, as we are not aware of a result showing what are the parameters of the uncertainty principle guaranteed by the subsampled randomized Hadamard transform. They exist however, as the transform is known to satisfy the RIP (Foucart & Rauhut, 2013), which is a stronger notion. ",
+        "bbox": [
+            174,
+            325,
+            825,
+            465
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 PRACTICAL CONSIDERATIONS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            483,
+            428,
+            497
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In practice, given a tight frame, the algorithm for computing Kashin’s representation is straightforward. It runs for $n$ iterations, and takes parameters $\\eta , \\delta$ as input. In a single iteration, one first computes the frame coefficients, projects them onto a $L _ { \\infty }$ ball, and reconstructs the error in the original domain. Another iteration proceeds starting with the reconstructed error and a smaller ball. We refer the reader to Lyubarskii & Vershynin (2010) for more details regarding $\\eta , \\delta$ and their relationship with the uncertainty principle. ",
+        "bbox": [
+            174,
+            510,
+            823,
+            593
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In our work, we use the randomized Hadamard transform as the initial tight frame (see Section A.1 for details on why this is possible). We also run the algorithm for just $n = 2$ iterations (as very often this provides most of the benefit), fixed $\\delta = 1$ , and used a variant of the algorithm which yields an exact representation (omitting the $L _ { \\infty }$ projection in the last iteration). Given this, the choice of $\\eta$ is irrelevant. The dominant part of the computation is then three applications of the fast Walsh-Hadamard transform, as opposed to a single one in Konecnˇ y et al. (2016b)). ´ ",
+        "bbox": [
+            174,
+            601,
+            823,
+            684
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "As a particular example, say we are to compress an 80-dimensional vector. We first pad the vector with zeros, so that its dimension is 128 (the closest larger power of 2). Then, we multiply the vector by a diagonal matrix with independent Rademacher random variables $( D \\in \\mathbb { R } ^ { 1 2 8 \\times 1 2 8 }$ ), followed by the application of the fast Walsh-Hadamard transform $( H \\in \\mathbb { R } ^ { 1 2 8 \\times 1 2 8 } ,$ ). The first 80 columns of the matrix $H D$ correspond to the tight frame used to find the Kashin’s representation. Nonetheless, we avoid representing this explicitly. ",
+        "bbox": [
+            174,
+            690,
+            825,
+            775
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Finally, note that, if the initial dimension was a power of 2, we need to pad zeros to the next power of 2 in order to realize any benefit over just using the Hadamard transform. ",
+        "bbox": [
+            174,
+            781,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.3 DOMINANCE OVER HADAMARD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            828,
+            437,
+            842
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Given the theoretical properties of Kashin’s representation, we hypothesize it should dominate the random Hadamard transform when it comes to the size vs. accuracy trade-off. A preliminary experiment to corroborate this hypothesis is the following: ",
+        "bbox": [
+            176,
+            854,
+            823,
+            897
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "1. We train an MNIST model until we get an accuracy of around $9 9 . 3 \\%$ ",
+        "bbox": [
+            174,
+            909,
+            648,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "2. We compress the original model using some linear transform, some subsampling ratio and some number of quantization bits. ",
+        "bbox": [
+            173,
+            103,
+            823,
+            131
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "3. We decompress the model and evaluate both its new accuracy and its $L _ { 2 }$ distance to the original model. ",
+        "bbox": [
+            173,
+            132,
+            821,
+            159
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "4. We repeat the previous two steps for different linear transforms (identity, random Hadamard transform and Kashin’s representation), subsampling ratios (0.25, 0.5 and 1.0) and quantization bits (1, 2, 4, 8, 16). ",
+        "bbox": [
+            176,
+            160,
+            823,
+            200
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "An important detail is that, whenever we use Kashin’s representation, we do a grid search over the best values for $n$ (from 1 to 10) and $\\eta$ . However, $\\delta$ is kept fixed as 1. ",
+        "bbox": [
+            174,
+            213,
+            821,
+            241
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The results of this experiment are shown in Figure 6. In the legend, R corresponds to rotation — I for identity, HD for randomized Hadamard, Kashin for Kashin based on the randomized Hadamard; and SR corresponds to subsampling ratio — the fraction of elements to be kept non-zero. In the top row, the figure shows the relationship of the accuracy of the compressed model vs. the number of bits used for quantization, and vs. the model’s size (in MB). In the bottom row, the $L _ { 2 }$ error incurred is plotted against the same. It is very clear then that Kashin’s representation does dominate the other two representations when it comes to the size vs. accuracy trade-off, making up the Pareto frontier for all combinations of subsampling ratio and quantization bits. Nevertheless, we did optimize over the parameters associated with Kashin’s algorithm, something that does not need to be done for the random Hadamard transform. In Section A.2, we propose a set of values that worked well enough for our experiments, but further exploration on how to easily determine these values is in order. ",
+        "bbox": [
+            173,
+            247,
+            825,
+            401
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/abf34396d464b2bb8d7b9fa4e35ae23bba50f9800a6aa216d47354b141f7e471.jpg",
+        "image_caption": [
+            "Figure 6: Compressing an already trained MNIST model with linear transform $^ +$ subsampling $^ +$ uniform quantization. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            415,
+            820,
+            681
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B MNIST EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            493,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For reasons of space, we have relegated the experimental results using MNIST (see Section 4) (Section A) to the Appendix. ",
+        "bbox": [
+            176,
+            133,
+            823,
+            161
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 shows the results of using our lossy compression on MNIST under the experimental setup presented in Section 4.2. Meanwhile, Figure 8 shows the results of using Federated Dropout (see Section 4.3 for details). Finally, Figure 9 shows the results of performing both lossy compression for downloads and uploads, as well as Federated Dropout, as described in Section 4.4. ",
+        "bbox": [
+            173,
+            169,
+            825,
+            226
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/76b0d59f01e9eb0328dc020065b77b91433f5eec90e30a1e84877121f28377b3.jpg",
+        "image_caption": [
+            "Figure 7: Effect of varying our lossy compression parameters on the convergence MNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            239,
+            821,
+            340
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/09ea07f075bbf2194d3e9f2122c5ec2706362418c4a9d51d750b66d8b33eca43.jpg",
+        "image_caption": [
+            "Figure 8: Effect of varying the percentage of neurons kept in each layer on MNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            354,
+            397,
+            642,
+            491
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/76fe8639e916dacc9a7c5c53cddd74600a52a6c614b2572b179f9d9a8004aa1c.jpg",
+        "image_caption": [
+            "Figure 9: Effect of using both lossy compression and Federated Dropout on MNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            547,
+            821,
+            648
+        ],
+        "page_idx": 12
+    }
+]

parse/train/yehlf2AvSD_/yehlf2AvSD_.md

@@ -0,0 +1,480 @@
+# Mean-Square Analysis with An Application to Optimal Dimension Dependence of Langevin Monte Carlo
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Sampling algorithms based on discretizations of Stochastic Differential Equations   
+2 (SDEs) compose a rich and popular subset of MCMC methods. This work pro  
+3 vides a general framework for the non-asymptotic analysis of sampling error in   
+4 2-Wasserstein distance, which also leads to a bound of mixing time. The method   
+5 applies to any consistent discretization of contractive SDEs. When applied to   
+6 Langevin Monte Carlo algorithm, it establishes $\widetilde { \mathcal { O } } \left( \sqrt { d } / \epsilon \right)$ mixing time, without   
+7 warm start, under the common log-smooth and log-strongly-convex conditions,   
+8 plus a growth condition on the potential of target measures at infinity. This bound   
+9 improves the best previously known $\widetilde { \mathcal { O } } \left( { d } / { \epsilon } \right)$ result and is optimal in both dimension   
+10 $d$ and accuracy tolerance $\epsilon$ for log-smooth and log-strongly-convex target measures.   
+11 Our theoretical analysis is further validated by numerical experiments.
+
+# 12 1 Introduction
+
+13 The problem of sampling statistical distributions has attracted considerable attention, not only in   
+14 the fields of statistics and scientific computing, but also in machine learning (Robert and Casella,   
+15 2013; Andrieu et al., 2003; Liu, 2008); for example, how various sampling algorithms scale with   
+16 the dimension of the target distribution is a popular recent topic in statistical deep learning (see   
+17 discussions below for references). For samplers that can be viewed as discretizations of SDEs, the   
+18 idea is to use an ergodic SDE whose equilibrium distribution agrees with the target distribution,   
+19 and employ an appropriate numerical algorithm that discretizes (the time of) the SDE. The iterates   
+20 of the numerical algorithm will approximately follow the target distribution when converged, and   
+21 can be used for various downstream applications such as Bayesian inference and inverse problem   
+22 (Dashti and Stuart, 2017). One notable example is the Langevin Monte Carlo algorithm (LMC),   
+23 which corresponds to Euler-Maruyama discretization of overdamped Langevin equation. Its study   
+24 dated back to at least the 90s (Roberts et al., 1996) but keeps on leading to important discoveries, for   
+25 example, on non-asymptotics and dimension dependence, which are relevant to machine learning   
+26 (e.g., Dalalyan (2017a,b); Cheng et al. (2018a); Durmus et al. (2019a,b); Vempala and Wibisono   
+27 (2019); Dalalyan and Riou-Durand (2020); Erdogdu and Hosseinzadeh (2020); Mou et al. (2019)).   
+28 LMC is closely related to SGD too (e.g., Mandt et al. (2017)). Many other examples exist, based   
+29 on alternative SDEs and different discretizations (e.g., Dalalyan and Riou-Durand (2020); Ma et al.   
+30 (2021); Mou et al. (2021); Li et al. (2020); Roberts and Rosenthal (1998); Chewi et al. (2020); Shen   
+31 and Lee (2019)).   
+32 Quantitatively characterizing the non-asymptotic sampling error of numerical algorithms is usually   
+33 critical for choosing the appropriate algorithm for a specific downstream application, for providing   
+34 practical guidance on hyperparameter selection and experiment design, and for designing improved   
+35 samplers. A powerful tool that dates back to (Jordan et al., 1998) is a paradigm of non-asymptotic   
+36 error analysis, namely to view sampling as optimization in probability space, and it led to many   
+37 important recent results (e.g., Liu and Wang (2016); Dalalyan (2017a); Wibisono (2018); Zhang   
+38 et al. (2018); Frogner and Poggio (2020); Chizat and Bach (2018); Chen et al. (2018); Ma et al.   
+39 (2021); Erdogdu and Hosseinzadeh (2020)). It works by choosing an objective functional, typically   
+40 some statistical distances/diverges, and showing that the law of the iterates of sampling algorithms   
+41 converges in that objective functional. However, the choice of the objective functional often needs to   
+42 be customized for different sampling algorithms. For example, KL divergence works for LMC (Cheng   
+43 and Bartlett, 2018), but a carefully hand-crafted cross term needs to be added to KL divergence for   
+44 analyzing KLMC (Ma et al., 2021). Even for the same underlying SDE, different discretization   
+45 schemes exist and lead to different sampling algorithms, and the analyses of them had usually been   
+46 case by case (e.g., Cheng et al. (2018b); Dalalyan and Riou-Durand (2020); Shen and Lee (2019)).   
+47 Therefore, it would be a desirable complement to have a unified, general framework to study the   
+48 non-asymptotic error of SDE-based sampling algorithms.   
+49 As an important member of the family of SDE-based sampling algorithms, Langevin Monte Carlo is   
+50 widely used in practice. Its stochastic gradient version is implemented in common machine learning   
+51 systems, such as Tensorflow (Abadi et al., 2016), and is the off-the-shelf algorithm for large scale   
+52 Bayesian inference. With the ever-growing size of parameter space, the non-asymptotic error of LMC   
+53 is of central theoretical and practical interest, in particular, its dependence on the dimension of the   
+54 sample space. The best current known upper bound of the mixing time in 2-Wasserstein distance for   
+55 LMC is $\begin{array} { r } { \bar { \mathcal { O } } \left( \frac { d } { \epsilon } \right) } \end{array}$ (Durmus et al., 2019b). Motivated by a recent result (Chewi et al., 2020) that shows   
+56 better dimension dependence for a Metropolis-Adjusted improvement of LMC, we wonder if the   
+57 current bound for (unadjusted) LMC is tight, and if not, what is the optimal dimension dependence?   
+58 Our contribution We study a broad family of numerical algorithms that discretize SDEs that   
+59 have a contraction property (possibly after a coordinate transformation). For this type of problems,   
+60 we revisit the classical mean-square analysis (Milstein and Tretyakov, 2013) in numerical SDE   
+61 literature and extend its the global error bound from finite time to infinite time. Same as in classical   
+62 mean-square analysis, we show the global error is only half order lower than the order of local strong   
+63 error $\left( p _ { 2 } \right)$ . We further obtain a $\tilde { \mathcal { O } } \left( C ^ { \frac { 1 } { p _ { 2 } - \frac { 1 } { 2 } } } \underbrace { \frac { 1 } { \epsilon ^ { \frac { 1 } { p _ { 2 } - \frac { 1 } { 2 } } } } } \right)$ mixing time upper bound in 2-Wasserstein   
+64 distance for the family of algorithms, where $C$ is a constant containing various information of the   
+65 underlying problem, e.g., the dimension $d$ .   
+66 As an application of the general mixing time result, we study the widely used Langevin Monte   
+67 Carlo algorithm (LMC) for sampling from a Gibbs distribution $\mu \propto \mathrm { e x p } \left( - f ( \pmb { x } ) \right)$ , which is an   
+68 Euler-Maruyama discretization of Langevin dynamics. Under the standard smoothness and strong  
+69 convexity assumptions, plus an additional linear growth condition on the third-order derivative of $f$ ,   
+70 we obtain a $\begin{array} { r } { \widetilde { \mathcal { O } } \left( \frac { \sqrt { d } } { \epsilon } \right) } \end{array}$ mixing time in 2-Wasserstein distance, which improves upon the previously best   
+71 known $\begin{array} { r } { \widetilde { \mathcal { O } } \left( \frac { d } { \epsilon } \right) } \end{array}$ result (Durmus et al., 2019b). For a comparison, note it was known that discretized   
+72 kinetic Langevin dynamics can lead to $\sqrt { d }$ dependence on dimension (Cheng and Bartlett, 2018;   
+73 Dalalyan and Riou-Durand, 2020) and some believe that it is the introduction of momentum that   
+74 improves the dimension dependence, but our result shows that discretized overdamped Langevin (no√   
+75 momentum) can also have mixing time scaling like $\sqrt { d }$ . In fact, it is important to mention that it was   
+76 recently shown that Metropolis-Adjusted Euler-Maruyama discretization of overdamped Langevin   
+77 (i.e., MALA) has an optimal dimension dependence of $\widetilde { \mathcal { O } } \left( \sqrt { d } \right)$ (Chewi et al., 2020), while what we   
+78 analyze here is the unadjusted version (i.e., LMC), and it has the same dimension dependence (note   
+79 however that our $\epsilon$ dependence is not as good as that for MALA; more discussion in Section 4). We   
+80 also constructed an example that shows that the mixing time of LMC is at least $\begin{array} { r } { \widetilde \Omega \left( \frac { \sqrt { d } } { \epsilon } \right) } \end{array}$ . Hence, our   
+81 mixing time bound has the optimal dependence on both $d$ and $\epsilon$ . Our theoretical analysis is further   
+82 validated by empirical investigation of numerical examples.
+
+# 83 2 Preliminaries
+
+84 Notation Use the symbol $_ { x }$ to denote a $d$ -dimensional vector, and the plain symbol $x$ to denote a   
+85 scalar variable. Use $\lVert \boldsymbol { x } \rVert$ to denote the Euclidean norm of vector $_ { x }$ . A numerical algorithm is denoted   
+86 by $\mathcal { A }$ and its $k$ -th iterate is denoted by $\bar { \pmb { x } } _ { k }$ . We slightly abuse notation by identifying measures with   
+87 their density function w.r.t. Lebesgue measure. We use the convention $\begin{array} { r } { \tilde { \mathcal { O } } \left( \cdot \right) = \mathbf { \bar { \mathcal { O } } } ( \cdot ) \log ^ { \mathcal { O } ( 1 ) } ( \cdot ) } \end{array}$   
+88 i.e., the $\widetilde { \mathcal { O } } \left( \cdot \right)$ notation ignores the dependence on logarithmic factors. We use the notation $\widetilde { \Omega } ( \cdot )$   
+89 similarly. Denote 2-Wasserstein distance by $\begin{array} { r } { W _ { 2 } ( \mu _ { 1 } , \mu _ { 2 } ) = \left( \operatorname* { i n f } _ { ( \pmb { X } , \pmb { Y } ) \sim \Pi ( \mu _ { 1 } , \mu _ { 2 } ) } \mathbb { E } \left\| \pmb { X } - \pmb { Y } \right\| ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } \end{array}$ ,   
+90 where $\Pi ( \mu _ { 1 } , \mu _ { 2 } )$ is the set of couplings, i.e. all joint measures with $X$ and $Y$ marginals being   
+91 $\mu _ { 1 }$ and $\mu _ { 2 }$ . Denote the target distribution by $\mu$ and the law of a random variable $\boldsymbol { X }$ by $\operatorname { L a w } ( X )$ .   
+92 Finally, denote the mixing time of an sampling algorithm $\mathcal { A }$ converging to its target distribution $\mu$ in   
+93 2-Wasserstein distance by $\begin{array} { r } { \tau _ { \operatorname* { m i x } } ( \epsilon ; W _ { 2 } ; A ) = \operatorname* { i n f } \{ k \geq 0 | W _ { 2 } ( \operatorname { L a w } ( \bar { x } _ { k } ) , \mu ) \leq \epsilon \} . } \end{array}$ .
+
+# 94 SDE for Sampling Consider a general SDE
+
+$$
+d { \pmb x } _ { t } = { \pmb b } ( t , { \pmb x } _ { t } ) d t + { \pmb \sigma } ( t , { \pmb x } _ { t } ) d { \pmb B } _ { t }
+$$
+
+95 where $\pmb { b } \in \mathbb { R } ^ { d }$ is a drift term, $\pmb { \sigma } \in \mathbb { R } ^ { d \times l }$ is a diffusion coefficient matrix and $\mathbf { \delta } _ { B _ { t } }$ is a $l$ -dimensional   
+96 Wiener process. Under mild condition (Pavliotis, 2014, Theorem 3.1), there exists a unique strong   
+97 solution $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ to Eq. (1). Some SDEs admit geometric ergodicity, so that their solutions converge   
+98 exponentially fast to a unique invariant distribution, and examples include the classical overdamped   
+99 and kinetic Langevin dynamics, but are not limited to those (e.g., Mou et al. (2021); Li et al. (2020)).   
+100 Such SDE are desired for sampling purposes, because one can set the target distribution to be the   
+101 invariant distribution by choosing an SDE with an appropriate potential, and then solve the solution   
+102 $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ of the SDE and push the time $t$ to infinity, so that (approximate) samples of the target distribution   
+103 can be obtained. Except for a few known cases, however, explicit solutions of Eq. (1) are elusive and   
+104 we have to resort to numerical schemes to simulate/integrate SDE. Such example schemes include,   
+105 but are not limited to Euler-Maruyama method, Milstein methods and Runge-Kutta method (e.g.,   
+106 Kloeden and Platen (1992); Milstein and Tretyakov (2013)). With constant stepsize $h$ and at $k$ -th   
+107 iteration, a typical numerical algorithm takes a previous iterate $\bar { \pmb { x } } _ { k - 1 }$ and outputs a new iterate $\bar { \pmb { x } } _ { k }$ as   
+108 an approximation of the solution $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ of Eq. (1) at time $t = k h$ .
+
+109 Langevin Monte Carlo Algorithm LMC algorithm is defined by the following update rule
+
+$$
+\bar { \pmb { x } } _ { k } = \bar { \pmb { x } } _ { k - 1 } - h \nabla f ( \bar { \pmb { x } } _ { k - 1 } ) + \sqrt { 2 h } \pmb { \xi } _ { k } , \quad k = 1 , 2 , \cdots
+$$
+
+where $\{ \pmb { \xi } _ { k } \} _ { k \in \mathbb { Z } _ { > 0 } }$ are i.i.d. standard $d$ -dimensional Gaussian vectors. LMC corresponds to an EulerMaruyama discretization of the continuous overdamped Langevin dynamics $d { \pmb x } _ { t } = - \nabla f ( { \pmb x } _ { t } ) d t +$ $\sqrt { 2 } d B _ { t }$ , which converges to an equilibrium distribution $\mu \sim \exp ( - f ( \pmb { x } ) )$ .
+
+13 Dalalyan (2017b) provided a non-asymptotic analysis of LMC. An $\begin{array} { r } { \widetilde { \mathcal { O } } \left( \frac { d } { \epsilon ^ { 2 } } \right) } \end{array}$ mixing time bound in   
+14 $W _ { 2 }$ for log-smooth and log-strongly-convex target measures (Dalalyan, 2017a; Cheng et al., 2018a;   
+15 Durmus et al., 2019a) has been established. It was further improved to $\begin{array} { r } { \widetilde { \mathcal { O } } \left( \frac { d } { \epsilon } \right) } \end{array}$ under additional   
+16 Lipschitz assumption on the Hessian of $f$ (Durmus et al., 2019b). Mixing time bounds of LMC   
+17 in other statistical distances/divergences have also been studied, including total variation distance   
+118 (Dalalyan, 2017b; Durmus et al., 2017) and KL divergence (Cheng and Bartlett, 2018).   
+119 Classical Mean-Square Analysis A powerful framework for quantifying the global discretization   
+120 error of a numerical algorithm for Eq. (1), i.e., $e _ { k } = \left\{ \mathbb { E } \left\| \pmb { x } _ { k h } - \bar { \pmb { x } } _ { k } \right\| \right\} ^ { \frac { 1 } { 2 } }$ , is mean-square analysis   
+121 (e.g., the monograph of Milstein and Tretyakov (2013)). Mean-square analysis studies how local   
+122 integration error propagate and accumulate into global integration error; in particular, if one-step   
+123 (local) weak error and strong error (both the exact solution $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ and the numerical approximation start   
+124 from the same initial value $_ { \textbf { \em x } }$ ) satisfy
+
+$$
+\begin{array} { r l } & { \left\| \mathbb { E } x _ { h } - \mathbb { E } \bar { { \bar { x } } } _ { 1 } \right\| \leq C _ { 1 } \left( 1 + \mathbb { E } \left\| x \right\| ^ { 2 } \right) ^ { \frac { 1 } { 2 } } h ^ { p _ { 1 } } , \quad \mathrm { ( l o c a l ~ w e a k ~ e r r o r ) } } \\ & { \left( \mathbb { E } \left\| x _ { h } - \bar { x } _ { 1 } \right\| ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \leq C _ { 2 } \left( 1 + \mathbb { E } \left\| x \right\| ^ { 2 } \right) ^ { \frac { 1 } { 2 } } h ^ { p _ { 2 } } , \quad \mathrm { ( l o c a l ~ s t r o n g ~ e r r o r ) } } \end{array}
+$$
+
+over a time interval 125 $[ 0 , K h ]$ for some constants $C _ { 1 } , C _ { 2 } > 0$ , $p _ { 2 } \geq { \frac { 1 } { 2 } }$ and $\begin{array} { r } { p _ { 1 } \geq p _ { 2 } + \frac { 1 } { 2 } } \end{array}$ , then the global error can be bounded by 126 $\begin{array} { r } { \displaystyle { e _ { k } \leq C \left( 1 + \mathbb { E } \left\| x _ { 0 } \right\| ^ { 2 } \right) ^ { \frac { 1 } { 2 } } h ^ { p _ { 2 } - \frac { 1 } { 2 } } } } \end{array}$ , $k = 1 , 2 , \cdots , K$ for some constant 127 $C > 0$ dependent on $K h$ .
+
+128 Although classical mean-square analysis is only concerned with numerical integration error, sampling   
+129 error can be also inferred. However, there is a limitation that prevents directly employing mean-square   
+130 analysis in the non-asymptotic analysis of sampling algorithms. The bound of global error only holds   
+131 in finite time because the constant $C$ can grow exponentially as $K$ increases, rendering the bound   
+132 useless when $K  \infty$ .
+
+# 3 Mean-Square Analysis of Samplers Based on Contractive SDE
+
+In order to prepare for the analysis of sampling error, we first show that the finite time limitation of integration error analysis can be lifted if the SDE being discretized is contractive.
+
+136 More precisely, one bottleneck that prevents the results of classical mean-square analysis from   
+137 extending to infinite time horizon, is the fact that the solution of a general SDE may not be bounded,   
+138 and neither is its discretization. Note that local error (Eq. (3)) depends on the initial value. To go   
+139 from local to global error, these ‘initial’ values correspond to iterates of numerical algorithms, which   
+140 change from iteration to iteration and can be unbounded, hence when accumulated together, it is   
+141 possible that the global error may blow up.   
+142 Samplers considered here, on the other hand, are based on stochastic differential equations, each of   
+143 which weakly converges to a limiting distributions. The solution of the underlying converging SDE,   
+144 as it converges to the invariant measure, gradually inherits boundedness properties from the target   
+145 measure. Thus, as long as the target measure has bounded 2nd-moment, a sampling algorithm based   
+146 on a reasonable discretization of the SDE should also have bounded 2nd-moment. Motivated by this   
+147 observation, we will assume the sampling algorithms we study are based on contractive SDEs, which   
+148 is a sufficient condition to ensure the underlying SDE converges to a statistical distribution.
+
+149 Definition 3.1. A stochastic differential equation is contractive if there exists a non-singular constant matrix 150 $A \in \mathbb { R } ^ { d \times d }$ , a constant $\beta > 0$ , such that any pair of solutions of the SDE satisfy
+
+$$
+\begin{array} { r } { \left( \mathbb { E } \left\| A \left( \pmb { x } _ { t } - \pmb { y } _ { t } \right) \right\| ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \leq \left\| A \left( \pmb { x } - \pmb { y } \right) \right\| \exp ( - \beta t ) , } \end{array}
+$$
+
+151 where $\mathbf { \boldsymbol { x } } _ { t } , \mathbf { \boldsymbol { y } } _ { t }$ are two solutions, driven by the same Brownian motion but evolved respectively from   
+152 initial conditions $_ { \textbf { \em x } }$ and $\textbf {  { y } }$ .   
+53 Remark. As long as $^ { b }$ and $\sigma$ in (1) are not explicitly dependent on time, it suffices to find an   
+54 arbitrarily small $t _ { 0 } > 0$ and show (4) holds for all $t < t _ { 0 }$ .
+
+Remark. Sometimes contraction is not easy to establish directly, but can be shown after an appropriate coordinate transformation, see (Dalalyan and Riou-Durand, 2020, Proposition 1) for such a treatment for kinetic Langevin dynamics. The introduction of $A$ permits such transformations.
+
+158 We now use contractivity to remove the finite time limitation. We will first need a lemma, which is a   
+159 local (short time) result.   
+160 Lemma 3.2. (Milstein and Tretyakov, 2013, Lemma 1.3) Suppose $^ { b }$ and $\pmb { \sigma }$ in Eq.(1) are Lipschitz   
+161 continuous. For two solutions $\mathbf { \boldsymbol { x } } _ { t } , \mathbf { \boldsymbol { y } } _ { t }$ of Eq. (1) starting from $\mathbf { \nabla } _ { \mathbf { x } , \mathbf { y } }$ respectively, denote $z : =$   
+162 $( { \pmb x } _ { t } - { \pmb x } ) - ( { \pmb y } _ { t } - { \pmb y } )$ , then there exist $C _ { 0 } > 0$ and $h _ { 0 } > 0$ such that
+
+$$
+\begin{array} { r } { \mathbb { E } \left\| z \right\| ^ { 2 } \leq C _ { 0 } \left\| x - y \right\| ^ { 2 } t , \quad \forall x , y , 0 < t \leq h _ { 0 } . } \end{array}
+$$
+
+163 Then we will have a sequence of results that connects sampling error (a statistical property) with   
+164 local integration error (a simulation property). This justifies our generic produce for non-asymptotic   
+165 sampling error analysis, which only requires bounding the orders of local weak and strong integration   
+166 errors (in addition to establishing contractivity of the continuous dynamics).   
+167 Theorem 3.3. (Global Integration Error, Infinite Time Version) Suppose Eq.(1) is contractive with   
+168 rate $\beta$ and with respect to a non-singular matrix $A \in \mathbb { R } ^ { d \times d }$ , with Lipschitz continuous $^ { b }$ and $\pmb { \sigma }$   
+169 and there is a numerical algorithm $\mathcal { A }$ with step size $h$ simulating the solution $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ of the $S D E$ , whose   
+170 iterates are denoted by $\bar { \boldsymbol { x } } _ { k } , k = 0 , 1 , \cdots$ . Suppose there exists $0 < h _ { 0 } \leq 1 , C _ { 1 } , C _ { 2 } > 0 , D _ { 1 } , D _ { 2 } \geq$   
+171 $0 , p _ { 1 } \geq 1 , \frac { 1 } { 2 } < p _ { 2 } \leq \dot { p _ { 1 } } - \frac { 1 } { 2 }$ such that for any $0 < h \leq h _ { 0 }$ , the algorithm $\mathcal { A }$ has, respectively, local   
+172 weak and strong error of order $p _ { 1 }$ and $p _ { 2 }$ , defined as
+
+$$
+\begin{array} { r l } & { \left\{ \left. \mathbb { E } \left( { \pmb x } _ { h } - { \bar { \pmb x } } _ { 1 } \right) \right. \right\} \leq \left( C _ { 1 } + D _ { 1 } \sqrt { \mathbb { E } \left. \pmb x \right. ^ { 2 } } \right) h ^ { p _ { 1 } } , } \\ & { \left( \mathbb { E } \left. \pmb x _ { h } - { \bar { \pmb x } } _ { 1 } \right. ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \leq \left( C _ { 2 } ^ { 2 } + D _ { 2 } ^ { 2 } \mathbb { E } \left. \pmb x \right. ^ { 2 } \right) ^ { \frac { 1 } { 2 } } h ^ { p _ { 2 } } , } \end{array}
+$$
+
+173 where $\scriptstyle { \mathbf { 2 } } ( { \mathbf { 2 } } \scriptscriptstyle { h }$ solves Eq.(1) with any initial value $_ { \textbf { \em x } }$ and $\bar { \mathbf { x } } _ { 1 }$ is the result of applying $\mathcal { A }$ to $_ { \textbf { \em x } }$ for one step.   
+174 If the solution of SDE $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ and algorithm $\mathcal { A }$ both start from $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ , then for $0 ~ < ~ h ~ \leq ~ h _ { 1 }$ ,   
+175 $\begin{array} { r l r } & { } & { \operatorname* { m i n } \left\{ h _ { 0 } , \frac { 1 } { 4 \beta } , \left( \frac { \sqrt { \beta } } { 4 \sqrt { 2 } \kappa _ { A } D _ { 2 } } \right) ^ { \frac { 1 } { p _ { 2 } - \frac { 1 } { 2 } } } , \left( \frac { \beta } { 8 \sqrt { 2 } \kappa _ { A } \left( D _ { 1 } + C _ { 0 } D _ { 2 } \right) } \right) ^ { \frac { 1 } { p _ { 2 } - \frac { 1 } { 2 } } } \right\} , t h e ~ g l o b a l ~ e r r } \\ & { } & { e _ { k } : = \left( { \mathbb E } \| x _ { k h } - \bar { x } _ { k } \| ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \leq C h ^ { p _ { 2 } - \frac { 1 } { 2 } } , \quad k = 0 , 1 , 2 , \cdots } \end{array}$ r $e _ { k }$ is bounded as   
+176
+
+177 where
+
+$$
+C = \frac { 2 } { \sqrt { \beta } } \kappa _ { A } ^ { 2 } \left( \frac { C _ { 1 } + C _ { 0 } C _ { 2 } + \sqrt { 2 } U ( D _ { 1 } + C _ { 0 } D _ { 2 } ) } { \sqrt { \beta } } + C _ { 2 } + \sqrt { 2 } D _ { 2 } U \right) ,
+$$
+
+178 $C _ { 0 }$ is from Eq. (5), $\kappa _ { A }$ is the condition number of matrix $A$ and $U ^ { 2 } \triangleq 4 \left\| \pmb { x } _ { 0 } \right\| ^ { 2 } + 5 \mathbb { E } _ { \mu } \left\| \pmb { x } \right\| ^ { 2 } .$ .
+
+179 Remark. We use the convention $1 / 0 = \infty$ when $D _ { 1 } = D _ { 2 } = 0$ . This is pertinent when a numerical   
+180 algorithm $\mathcal { A }$ , e.g. LMC (Lemma $D . 3$ ), produces bounded iterates. In such cases, the initial value in   
+181 Eq. (6) are iterations of $\mathcal { A }$ and will be bounded, it then can be absorbed into $C _ { 1 } , C _ { 2 }$ and we may set   
+182 $D _ { 1 } = D _ { 2 } = 0$ .
+
+83 Following Theorem 3.3, we obtain the following non-asymptotic bound of the sampling error in $W _ { 2 }$ :
+
+84 Theorem 3.4. (Non-Asymptotic Sampling Error Bound: General Case) Under the same assump  
+185 tion and with the same notation of Theorem 3.3, we have
+
+$$
+W _ { 2 } ( L a w ( \bar { x } _ { k } ) , \mu ) \leq \sqrt { 2 } e ^ { - \beta k h } W _ { 2 } ( L a w ( x _ { 0 } ) , \mu ) + \sqrt { 2 } C h ^ { p _ { 2 } - \frac { 1 } { 2 } } , \quad \forall 0 < h \leq h _ { 1 } .
+$$
+
+186 A corollary of Theorem 3.4 is a bound on the mixing time of the sampling algorithm:
+
+187 Corollary 3.5. (Upper Bound of Mixing Time: General Case) Under the same assumption and   
+188 with the same notation of Theorem 3.3, we have
+
+$$
+\tau _ { \mathrm { m i x } } ( \epsilon ; W _ { 2 } ; A ) \leq \operatorname* { m a x } \left\{ \frac { 1 } { \beta h _ { 1 } } , \frac { 1 } { \beta } \left( \frac { 2 C } { \epsilon } \right) ^ { \frac { 1 } { p _ { 2 } - \frac { 1 } { 2 } } } \right\} \log \frac { 2 \sqrt { 2 } W _ { 2 } ( L a w ( x _ { 0 } ) \mu ) } { \epsilon }
+$$
+
+In particular, when high accuracy is needed, i.e.,  < 2Chp2− 121189 , we have
+
+$$
+\tau _ { \mathrm { m i x } } ( \epsilon ; W _ { 2 } ; A ) \leq \frac { ( 2 C ) ^ { \frac { 1 } { p _ { 2 } - \frac { 1 } { 2 } } } } { \beta } \frac { 1 } { \epsilon ^ { \frac { 1 } { p _ { 2 } - \frac { 1 } { 2 } } } } \log \frac { 2 \sqrt { 2 } W _ { 2 } ( L a w ( x _ { 0 } ) , \mu ) } { \epsilon } = \widetilde { \mathcal { O } } \left( \frac { C ^ { \frac { 1 } { p _ { 2 } - \frac { 1 } { 2 } } } } { \beta } \frac { 1 } { \epsilon ^ { \frac { 1 } { p _ { 2 } - \frac { 1 } { 2 } } } } \right)
+$$
+
+190 Corollary 3.5 states how mixing time depends on the order of local (strong) error (i.e., $p _ { 2 } ^ { \phantom { } } ,$ ) of a   
+191 numerical algorithm. The larger $p _ { 2 }$ is, the shorter the mixing time of the algorithm is, in term of   
+192 the dependence on accuracy tolerance parameter $\epsilon$ . It is important to note that for constant stepsize   
+193 discretizations that are deterministic on the filtration of the driving Brownian motion and use only its   
+194 increments, there is a strong order barrier, namely $p _ { 2 } \leq 1 . 5$ (Rüemelin, 1982); however, methods   
+195 involving multiple stochastic integrals (e.g., Kloeden and Platen (1992); Milstein and Tretyakov   
+196 (2013)) and randomization (e.g., Shen and Lee (2019)) can yield a larger $p _ { 2 }$ .   
+197 The constant $C$ defined in Eq. (7) typically contains rich information about the underlying SDE, e.g.   
+198 dimension, Lipschitz constant of drift and noise diffusion, and the initial value $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ of the sampling   
+199 algorithm. Through $C$ , we can uncover the dependence of mixing time bound on various parameters,   
+200 such as the dimension $d$ . This will be exemplified with Langevin Monte Carlo in the next section.
+
+# 201 4 Non-Asymptotic Analysis of Langevin Monte Carlo Algorithm
+
+202 This section quantifies how LMC samples from Gibbs target distribution $\mu \sim \exp \left( - f ( \pmb { x } ) \right)$ that has   
+203 a finite second moment, i.e., $\begin{array} { r } { \int _ { \mathbb { R } ^ { d } } \left\| \pmb { x } \right\| ^ { 2 } d \mu < \infty } \end{array}$ . Assume without loss of generality that the origin is   
+204 a local minimizer of $f$ , i.e. $\nabla f ( \mathbf { 0 } ) = \mathbf { 0 }$ ; this is for notational convenience in the analysis and can   
+205 be realized via a simple coordinate shift, and it is not needed in the practical implementation. In   
+206 addition, we assume the following two conditions hold:
+
+A 1. (Smoothness and Strong Convexity) Assume 7 $f \in { \mathcal { C } } ^ { 2 }$ and is $L$ -smooth and $m$ -strongly-convex, i.e. there exists 08 $0 < m \le L$ such that $m \bar { I } _ { d } \preccurlyeq \nabla ^ { 2 } f ( \acute { \pmb x } ) \preccurlyeq L I _ { d } , \quad$ $\forall \pmb { x } \in \mathbb { R } ^ { d }$ .
+
+Denote the condition number of 209 the standard assumption in the lit $f$ by atu $\kappa \triangleq { \frac { L } { m } }$ . The smoothness and strong-convexity assumption isalyzing LMC algorithm (Dalalyan, 2017a,b; Cheng and Bartlett, 2018; Durmus et al., 2019a,b).
+
+A 2. (Linear Growth of the 3rd-order derivative) Assume $f \in \mathcal { C } ^ { 3 }$ and the operator $\nabla ( \Delta f )$ grows at most linearly, i.e., there exists a constant $G > 0$ such that $\begin{array} { r } { \left\| \nabla ( \Delta f ( \pmb { x } ) ) \right\| \leq G \left( 1 + \| \pmb { x } \| \right) } \end{array}$ .
+
+Remark. The linear growth (at infinity) condition on $\nabla \Delta f$ is actually not as restrictive as it appears, and in some sense even weaker than some classical condition for the existence of solutions to SDE. For example, a standard condition for ensuring the existence and uniqueness of a global solution to SDE is at most a linear growth (at infinity) of the drift (Pavliotis, 2014, Theorem 3.1). If we consider monomial potentials, i.e., $f ( x ) = x ^ { p } , p \in \mathbb { N } _ { + }$ , then the linear growth condition on $\nabla \Delta f$ is met when $p \leq 4$ , whereas the classical condition for the existence of solutions holds only when $p \leq 2$ .
+
+220 To apply mean-square analysis to study LMC algorithm, we will need to ensure the underlying   
+221 Langevin dynamics is contractive, which we verify in Section C and D in the appendix. In addition,   
+222 we work out all required constants to determine the $C$ in Eq. 7 explicitly in the appendix. With all   
+223 these necessary ingredients, we now invoke Theorem 3.4 and obtain the following result:
+
+224 Theorem 4.1. (Non-Asymptotic Error Bound: LMC) Suppose Assumption √ $\cdot$ and 2 hold. LMC iteration 225 $\bar { \pmb { x } } _ { k + 1 } = \bar { \pmb { x } } _ { k } - h \nabla f ( \bar { \pmb { x } } _ { k } ) + \sqrt { 2 h } \xi _ { k }$ satisfies
+
+$$
+\begin{array} { l } { { \displaystyle { W _ { 2 } } \big ( L a w ( \bar { x } _ { k } ) , \mu \big ) \le \sqrt 2 e ^ { - m k h } W _ { 2 } \big ( L a w ( x _ { 0 } ) , \mu \big ) + \sqrt 2 C _ { L M C } h , \quad 0 < h \le \frac 1 { 4 \kappa L } , k \in \mathbb N } } \\ { ~ } \\ { \displaystyle { { e ^ { } C _ { L M C } } = \frac { 1 0 ( L ^ { 2 } + G ) } { m ^ { \frac 3 2 } } \sqrt { 2 d + m \left( \left. x _ { 0 } \right. ^ { 2 } + 1 \right) } = \mathcal { O } ( \sqrt d ) . } } \end{array}
+$$
+
+27 Corollary 3.5 combined with the above result gives the following bound on the mixing time of LMC:
+
+28 Theorem 4.2. (Upper Bound of Mixing Time: LMC) Suppose Assumption $I$ and 2 hold. If running   
+29 LMC from $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ , we then have
+
+$$
+\tau _ { \operatorname* { m i x } } ( \epsilon ; W _ { 2 } ; \mathrm { L M C } ) \le \operatorname* { m a x } \{ 4 \kappa ^ { 2 } , \frac { 2 C _ { L M C } } { m } \frac { 1 } { \epsilon } \} \log \frac { 2 \sqrt { 2 } W _ { 2 } ( L a w ( x _ { 0 } ) , \mu ) } { \epsilon }
+$$
+
+where 230 $C _ { L M C }$ is the same in Theorem 4.1. When high accuracy is needed, i.e., $\begin{array} { r } { \epsilon \leq \frac { C _ { L M C } } { 2 m \kappa ^ { 2 } } } \end{array}$ CLMC 2mκ2 , we have
+
+$$
+\tau _ { \mathrm { m i x } } ( \epsilon ; W _ { 2 } ; \mathrm { L M C } ) \leq \frac { 2 C _ { L M C } } { m } \frac { 1 } { \epsilon } \log \frac { 2 \sqrt { 2 } W _ { 2 } ( L a w ( { \pmb x } _ { 0 } ) , \mu ) } { \epsilon } = \widetilde { \mathcal { O } } \left( \frac { \sqrt { d } } { \epsilon } \right) .
+$$
+
+231 The $\begin{array} { r } { \widetilde { \mathcal { O } } \left( \frac { \sqrt { d } } { \epsilon } \right) } \end{array}$ mixing time bound in 2-Wasserstein distance improves upon the previous ones   
+232 (Dalalyan, 2017a; Cheng and Bartlett, 2018; Durmus et al., 2019b,a) in the dependence of √ $d$ and/or $\epsilon$ .   
+233 If further assuming $G = \mathcal { O } ( L ^ { 2 } )$ , we then have $C _ { \mathrm { L M C } } = \mathcal { O } ( \kappa ^ { 2 } \sqrt { m } \sqrt { d } )$ and Thm.4.2 shows the mixing   
+234 time is $\widetilde { \mathcal { O } } \left( \frac { \kappa ^ { 2 } } { \sqrt { m } } \frac { \sqrt { d } } { \epsilon } \right)$ , which also improves the $\kappa$ dependence in some previous results (Dalalyan,   
+235 $2 0 1 7 \mathrm { a }$ ; Cheng and Bartlett, 2018) in the regime $m \leq 1$ . A brief comparison is summarized in Table 1.   
+236 Optimality In fact, the $\begin{array} { r } { \widetilde { \mathcal { O } } \left( \frac { \sqrt { d } } { \epsilon } \right) } \end{array}$ mixing time of LMC has the optimal scaling one can expect. This   
+237 is in terms of the dependence on $d$ and $\epsilon$ , over the class of all log-smooth and log-strongly-convex   
+238 target measures. To illustrate this, consider the following Gaussian target distribution whose potential   
+239 is
+
+$$
+f ( \pmb { x } ) = \frac { m } { 2 } \sum _ { i = 1 } ^ { d } x _ { i } ^ { 2 } + \frac { L } { 2 } \sum _ { i = d + 1 } ^ { 2 d } x _ { i } ^ { 2 } , \quad \mathrm { { w i t h } } m = 1 , L \geq 4 m .
+$$
+
+240 We now establish a lower bound on the mixing time of LMC algorithm for this target measure.
+
+241 Theorem 4.3. (Lower Bound of Mixing Time) Suppose we run LMC for the target measure defined   
+242 in Eq. (11) from ${ \pmb x } _ { 0 } = { \bf 1 } _ { 2 d }$ , then for any choice of step size $h > 0$ within stability limit, we have
+
+$$
+\tau _ { \mathrm { m i x } } ( \epsilon ; W _ { 2 } ; \mathrm { L M C } ) \geq \frac { \sqrt { d } } { 8 \epsilon } \log \frac { \sqrt { d } } { \epsilon } = \widetilde \Omega \left( \frac { \sqrt { d } } { \epsilon } \right) .
+$$
+
+Table 1: Comparison of mixing time results in 2-Wassertein distance of LMC with $L$ -smooth and $m$ -strongly-convex potential. Constant step size is used and accuracy tolerance $\epsilon$ is small enough.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>mixing time</td><td rowspan=1 colspan=1>Additional Assumption</td></tr><tr><td rowspan=1 colspan=1>(Dalalyan,2017a, Theorem 1)</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>K²</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>N/A</td></tr><tr><td rowspan=1 colspan=1>(Cheng and Bartlett, 2018, Theorem 1)</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>N/A</td></tr><tr><td rowspan=1 colspan=1>(Durmus et al., 2019a, Corollary 10)</td><td rowspan=1 colspan=3>8（）</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td rowspan=1 colspan=1>(Durmus et al., 2019b, Theorem 8)</td><td rowspan=1 colspan=3></td><td rowspan=1 colspan=1>|/V²f(x)-V²f(y)||≤L|x- yll</td></tr><tr><td rowspan=1 colspan=1>This work (Theorem 4.2)</td><td rowspan=1 colspan=3>6Em</td><td rowspan=1 colspan=1>Assumption 2 and G = O(L²)2</td></tr></table>
+
+243 Combining Theorem 4.2 and 4.3, we see that mean-square analysis provides a tight bound for LMC.
+
+244 However, there is one limitation of our result – Assumption 2, which is, although mild, still extra to the   
+245 standard setup. Therefore, the gap between the upper bound and the lower bound of LMC algorithm   
+246 over the entire family of log-smooth and log-strongly-convex target measures is not completely   
+247 closed. We tend to believe that Assumption 2 may not be essential, but rather than an artifact of our   
+248 proof technique. We hope to lift this restriction in future work.   
+249 Comparison At least two sampling algorithms are closely related to LMC. One is Kinetic Langevin   
+250 Monte Carlo algorithm (KLMC), which is discretized kinetic/underdamped Langevin dynamics, and   
+251 the other is Metropolis-Adjusted Langevin Algorithm (MALA) which uses the one-step update of   
+252 LMC as a proposal and then accepts/rejects them with a Metropolis-Hastings algorithm.
+
+The $\begin{array} { r } { \widetilde { \mathcal { O } } \left( \frac { \sqrt { d } } { \epsilon } \right) } \end{array}$ mixing time in 2-Wasserstein distance of KLMC has been established for log-smooth and log-strongly-convex target measures in existing literature (Cheng et al., 2018b; Dalalyan and Riou-Durand, 2020). Due to its better dimension dependence over previously best known results of LMC, KLMC is understood to be the analog of Nesterov’s accelerated gradient method for sampling (Ma et al., 2021). Our findings show that LMC is able to achieve the same mixing time, albeit under an additional growth-at-infinity condition. However, this does not say anything about whether/how KLMC accelerates LMC, as the optimality of KLMC bound is not yet clear. We also note KLMC has better condition number dependence, although the $\kappa$ dependence in our bound may not be tight.
+
+261 In terms of MALA, a recent work (Chewi et al., 2020) establishes a $\widetilde { \mathcal { O } } \left( \sqrt { d } \right)$ mixing time in 2-   
+262 Wasserstein distance with warm start, and the dimension dependence is shown to be optimal. We see   
+263 that without the Metropolis adjustment, LMC can also achieve the optimal dimension dependence as   
+264 MALA. But unlike LMC, MALA only has logarithmic dependence on $\frac { 1 } { \epsilon }$ . Under warm-start condition,   
+265 is it possible/how to improve the dependence of $\frac { 1 } { \epsilon }$ for LMC, from polynomial to logarithmic? This   
+266 question is beyond the scope of this paper but worth further investigation.
+
+# 267 5 Numerical Examples
+
+268 This section numerically verifies our theoretical findings for LMC in Section 4, with a particular   
+269 focus on the dependence of the discretization error in Theorem 4.1 on dimension $d$ and step size $h$ .   
+270 To this end, we consider two target measures specified by the following two potentials:
+
+$$
+f _ { 1 } ( \boldsymbol { x } ) = \frac { 1 } { 2 } \left\| \boldsymbol { x } \right\| ^ { 2 } + \log \left( \sum _ { i = 1 } ^ { d } e ^ { x _ { i } } \right) \quad \mathrm { a n d } \quad f _ { 2 } ( \boldsymbol { x } ) = \frac { 1 } { 2 } \left\| \boldsymbol { x } \right\| ^ { 2 } - \frac { 1 } { 2 d ^ { \frac { 1 } { 2 } } } \sum _ { i = 1 } ^ { d } \cos \left( d ^ { \frac { 1 } { 4 } } \boldsymbol { x } _ { i } \right) .
+$$
+
+It is not hard to see that 271 $f _ { 1 }$ is 2-smooth and 1-strongly convex, $f _ { 2 }$ is 3 -smooth and 1-strongly272 convex. $f _ { 2 }$ is also used in (Chewi et al., 2020) to illustrate the optimal dimension dependence 273 of MALA. Explicit expression of 2-Wasserstein distance between non-Gaussian distributions is
+
+274 typically not available, instead, we use the Euclidean norm of the mean error as a surrogate because   
+275 $\bigl | \bigl | \hat { \mathbb { E } } \bar { \pmb { x } } _ { k } - \mathbb { E } _ { \mu } \pmb { x } \bigr | \bigr | \leq W _ { 2 } \bigl ( \mathrm { L a w } ( \bar { \pmb { x } } _ { k } ) , \mu \bigr )$ due to Jensen’s inequality. To obtain an accurate estimate of the   
+276 ground truth, we run $1 0 ^ { 8 }$ independent LMC realizations using a tiny step size $\mathrm { { h } = 0 . 0 0 1 }$ ), each till a   
+277 fixed, long enough time, and use the empirical average to approximate $\mathbb { E } _ { \mu } \pmb { x }$ .   
+278 To study the dimension dependence of sampling error, we fix step size $h = 0 . 1$ , and for each   
+279 $d \in \{ 1 , 2 , 5 , 1 0 , 2 0 , 5 0 , 1 0 0 , 2 0 0 , 5 0 0 , 1 0 0 0 \}$ , we simulate $1 0 ^ { 4 }$ independent Markov chains using   
+280 LMC algorithm for 100 iterations, which is long enough for the chain to be well-mixed. The mean   
+281 and the standard deviation of the sampling error corresponding to the last 10 iterates are recorded.   
+282 To study step size dependence of sampling error, dimension is fixed to be $d = 1 0$ . We experiment with   
+283 step size $h \in \{ 1 , 2 , \bar { 3 } , 4 , 5 , 6 , 7 , 8 , 9 , \bar { 1 0 } \} \times 1 0 ^ { - 1 }$ . We fix a continuous time $T = 2 0$ , and run LMC   
+284 algorithm for $\left\lceil { \frac { T } { h } } \right\rceil$ iterations for each $h$ . The procedure is repeated $1 0 ^ { 4 }$ times with different random   
+285 seeds to obtain independent samples. When the corresponding continuous time $t = k h > 1 0$ , we see   
+286 from Eq. (10) that LMC is well converged and the sampling error is saturated by the discretization   
+287 error. Therefore, for each $h$ , we take the last $\lceil \frac { 1 0 } { h } \rceil$ iterates and record the mean and standard deviation   
+288 of their sampling error.   
+289 The experiment results shown in Figure 1 are consistent with our theoretical analysis of the sampling√   
+290 error.Both linear dependence on $\sqrt { d }$ and $h$ can be identified in and supported by the empirical   
+291 evidence. Note results with smaller $h$ are less accurate because one starts to see the error of empirical   
+292 approximation due to finite samples. Experiments were conducted on a machine with a 2.20GHz   
+293 Intel(R) Xeon(R) E5-2630 v4 CPU and an Nvidia GeForce GTX 1080 GPU.
+
+![](images/3259972bc31cdc763783ea24599784e0958d3e4f5af33daba50f010695753e2c.jpg)  
+Figure 1: (a) Dependence of the sampling error of LMC on dimension $d$ and step size $h$ for $f _ { 1 }$ and $f _ { 2 }$ Both axes in Figure 1a and 1b are in log scale. The shaded areas in Figure 1a and 1b represent one standard deviation of the last 10 iterates. The shaded areas in Figure 1c and 1d represent one standard deviation of the last $\lceil \frac { 1 0 } { h } \rceil$ iterations.
+
+# 6 Conclusion
+
+This paper extends the mean-square analysis framework for analyzing the integration error of SDE to analyzing the sampling error in 2-Wasserstein distance. Corresponding mixing time bound unveils how a high-order numerical algorithm can help improve dependence on accuracy tolerance , and potentially other parameters, such as the dimension. When applied to Langevin Monte Carlo algorithm, it obtains an improved and optimal $\widetilde { \mathcal { O } } \left( \sqrt { d } / \epsilon \right)$ bound, which was previously thought to be obtainable only with the addition of momentum.
+
+01 Here are some possible directions worth further investigations. (i) In data-intensive applications,   
+02 stochastic gradients are typically used for better scalability. It seems natural to apply the mean-square   
+03 analysis framework to study SDE-basd stochastic gradient MCMC methods; (ii) Assumption 2 is   
+04 likely to be an artifact of our analysis; how to establish the optimal mixing time bound in the standard   
+05 log-smooth and log-strongly-convex setup is still an open question; (iii) Motivated by the recent   
+06 result of MALA (Chewi et al., 2020), it would be interesting to know whether the dependence on $\frac { 1 } { \epsilon }$   
+07 can be improved to logarithmic, for example if LMC is initialized at a warm start.
+
+# References
+
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] Limitation is discussed in the paragraph below Table 1.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] To the best of our knowledge, this work does not have any potential negative societal impact.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] Assumptions are fully, clearly stated in theorems.   
+(b) Did you include complete proofs of all theoretical results? [Yes] Complete proofs of all theoretical results are provided in supplementary materials.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] All are provided in supplementary materials.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Our experiments do not need training. We specify all the hyperparameters needed to run the experiments.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Figure 1.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We report the hardware used to run our experiments.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [N/A]   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
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+    {
+        "type": "text",
+        "text": "Mean-Square Analysis with An Application to Optimal Dimension Dependence of Langevin Monte Carlo ",
+        "text_level": 1,
+        "bbox": [
+            165,
+            122,
+            823,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            226,
+            580,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+            318,
+            535,
+            334
+        ],
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+    },
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+        "type": "text",
+        "text": "1 Sampling algorithms based on discretizations of Stochastic Differential Equations   \n2 (SDEs) compose a rich and popular subset of MCMC methods. This work pro  \n3 vides a general framework for the non-asymptotic analysis of sampling error in   \n4 2-Wasserstein distance, which also leads to a bound of mixing time. The method   \n5 applies to any consistent discretization of contractive SDEs. When applied to   \n6 Langevin Monte Carlo algorithm, it establishes $\\widetilde { \\mathcal { O } } \\left( \\sqrt { d } / \\epsilon \\right)$ mixing time, without   \n7 warm start, under the common log-smooth and log-strongly-convex conditions,   \n8 plus a growth condition on the potential of target measures at infinity. This bound   \n9 improves the best previously known $\\widetilde { \\mathcal { O } } \\left( { d } / { \\epsilon } \\right)$ result and is optimal in both dimension   \n10 $d$ and accuracy tolerance $\\epsilon$ for log-smooth and log-strongly-convex target measures.   \n11 Our theoretical analysis is further validated by numerical experiments. ",
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "12 1 Introduction ",
+        "text_level": 1,
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+            148,
+            535,
+            312,
+            551
+        ],
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+    },
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+        "text": "13 The problem of sampling statistical distributions has attracted considerable attention, not only in   \n14 the fields of statistics and scientific computing, but also in machine learning (Robert and Casella,   \n15 2013; Andrieu et al., 2003; Liu, 2008); for example, how various sampling algorithms scale with   \n16 the dimension of the target distribution is a popular recent topic in statistical deep learning (see   \n17 discussions below for references). For samplers that can be viewed as discretizations of SDEs, the   \n18 idea is to use an ergodic SDE whose equilibrium distribution agrees with the target distribution,   \n19 and employ an appropriate numerical algorithm that discretizes (the time of) the SDE. The iterates   \n20 of the numerical algorithm will approximately follow the target distribution when converged, and   \n21 can be used for various downstream applications such as Bayesian inference and inverse problem   \n22 (Dashti and Stuart, 2017). One notable example is the Langevin Monte Carlo algorithm (LMC),   \n23 which corresponds to Euler-Maruyama discretization of overdamped Langevin equation. Its study   \n24 dated back to at least the 90s (Roberts et al., 1996) but keeps on leading to important discoveries, for   \n25 example, on non-asymptotics and dimension dependence, which are relevant to machine learning   \n26 (e.g., Dalalyan (2017a,b); Cheng et al. (2018a); Durmus et al. (2019a,b); Vempala and Wibisono   \n27 (2019); Dalalyan and Riou-Durand (2020); Erdogdu and Hosseinzadeh (2020); Mou et al. (2019)).   \n28 LMC is closely related to SGD too (e.g., Mandt et al. (2017)). Many other examples exist, based   \n29 on alternative SDEs and different discretizations (e.g., Dalalyan and Riou-Durand (2020); Ma et al.   \n30 (2021); Mou et al. (2021); Li et al. (2020); Roberts and Rosenthal (1998); Chewi et al. (2020); Shen   \n31 and Lee (2019)).   \n32 Quantitatively characterizing the non-asymptotic sampling error of numerical algorithms is usually   \n33 critical for choosing the appropriate algorithm for a specific downstream application, for providing   \n34 practical guidance on hyperparameter selection and experiment design, and for designing improved   \n35 samplers. A powerful tool that dates back to (Jordan et al., 1998) is a paradigm of non-asymptotic   \n36 error analysis, namely to view sampling as optimization in probability space, and it led to many   \n37 important recent results (e.g., Liu and Wang (2016); Dalalyan (2017a); Wibisono (2018); Zhang   \n38 et al. (2018); Frogner and Poggio (2020); Chizat and Bach (2018); Chen et al. (2018); Ma et al.   \n39 (2021); Erdogdu and Hosseinzadeh (2020)). It works by choosing an objective functional, typically   \n40 some statistical distances/diverges, and showing that the law of the iterates of sampling algorithms   \n41 converges in that objective functional. However, the choice of the objective functional often needs to   \n42 be customized for different sampling algorithms. For example, KL divergence works for LMC (Cheng   \n43 and Bartlett, 2018), but a carefully hand-crafted cross term needs to be added to KL divergence for   \n44 analyzing KLMC (Ma et al., 2021). Even for the same underlying SDE, different discretization   \n45 schemes exist and lead to different sampling algorithms, and the analyses of them had usually been   \n46 case by case (e.g., Cheng et al. (2018b); Dalalyan and Riou-Durand (2020); Shen and Lee (2019)).   \n47 Therefore, it would be a desirable complement to have a unified, general framework to study the   \n48 non-asymptotic error of SDE-based sampling algorithms.   \n49 As an important member of the family of SDE-based sampling algorithms, Langevin Monte Carlo is   \n50 widely used in practice. Its stochastic gradient version is implemented in common machine learning   \n51 systems, such as Tensorflow (Abadi et al., 2016), and is the off-the-shelf algorithm for large scale   \n52 Bayesian inference. With the ever-growing size of parameter space, the non-asymptotic error of LMC   \n53 is of central theoretical and practical interest, in particular, its dependence on the dimension of the   \n54 sample space. The best current known upper bound of the mixing time in 2-Wasserstein distance for   \n55 LMC is $\\begin{array} { r } { \\bar { \\mathcal { O } } \\left( \\frac { d } { \\epsilon } \\right) } \\end{array}$ (Durmus et al., 2019b). Motivated by a recent result (Chewi et al., 2020) that shows   \n56 better dimension dependence for a Metropolis-Adjusted improvement of LMC, we wonder if the   \n57 current bound for (unadjusted) LMC is tight, and if not, what is the optimal dimension dependence?   \n58 Our contribution We study a broad family of numerical algorithms that discretize SDEs that   \n59 have a contraction property (possibly after a coordinate transformation). For this type of problems,   \n60 we revisit the classical mean-square analysis (Milstein and Tretyakov, 2013) in numerical SDE   \n61 literature and extend its the global error bound from finite time to infinite time. Same as in classical   \n62 mean-square analysis, we show the global error is only half order lower than the order of local strong   \n63 error $\\left( p _ { 2 } \\right)$ . We further obtain a $\\tilde { \\mathcal { O } } \\left( C ^ { \\frac { 1 } { p _ { 2 } - \\frac { 1 } { 2 } } } \\underbrace { \\frac { 1 } { \\epsilon ^ { \\frac { 1 } { p _ { 2 } - \\frac { 1 } { 2 } } } } } \\right)$ mixing time upper bound in 2-Wasserstein   \n64 distance for the family of algorithms, where $C$ is a constant containing various information of the   \n65 underlying problem, e.g., the dimension $d$ .   \n66 As an application of the general mixing time result, we study the widely used Langevin Monte   \n67 Carlo algorithm (LMC) for sampling from a Gibbs distribution $\\mu \\propto \\mathrm { e x p } \\left( - f ( \\pmb { x } ) \\right)$ , which is an   \n68 Euler-Maruyama discretization of Langevin dynamics. Under the standard smoothness and strong  \n69 convexity assumptions, plus an additional linear growth condition on the third-order derivative of $f$ ,   \n70 we obtain a $\\begin{array} { r } { \\widetilde { \\mathcal { O } } \\left( \\frac { \\sqrt { d } } { \\epsilon } \\right) } \\end{array}$ mixing time in 2-Wasserstein distance, which improves upon the previously best   \n71 known $\\begin{array} { r } { \\widetilde { \\mathcal { O } } \\left( \\frac { d } { \\epsilon } \\right) } \\end{array}$ result (Durmus et al., 2019b). For a comparison, note it was known that discretized   \n72 kinetic Langevin dynamics can lead to $\\sqrt { d }$ dependence on dimension (Cheng and Bartlett, 2018;   \n73 Dalalyan and Riou-Durand, 2020) and some believe that it is the introduction of momentum that   \n74 improves the dimension dependence, but our result shows that discretized overdamped Langevin (no√   \n75 momentum) can also have mixing time scaling like $\\sqrt { d }$ . In fact, it is important to mention that it was   \n76 recently shown that Metropolis-Adjusted Euler-Maruyama discretization of overdamped Langevin   \n77 (i.e., MALA) has an optimal dimension dependence of $\\widetilde { \\mathcal { O } } \\left( \\sqrt { d } \\right)$ (Chewi et al., 2020), while what we   \n78 analyze here is the unadjusted version (i.e., LMC), and it has the same dimension dependence (note   \n79 however that our $\\epsilon$ dependence is not as good as that for MALA; more discussion in Section 4). We   \n80 also constructed an example that shows that the mixing time of LMC is at least $\\begin{array} { r } { \\widetilde \\Omega \\left( \\frac { \\sqrt { d } } { \\epsilon } \\right) } \\end{array}$ . Hence, our   \n81 mixing time bound has the optimal dependence on both $d$ and $\\epsilon$ . Our theoretical analysis is further   \n82 validated by empirical investigation of numerical examples. ",
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+        "text": "83 2 Preliminaries ",
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+        "text": "84 Notation Use the symbol $_ { x }$ to denote a $d$ -dimensional vector, and the plain symbol $x$ to denote a   \n85 scalar variable. Use $\\lVert \\boldsymbol { x } \\rVert$ to denote the Euclidean norm of vector $_ { x }$ . A numerical algorithm is denoted   \n86 by $\\mathcal { A }$ and its $k$ -th iterate is denoted by $\\bar { \\pmb { x } } _ { k }$ . We slightly abuse notation by identifying measures with   \n87 their density function w.r.t. Lebesgue measure. We use the convention $\\begin{array} { r } { \\tilde { \\mathcal { O } } \\left( \\cdot \\right) = \\mathbf { \\bar { \\mathcal { O } } } ( \\cdot ) \\log ^ { \\mathcal { O } ( 1 ) } ( \\cdot ) } \\end{array}$   \n88 i.e., the $\\widetilde { \\mathcal { O } } \\left( \\cdot \\right)$ notation ignores the dependence on logarithmic factors. We use the notation $\\widetilde { \\Omega } ( \\cdot )$   \n89 similarly. Denote 2-Wasserstein distance by $\\begin{array} { r } { W _ { 2 } ( \\mu _ { 1 } , \\mu _ { 2 } ) = \\left( \\operatorname* { i n f } _ { ( \\pmb { X } , \\pmb { Y } ) \\sim \\Pi ( \\mu _ { 1 } , \\mu _ { 2 } ) } \\mathbb { E } \\left\\| \\pmb { X } - \\pmb { Y } \\right\\| ^ { 2 } \\right) ^ { \\frac { 1 } { 2 } } } \\end{array}$ ,   \n90 where $\\Pi ( \\mu _ { 1 } , \\mu _ { 2 } )$ is the set of couplings, i.e. all joint measures with $X$ and $Y$ marginals being   \n91 $\\mu _ { 1 }$ and $\\mu _ { 2 }$ . Denote the target distribution by $\\mu$ and the law of a random variable $\\boldsymbol { X }$ by $\\operatorname { L a w } ( X )$ .   \n92 Finally, denote the mixing time of an sampling algorithm $\\mathcal { A }$ converging to its target distribution $\\mu$ in   \n93 2-Wasserstein distance by $\\begin{array} { r } { \\tau _ { \\operatorname* { m i x } } ( \\epsilon ; W _ { 2 } ; A ) = \\operatorname* { i n f } \\{ k \\geq 0 | W _ { 2 } ( \\operatorname { L a w } ( \\bar { x } _ { k } ) , \\mu ) \\leq \\epsilon \\} . } \\end{array}$ . ",
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+        "text": "94 SDE for Sampling Consider a general SDE ",
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+        "text": "$$\nd { \\pmb x } _ { t } = { \\pmb b } ( t , { \\pmb x } _ { t } ) d t + { \\pmb \\sigma } ( t , { \\pmb x } _ { t } ) d { \\pmb B } _ { t }\n$$",
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+        "text": "95 where $\\pmb { b } \\in \\mathbb { R } ^ { d }$ is a drift term, $\\pmb { \\sigma } \\in \\mathbb { R } ^ { d \\times l }$ is a diffusion coefficient matrix and $\\mathbf { \\delta } _ { B _ { t } }$ is a $l$ -dimensional   \n96 Wiener process. Under mild condition (Pavliotis, 2014, Theorem 3.1), there exists a unique strong   \n97 solution $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ to Eq. (1). Some SDEs admit geometric ergodicity, so that their solutions converge   \n98 exponentially fast to a unique invariant distribution, and examples include the classical overdamped   \n99 and kinetic Langevin dynamics, but are not limited to those (e.g., Mou et al. (2021); Li et al. (2020)).   \n100 Such SDE are desired for sampling purposes, because one can set the target distribution to be the   \n101 invariant distribution by choosing an SDE with an appropriate potential, and then solve the solution   \n102 $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ of the SDE and push the time $t$ to infinity, so that (approximate) samples of the target distribution   \n103 can be obtained. Except for a few known cases, however, explicit solutions of Eq. (1) are elusive and   \n104 we have to resort to numerical schemes to simulate/integrate SDE. Such example schemes include,   \n105 but are not limited to Euler-Maruyama method, Milstein methods and Runge-Kutta method (e.g.,   \n106 Kloeden and Platen (1992); Milstein and Tretyakov (2013)). With constant stepsize $h$ and at $k$ -th   \n107 iteration, a typical numerical algorithm takes a previous iterate $\\bar { \\pmb { x } } _ { k - 1 }$ and outputs a new iterate $\\bar { \\pmb { x } } _ { k }$ as   \n108 an approximation of the solution $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ of Eq. (1) at time $t = k h$ . ",
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+        "type": "text",
+        "text": "109 Langevin Monte Carlo Algorithm LMC algorithm is defined by the following update rule ",
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+        "text": "$$\n\\bar { \\pmb { x } } _ { k } = \\bar { \\pmb { x } } _ { k - 1 } - h \\nabla f ( \\bar { \\pmb { x } } _ { k - 1 } ) + \\sqrt { 2 h } \\pmb { \\xi } _ { k } , \\quad k = 1 , 2 , \\cdots\n$$",
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+        "text": "where $\\{ \\pmb { \\xi } _ { k } \\} _ { k \\in \\mathbb { Z } _ { > 0 } }$ are i.i.d. standard $d$ -dimensional Gaussian vectors. LMC corresponds to an EulerMaruyama discretization of the continuous overdamped Langevin dynamics $d { \\pmb x } _ { t } = - \\nabla f ( { \\pmb x } _ { t } ) d t +$ $\\sqrt { 2 } d B _ { t }$ , which converges to an equilibrium distribution $\\mu \\sim \\exp ( - f ( \\pmb { x } ) )$ . ",
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+        "text": "13 Dalalyan (2017b) provided a non-asymptotic analysis of LMC. An $\\begin{array} { r } { \\widetilde { \\mathcal { O } } \\left( \\frac { d } { \\epsilon ^ { 2 } } \\right) } \\end{array}$ mixing time bound in   \n14 $W _ { 2 }$ for log-smooth and log-strongly-convex target measures (Dalalyan, 2017a; Cheng et al., 2018a;   \n15 Durmus et al., 2019a) has been established. It was further improved to $\\begin{array} { r } { \\widetilde { \\mathcal { O } } \\left( \\frac { d } { \\epsilon } \\right) } \\end{array}$ under additional   \n16 Lipschitz assumption on the Hessian of $f$ (Durmus et al., 2019b). Mixing time bounds of LMC   \n17 in other statistical distances/divergences have also been studied, including total variation distance   \n118 (Dalalyan, 2017b; Durmus et al., 2017) and KL divergence (Cheng and Bartlett, 2018).   \n119 Classical Mean-Square Analysis A powerful framework for quantifying the global discretization   \n120 error of a numerical algorithm for Eq. (1), i.e., $e _ { k } = \\left\\{ \\mathbb { E } \\left\\| \\pmb { x } _ { k h } - \\bar { \\pmb { x } } _ { k } \\right\\| \\right\\} ^ { \\frac { 1 } { 2 } }$ , is mean-square analysis   \n121 (e.g., the monograph of Milstein and Tretyakov (2013)). Mean-square analysis studies how local   \n122 integration error propagate and accumulate into global integration error; in particular, if one-step   \n123 (local) weak error and strong error (both the exact solution $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ and the numerical approximation start   \n124 from the same initial value $_ { \\textbf { \\em x } }$ ) satisfy ",
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+        "text": "$$\n\\begin{array} { r l } & { \\left\\| \\mathbb { E } x _ { h } - \\mathbb { E } \\bar { { \\bar { x } } } _ { 1 } \\right\\| \\leq C _ { 1 } \\left( 1 + \\mathbb { E } \\left\\| x \\right\\| ^ { 2 } \\right) ^ { \\frac { 1 } { 2 } } h ^ { p _ { 1 } } , \\quad \\mathrm { ( l o c a l ~ w e a k ~ e r r o r ) } } \\\\ & { \\left( \\mathbb { E } \\left\\| x _ { h } - \\bar { x } _ { 1 } \\right\\| ^ { 2 } \\right) ^ { \\frac { 1 } { 2 } } \\leq C _ { 2 } \\left( 1 + \\mathbb { E } \\left\\| x \\right\\| ^ { 2 } \\right) ^ { \\frac { 1 } { 2 } } h ^ { p _ { 2 } } , \\quad \\mathrm { ( l o c a l ~ s t r o n g ~ e r r o r ) } } \\end{array}\n$$",
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+        "text": "over a time interval 125 $[ 0 , K h ]$ for some constants $C _ { 1 } , C _ { 2 } > 0$ , $p _ { 2 } \\geq { \\frac { 1 } { 2 } }$ and $\\begin{array} { r } { p _ { 1 } \\geq p _ { 2 } + \\frac { 1 } { 2 } } \\end{array}$ , then the global error can be bounded by 126 $\\begin{array} { r } { \\displaystyle { e _ { k } \\leq C \\left( 1 + \\mathbb { E } \\left\\| x _ { 0 } \\right\\| ^ { 2 } \\right) ^ { \\frac { 1 } { 2 } } h ^ { p _ { 2 } - \\frac { 1 } { 2 } } } } \\end{array}$ , $k = 1 , 2 , \\cdots , K$ for some constant 127 $C > 0$ dependent on $K h$ . ",
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+        "text": "128 Although classical mean-square analysis is only concerned with numerical integration error, sampling   \n129 error can be also inferred. However, there is a limitation that prevents directly employing mean-square   \n130 analysis in the non-asymptotic analysis of sampling algorithms. The bound of global error only holds   \n131 in finite time because the constant $C$ can grow exponentially as $K$ increases, rendering the bound   \n132 useless when $K  \\infty$ . ",
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+        "text": "3 Mean-Square Analysis of Samplers Based on Contractive SDE ",
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+        "text": "In order to prepare for the analysis of sampling error, we first show that the finite time limitation of integration error analysis can be lifted if the SDE being discretized is contractive. ",
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+        "text": "136 More precisely, one bottleneck that prevents the results of classical mean-square analysis from   \n137 extending to infinite time horizon, is the fact that the solution of a general SDE may not be bounded,   \n138 and neither is its discretization. Note that local error (Eq. (3)) depends on the initial value. To go   \n139 from local to global error, these ‘initial’ values correspond to iterates of numerical algorithms, which   \n140 change from iteration to iteration and can be unbounded, hence when accumulated together, it is   \n141 possible that the global error may blow up.   \n142 Samplers considered here, on the other hand, are based on stochastic differential equations, each of   \n143 which weakly converges to a limiting distributions. The solution of the underlying converging SDE,   \n144 as it converges to the invariant measure, gradually inherits boundedness properties from the target   \n145 measure. Thus, as long as the target measure has bounded 2nd-moment, a sampling algorithm based   \n146 on a reasonable discretization of the SDE should also have bounded 2nd-moment. Motivated by this   \n147 observation, we will assume the sampling algorithms we study are based on contractive SDEs, which   \n148 is a sufficient condition to ensure the underlying SDE converges to a statistical distribution. ",
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+        "text": "149 Definition 3.1. A stochastic differential equation is contractive if there exists a non-singular constant matrix 150 $A \\in \\mathbb { R } ^ { d \\times d }$ , a constant $\\beta > 0$ , such that any pair of solutions of the SDE satisfy ",
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+        "text": "$$\n\\begin{array} { r } { \\left( \\mathbb { E } \\left\\| A \\left( \\pmb { x } _ { t } - \\pmb { y } _ { t } \\right) \\right\\| ^ { 2 } \\right) ^ { \\frac { 1 } { 2 } } \\leq \\left\\| A \\left( \\pmb { x } - \\pmb { y } \\right) \\right\\| \\exp ( - \\beta t ) , } \\end{array}\n$$",
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+        "text": "151 where $\\mathbf { \\boldsymbol { x } } _ { t } , \\mathbf { \\boldsymbol { y } } _ { t }$ are two solutions, driven by the same Brownian motion but evolved respectively from   \n152 initial conditions $_ { \\textbf { \\em x } }$ and $\\textbf {  { y } }$ .   \n53 Remark. As long as $^ { b }$ and $\\sigma$ in (1) are not explicitly dependent on time, it suffices to find an   \n54 arbitrarily small $t _ { 0 } > 0$ and show (4) holds for all $t < t _ { 0 }$ . ",
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+        "text": "Remark. Sometimes contraction is not easy to establish directly, but can be shown after an appropriate coordinate transformation, see (Dalalyan and Riou-Durand, 2020, Proposition 1) for such a treatment for kinetic Langevin dynamics. The introduction of $A$ permits such transformations. ",
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+        "text": "158 We now use contractivity to remove the finite time limitation. We will first need a lemma, which is a   \n159 local (short time) result.   \n160 Lemma 3.2. (Milstein and Tretyakov, 2013, Lemma 1.3) Suppose $^ { b }$ and $\\pmb { \\sigma }$ in Eq.(1) are Lipschitz   \n161 continuous. For two solutions $\\mathbf { \\boldsymbol { x } } _ { t } , \\mathbf { \\boldsymbol { y } } _ { t }$ of Eq. (1) starting from $\\mathbf { \\nabla } _ { \\mathbf { x } , \\mathbf { y } }$ respectively, denote $z : =$   \n162 $( { \\pmb x } _ { t } - { \\pmb x } ) - ( { \\pmb y } _ { t } - { \\pmb y } )$ , then there exist $C _ { 0 } > 0$ and $h _ { 0 } > 0$ such that ",
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+        "type": "equation",
+        "img_path": "images/502bc98b0f941e127ebc54bfd4ab8af6cf10be50beb0da784fa62cc77a6002e9.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\left\\| z \\right\\| ^ { 2 } \\leq C _ { 0 } \\left\\| x - y \\right\\| ^ { 2 } t , \\quad \\forall x , y , 0 < t \\leq h _ { 0 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "163 Then we will have a sequence of results that connects sampling error (a statistical property) with   \n164 local integration error (a simulation property). This justifies our generic produce for non-asymptotic   \n165 sampling error analysis, which only requires bounding the orders of local weak and strong integration   \n166 errors (in addition to establishing contractivity of the continuous dynamics).   \n167 Theorem 3.3. (Global Integration Error, Infinite Time Version) Suppose Eq.(1) is contractive with   \n168 rate $\\beta$ and with respect to a non-singular matrix $A \\in \\mathbb { R } ^ { d \\times d }$ , with Lipschitz continuous $^ { b }$ and $\\pmb { \\sigma }$   \n169 and there is a numerical algorithm $\\mathcal { A }$ with step size $h$ simulating the solution $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ of the $S D E$ , whose   \n170 iterates are denoted by $\\bar { \\boldsymbol { x } } _ { k } , k = 0 , 1 , \\cdots$ . Suppose there exists $0 < h _ { 0 } \\leq 1 , C _ { 1 } , C _ { 2 } > 0 , D _ { 1 } , D _ { 2 } \\geq$   \n171 $0 , p _ { 1 } \\geq 1 , \\frac { 1 } { 2 } < p _ { 2 } \\leq \\dot { p _ { 1 } } - \\frac { 1 } { 2 }$ such that for any $0 < h \\leq h _ { 0 }$ , the algorithm $\\mathcal { A }$ has, respectively, local   \n172 weak and strong error of order $p _ { 1 }$ and $p _ { 2 }$ , defined as ",
+        "bbox": [
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/79f58d7f4ff7f79206842025d61371e5c0c8e13ca028745690cf349ff5fbe675.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\left\\{ \\left. \\mathbb { E } \\left( { \\pmb x } _ { h } - { \\bar { \\pmb x } } _ { 1 } \\right) \\right. \\right\\} \\leq \\left( C _ { 1 } + D _ { 1 } \\sqrt { \\mathbb { E } \\left. \\pmb x \\right. ^ { 2 } } \\right) h ^ { p _ { 1 } } , } \\\\ & { \\left( \\mathbb { E } \\left. \\pmb x _ { h } - { \\bar { \\pmb x } } _ { 1 } \\right. ^ { 2 } \\right) ^ { \\frac { 1 } { 2 } } \\leq \\left( C _ { 2 } ^ { 2 } + D _ { 2 } ^ { 2 } \\mathbb { E } \\left. \\pmb x \\right. ^ { 2 } \\right) ^ { \\frac { 1 } { 2 } } h ^ { p _ { 2 } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "173 where $\\scriptstyle { \\mathbf { 2 } } ( { \\mathbf { 2 } } \\scriptscriptstyle { h }$ solves Eq.(1) with any initial value $_ { \\textbf { \\em x } }$ and $\\bar { \\mathbf { x } } _ { 1 }$ is the result of applying $\\mathcal { A }$ to $_ { \\textbf { \\em x } }$ for one step.   \n174 If the solution of SDE $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ and algorithm $\\mathcal { A }$ both start from $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ , then for $0 ~ < ~ h ~ \\leq ~ h _ { 1 }$ ,   \n175 $\\begin{array} { r l r } & { } & { \\operatorname* { m i n } \\left\\{ h _ { 0 } , \\frac { 1 } { 4 \\beta } , \\left( \\frac { \\sqrt { \\beta } } { 4 \\sqrt { 2 } \\kappa _ { A } D _ { 2 } } \\right) ^ { \\frac { 1 } { p _ { 2 } - \\frac { 1 } { 2 } } } , \\left( \\frac { \\beta } { 8 \\sqrt { 2 } \\kappa _ { A } \\left( D _ { 1 } + C _ { 0 } D _ { 2 } \\right) } \\right) ^ { \\frac { 1 } { p _ { 2 } - \\frac { 1 } { 2 } } } \\right\\} , t h e ~ g l o b a l ~ e r r } \\\\ & { } & { e _ { k } : = \\left( { \\mathbb E } \\| x _ { k h } - \\bar { x } _ { k } \\| ^ { 2 } \\right) ^ { \\frac { 1 } { 2 } } \\leq C h ^ { p _ { 2 } - \\frac { 1 } { 2 } } , \\quad k = 0 , 1 , 2 , \\cdots } \\end{array}$ r $e _ { k }$ is bounded as   \n176 ",
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+    {
+        "type": "text",
+        "text": "177 where ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/c9367b955513434e5103ae08905143f0f2503b030c945f799cd94edca7c62cd8.jpg",
+        "text": "$$\nC = \\frac { 2 } { \\sqrt { \\beta } } \\kappa _ { A } ^ { 2 } \\left( \\frac { C _ { 1 } + C _ { 0 } C _ { 2 } + \\sqrt { 2 } U ( D _ { 1 } + C _ { 0 } D _ { 2 } ) } { \\sqrt { \\beta } } + C _ { 2 } + \\sqrt { 2 } D _ { 2 } U \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "178 $C _ { 0 }$ is from Eq. (5), $\\kappa _ { A }$ is the condition number of matrix $A$ and $U ^ { 2 } \\triangleq 4 \\left\\| \\pmb { x } _ { 0 } \\right\\| ^ { 2 } + 5 \\mathbb { E } _ { \\mu } \\left\\| \\pmb { x } \\right\\| ^ { 2 } .$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "179 Remark. We use the convention $1 / 0 = \\infty$ when $D _ { 1 } = D _ { 2 } = 0$ . This is pertinent when a numerical   \n180 algorithm $\\mathcal { A }$ , e.g. LMC (Lemma $D . 3$ ), produces bounded iterates. In such cases, the initial value in   \n181 Eq. (6) are iterations of $\\mathcal { A }$ and will be bounded, it then can be absorbed into $C _ { 1 } , C _ { 2 }$ and we may set   \n182 $D _ { 1 } = D _ { 2 } = 0$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "83 Following Theorem 3.3, we obtain the following non-asymptotic bound of the sampling error in $W _ { 2 }$ : ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "84 Theorem 3.4. (Non-Asymptotic Sampling Error Bound: General Case) Under the same assump  \n185 tion and with the same notation of Theorem 3.3, we have ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/afcb3ca80cf0a4a26bbd44dccf992e286b41ee63bf8ff6eb9958d3ca2b6afb27.jpg",
+        "text": "$$\nW _ { 2 } ( L a w ( \\bar { x } _ { k } ) , \\mu ) \\leq \\sqrt { 2 } e ^ { - \\beta k h } W _ { 2 } ( L a w ( x _ { 0 } ) , \\mu ) + \\sqrt { 2 } C h ^ { p _ { 2 } - \\frac { 1 } { 2 } } , \\quad \\forall 0 < h \\leq h _ { 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "186 A corollary of Theorem 3.4 is a bound on the mixing time of the sampling algorithm: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "187 Corollary 3.5. (Upper Bound of Mixing Time: General Case) Under the same assumption and   \n188 with the same notation of Theorem 3.3, we have ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ddd07dea97d6424c0245f93bf9ab36b5a9191c4b064d8a0b993886232fc55507.jpg",
+        "text": "$$\n\\tau _ { \\mathrm { m i x } } ( \\epsilon ; W _ { 2 } ; A ) \\leq \\operatorname* { m a x } \\left\\{ \\frac { 1 } { \\beta h _ { 1 } } , \\frac { 1 } { \\beta } \\left( \\frac { 2 C } { \\epsilon } \\right) ^ { \\frac { 1 } { p _ { 2 } - \\frac { 1 } { 2 } } } \\right\\} \\log \\frac { 2 \\sqrt { 2 } W _ { 2 } ( L a w ( x _ { 0 } ) \\mu ) } { \\epsilon }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In particular, when high accuracy is needed, i.e., \u000f < 2Chp2− 121189 , we have ",
+        "bbox": [
+            142,
+            540,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a43f2644234f715141d271c2214c07ef17e77cdf67b26d060a0840247ec4c92d.jpg",
+        "text": "$$\n\\tau _ { \\mathrm { m i x } } ( \\epsilon ; W _ { 2 } ; A ) \\leq \\frac { ( 2 C ) ^ { \\frac { 1 } { p _ { 2 } - \\frac { 1 } { 2 } } } } { \\beta } \\frac { 1 } { \\epsilon ^ { \\frac { 1 } { p _ { 2 } - \\frac { 1 } { 2 } } } } \\log \\frac { 2 \\sqrt { 2 } W _ { 2 } ( L a w ( x _ { 0 } ) , \\mu ) } { \\epsilon } = \\widetilde { \\mathcal { O } } \\left( \\frac { C ^ { \\frac { 1 } { p _ { 2 } - \\frac { 1 } { 2 } } } } { \\beta } \\frac { 1 } { \\epsilon ^ { \\frac { 1 } { p _ { 2 } - \\frac { 1 } { 2 } } } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "190 Corollary 3.5 states how mixing time depends on the order of local (strong) error (i.e., $p _ { 2 } ^ { \\phantom { } } ,$ ) of a   \n191 numerical algorithm. The larger $p _ { 2 }$ is, the shorter the mixing time of the algorithm is, in term of   \n192 the dependence on accuracy tolerance parameter $\\epsilon$ . It is important to note that for constant stepsize   \n193 discretizations that are deterministic on the filtration of the driving Brownian motion and use only its   \n194 increments, there is a strong order barrier, namely $p _ { 2 } \\leq 1 . 5$ (Rüemelin, 1982); however, methods   \n195 involving multiple stochastic integrals (e.g., Kloeden and Platen (1992); Milstein and Tretyakov   \n196 (2013)) and randomization (e.g., Shen and Lee (2019)) can yield a larger $p _ { 2 }$ .   \n197 The constant $C$ defined in Eq. (7) typically contains rich information about the underlying SDE, e.g.   \n198 dimension, Lipschitz constant of drift and noise diffusion, and the initial value $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ of the sampling   \n199 algorithm. Through $C$ , we can uncover the dependence of mixing time bound on various parameters,   \n200 such as the dimension $d$ . This will be exemplified with Langevin Monte Carlo in the next section. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "201 4 Non-Asymptotic Analysis of Langevin Monte Carlo Algorithm ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "202 This section quantifies how LMC samples from Gibbs target distribution $\\mu \\sim \\exp \\left( - f ( \\pmb { x } ) \\right)$ that has   \n203 a finite second moment, i.e., $\\begin{array} { r } { \\int _ { \\mathbb { R } ^ { d } } \\left\\| \\pmb { x } \\right\\| ^ { 2 } d \\mu < \\infty } \\end{array}$ . Assume without loss of generality that the origin is   \n204 a local minimizer of $f$ , i.e. $\\nabla f ( \\mathbf { 0 } ) = \\mathbf { 0 }$ ; this is for notational convenience in the analysis and can   \n205 be realized via a simple coordinate shift, and it is not needed in the practical implementation. In   \n206 addition, we assume the following two conditions hold: ",
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+    },
+    {
+        "type": "text",
+        "text": "A 1. (Smoothness and Strong Convexity) Assume 7 $f \\in { \\mathcal { C } } ^ { 2 }$ and is $L$ -smooth and $m$ -strongly-convex, i.e. there exists 08 $0 < m \\le L$ such that $m \\bar { I } _ { d } \\preccurlyeq \\nabla ^ { 2 } f ( \\acute { \\pmb x } ) \\preccurlyeq L I _ { d } , \\quad$ $\\forall \\pmb { x } \\in \\mathbb { R } ^ { d }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Denote the condition number of 209 the standard assumption in the lit $f$ by atu $\\kappa \\triangleq { \\frac { L } { m } }$ . The smoothness and strong-convexity assumption isalyzing LMC algorithm (Dalalyan, 2017a,b; Cheng and Bartlett, 2018; Durmus et al., 2019a,b). ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "A 2. (Linear Growth of the 3rd-order derivative) Assume $f \\in \\mathcal { C } ^ { 3 }$ and the operator $\\nabla ( \\Delta f )$ grows at most linearly, i.e., there exists a constant $G > 0$ such that $\\begin{array} { r } { \\left\\| \\nabla ( \\Delta f ( \\pmb { x } ) ) \\right\\| \\leq G \\left( 1 + \\| \\pmb { x } \\| \\right) } \\end{array}$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark. The linear growth (at infinity) condition on $\\nabla \\Delta f$ is actually not as restrictive as it appears, and in some sense even weaker than some classical condition for the existence of solutions to SDE. For example, a standard condition for ensuring the existence and uniqueness of a global solution to SDE is at most a linear growth (at infinity) of the drift (Pavliotis, 2014, Theorem 3.1). If we consider monomial potentials, i.e., $f ( x ) = x ^ { p } , p \\in \\mathbb { N } _ { + }$ , then the linear growth condition on $\\nabla \\Delta f$ is met when $p \\leq 4$ , whereas the classical condition for the existence of solutions holds only when $p \\leq 2$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "220 To apply mean-square analysis to study LMC algorithm, we will need to ensure the underlying   \n221 Langevin dynamics is contractive, which we verify in Section C and D in the appendix. In addition,   \n222 we work out all required constants to determine the $C$ in Eq. 7 explicitly in the appendix. With all   \n223 these necessary ingredients, we now invoke Theorem 3.4 and obtain the following result: ",
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+    },
+    {
+        "type": "text",
+        "text": "224 Theorem 4.1. (Non-Asymptotic Error Bound: LMC) Suppose Assumption √ $\\cdot$ and 2 hold. LMC iteration 225 $\\bar { \\pmb { x } } _ { k + 1 } = \\bar { \\pmb { x } } _ { k } - h \\nabla f ( \\bar { \\pmb { x } } _ { k } ) + \\sqrt { 2 h } \\xi _ { k }$ satisfies ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f662137aa44dc2ca2777033cc0245c422dc2de469ad0402646baf56ce9fe2607.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { W _ { 2 } } \\big ( L a w ( \\bar { x } _ { k } ) , \\mu \\big ) \\le \\sqrt 2 e ^ { - m k h } W _ { 2 } \\big ( L a w ( x _ { 0 } ) , \\mu \\big ) + \\sqrt 2 C _ { L M C } h , \\quad 0 < h \\le \\frac 1 { 4 \\kappa L } , k \\in \\mathbb N } } \\\\ { ~ } \\\\ { \\displaystyle { { e ^ { } C _ { L M C } } = \\frac { 1 0 ( L ^ { 2 } + G ) } { m ^ { \\frac 3 2 } } \\sqrt { 2 d + m \\left( \\left. x _ { 0 } \\right. ^ { 2 } + 1 \\right) } = \\mathcal { O } ( \\sqrt d ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "27 Corollary 3.5 combined with the above result gives the following bound on the mixing time of LMC: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "28 Theorem 4.2. (Upper Bound of Mixing Time: LMC) Suppose Assumption $I$ and 2 hold. If running   \n29 LMC from $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ , we then have ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bc4c21186fe94b70d8283897e0087f639f56b31f1eb9719e3a9e0e61cfa3001d.jpg",
+        "text": "$$\n\\tau _ { \\operatorname* { m i x } } ( \\epsilon ; W _ { 2 } ; \\mathrm { L M C } ) \\le \\operatorname* { m a x } \\{ 4 \\kappa ^ { 2 } , \\frac { 2 C _ { L M C } } { m } \\frac { 1 } { \\epsilon } \\} \\log \\frac { 2 \\sqrt { 2 } W _ { 2 } ( L a w ( x _ { 0 } ) , \\mu ) } { \\epsilon }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where 230 $C _ { L M C }$ is the same in Theorem 4.1. When high accuracy is needed, i.e., $\\begin{array} { r } { \\epsilon \\leq \\frac { C _ { L M C } } { 2 m \\kappa ^ { 2 } } } \\end{array}$ CLMC 2mκ2 , we have ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7ed887bd56ae233180d19684d51da84b8bde8d9e4c04ba81914ef85e919e01c9.jpg",
+        "text": "$$\n\\tau _ { \\mathrm { m i x } } ( \\epsilon ; W _ { 2 } ; \\mathrm { L M C } ) \\leq \\frac { 2 C _ { L M C } } { m } \\frac { 1 } { \\epsilon } \\log \\frac { 2 \\sqrt { 2 } W _ { 2 } ( L a w ( { \\pmb x } _ { 0 } ) , \\mu ) } { \\epsilon } = \\widetilde { \\mathcal { O } } \\left( \\frac { \\sqrt { d } } { \\epsilon } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "231 The $\\begin{array} { r } { \\widetilde { \\mathcal { O } } \\left( \\frac { \\sqrt { d } } { \\epsilon } \\right) } \\end{array}$ mixing time bound in 2-Wasserstein distance improves upon the previous ones   \n232 (Dalalyan, 2017a; Cheng and Bartlett, 2018; Durmus et al., 2019b,a) in the dependence of √ $d$ and/or $\\epsilon$ .   \n233 If further assuming $G = \\mathcal { O } ( L ^ { 2 } )$ , we then have $C _ { \\mathrm { L M C } } = \\mathcal { O } ( \\kappa ^ { 2 } \\sqrt { m } \\sqrt { d } )$ and Thm.4.2 shows the mixing   \n234 time is $\\widetilde { \\mathcal { O } } \\left( \\frac { \\kappa ^ { 2 } } { \\sqrt { m } } \\frac { \\sqrt { d } } { \\epsilon } \\right)$ , which also improves the $\\kappa$ dependence in some previous results (Dalalyan,   \n235 $2 0 1 7 \\mathrm { a }$ ; Cheng and Bartlett, 2018) in the regime $m \\leq 1$ . A brief comparison is summarized in Table 1.   \n236 Optimality In fact, the $\\begin{array} { r } { \\widetilde { \\mathcal { O } } \\left( \\frac { \\sqrt { d } } { \\epsilon } \\right) } \\end{array}$ mixing time of LMC has the optimal scaling one can expect. This   \n237 is in terms of the dependence on $d$ and $\\epsilon$ , over the class of all log-smooth and log-strongly-convex   \n238 target measures. To illustrate this, consider the following Gaussian target distribution whose potential   \n239 is ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/db696df16ecb1527f8bc0f2a845ef70701977e77846c75b9bdbd7b02786418b6.jpg",
+        "text": "$$\nf ( \\pmb { x } ) = \\frac { m } { 2 } \\sum _ { i = 1 } ^ { d } x _ { i } ^ { 2 } + \\frac { L } { 2 } \\sum _ { i = d + 1 } ^ { 2 d } x _ { i } ^ { 2 } , \\quad \\mathrm { { w i t h } } m = 1 , L \\geq 4 m .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "240 We now establish a lower bound on the mixing time of LMC algorithm for this target measure. ",
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+        "type": "text",
+        "text": "241 Theorem 4.3. (Lower Bound of Mixing Time) Suppose we run LMC for the target measure defined   \n242 in Eq. (11) from ${ \\pmb x } _ { 0 } = { \\bf 1 } _ { 2 d }$ , then for any choice of step size $h > 0$ within stability limit, we have ",
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+        "text": "$$\n\\tau _ { \\mathrm { m i x } } ( \\epsilon ; W _ { 2 } ; \\mathrm { L M C } ) \\geq \\frac { \\sqrt { d } } { 8 \\epsilon } \\log \\frac { \\sqrt { d } } { \\epsilon } = \\widetilde \\Omega \\left( \\frac { \\sqrt { d } } { \\epsilon } \\right) .\n$$",
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+            "Table 1: Comparison of mixing time results in 2-Wassertein distance of LMC with $L$ -smooth and $m$ -strongly-convex potential. Constant step size is used and accuracy tolerance $\\epsilon$ is small enough. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>mixing time</td><td rowspan=1 colspan=1>Additional Assumption</td></tr><tr><td rowspan=1 colspan=1>(Dalalyan,2017a, Theorem 1)</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>K²</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>N/A</td></tr><tr><td rowspan=1 colspan=1>(Cheng and Bartlett, 2018, Theorem 1)</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>N/A</td></tr><tr><td rowspan=1 colspan=1>(Durmus et al., 2019a, Corollary 10)</td><td rowspan=1 colspan=3>8（）</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td rowspan=1 colspan=1>(Durmus et al., 2019b, Theorem 8)</td><td rowspan=1 colspan=3></td><td rowspan=1 colspan=1>|/V²f(x)-V²f(y)||≤L|x- yll</td></tr><tr><td rowspan=1 colspan=1>This work (Theorem 4.2)</td><td rowspan=1 colspan=3>6Em</td><td rowspan=1 colspan=1>Assumption 2 and G = O(L²)2</td></tr></table>",
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+        "type": "text",
+        "text": "243 Combining Theorem 4.2 and 4.3, we see that mean-square analysis provides a tight bound for LMC. ",
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+        "type": "text",
+        "text": "244 However, there is one limitation of our result – Assumption 2, which is, although mild, still extra to the   \n245 standard setup. Therefore, the gap between the upper bound and the lower bound of LMC algorithm   \n246 over the entire family of log-smooth and log-strongly-convex target measures is not completely   \n247 closed. We tend to believe that Assumption 2 may not be essential, but rather than an artifact of our   \n248 proof technique. We hope to lift this restriction in future work.   \n249 Comparison At least two sampling algorithms are closely related to LMC. One is Kinetic Langevin   \n250 Monte Carlo algorithm (KLMC), which is discretized kinetic/underdamped Langevin dynamics, and   \n251 the other is Metropolis-Adjusted Langevin Algorithm (MALA) which uses the one-step update of   \n252 LMC as a proposal and then accepts/rejects them with a Metropolis-Hastings algorithm. ",
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+        "type": "text",
+        "text": "The $\\begin{array} { r } { \\widetilde { \\mathcal { O } } \\left( \\frac { \\sqrt { d } } { \\epsilon } \\right) } \\end{array}$ mixing time in 2-Wasserstein distance of KLMC has been established for log-smooth and log-strongly-convex target measures in existing literature (Cheng et al., 2018b; Dalalyan and Riou-Durand, 2020). Due to its better dimension dependence over previously best known results of LMC, KLMC is understood to be the analog of Nesterov’s accelerated gradient method for sampling (Ma et al., 2021). Our findings show that LMC is able to achieve the same mixing time, albeit under an additional growth-at-infinity condition. However, this does not say anything about whether/how KLMC accelerates LMC, as the optimality of KLMC bound is not yet clear. We also note KLMC has better condition number dependence, although the $\\kappa$ dependence in our bound may not be tight. ",
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+        "type": "text",
+        "text": "261 In terms of MALA, a recent work (Chewi et al., 2020) establishes a $\\widetilde { \\mathcal { O } } \\left( \\sqrt { d } \\right)$ mixing time in 2-   \n262 Wasserstein distance with warm start, and the dimension dependence is shown to be optimal. We see   \n263 that without the Metropolis adjustment, LMC can also achieve the optimal dimension dependence as   \n264 MALA. But unlike LMC, MALA only has logarithmic dependence on $\\frac { 1 } { \\epsilon }$ . Under warm-start condition,   \n265 is it possible/how to improve the dependence of $\\frac { 1 } { \\epsilon }$ for LMC, from polynomial to logarithmic? This   \n266 question is beyond the scope of this paper but worth further investigation. ",
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+        "type": "text",
+        "text": "267 5 Numerical Examples ",
+        "text_level": 1,
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+        "text": "268 This section numerically verifies our theoretical findings for LMC in Section 4, with a particular   \n269 focus on the dependence of the discretization error in Theorem 4.1 on dimension $d$ and step size $h$ .   \n270 To this end, we consider two target measures specified by the following two potentials: ",
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+        "text": "$$\nf _ { 1 } ( \\boldsymbol { x } ) = \\frac { 1 } { 2 } \\left\\| \\boldsymbol { x } \\right\\| ^ { 2 } + \\log \\left( \\sum _ { i = 1 } ^ { d } e ^ { x _ { i } } \\right) \\quad \\mathrm { a n d } \\quad f _ { 2 } ( \\boldsymbol { x } ) = \\frac { 1 } { 2 } \\left\\| \\boldsymbol { x } \\right\\| ^ { 2 } - \\frac { 1 } { 2 d ^ { \\frac { 1 } { 2 } } } \\sum _ { i = 1 } ^ { d } \\cos \\left( d ^ { \\frac { 1 } { 4 } } \\boldsymbol { x } _ { i } \\right) .\n$$",
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+        "type": "text",
+        "text": "It is not hard to see that 271 $f _ { 1 }$ is 2-smooth and 1-strongly convex, $f _ { 2 }$ is 3 -smooth and 1-strongly272 convex. $f _ { 2 }$ is also used in (Chewi et al., 2020) to illustrate the optimal dimension dependence 273 of MALA. Explicit expression of 2-Wasserstein distance between non-Gaussian distributions is ",
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+        "type": "text",
+        "text": "274 typically not available, instead, we use the Euclidean norm of the mean error as a surrogate because   \n275 $\\bigl | \\bigl | \\hat { \\mathbb { E } } \\bar { \\pmb { x } } _ { k } - \\mathbb { E } _ { \\mu } \\pmb { x } \\bigr | \\bigr | \\leq W _ { 2 } \\bigl ( \\mathrm { L a w } ( \\bar { \\pmb { x } } _ { k } ) , \\mu \\bigr )$ due to Jensen’s inequality. To obtain an accurate estimate of the   \n276 ground truth, we run $1 0 ^ { 8 }$ independent LMC realizations using a tiny step size $\\mathrm { { h } = 0 . 0 0 1 }$ ), each till a   \n277 fixed, long enough time, and use the empirical average to approximate $\\mathbb { E } _ { \\mu } \\pmb { x }$ .   \n278 To study the dimension dependence of sampling error, we fix step size $h = 0 . 1$ , and for each   \n279 $d \\in \\{ 1 , 2 , 5 , 1 0 , 2 0 , 5 0 , 1 0 0 , 2 0 0 , 5 0 0 , 1 0 0 0 \\}$ , we simulate $1 0 ^ { 4 }$ independent Markov chains using   \n280 LMC algorithm for 100 iterations, which is long enough for the chain to be well-mixed. The mean   \n281 and the standard deviation of the sampling error corresponding to the last 10 iterates are recorded.   \n282 To study step size dependence of sampling error, dimension is fixed to be $d = 1 0$ . We experiment with   \n283 step size $h \\in \\{ 1 , 2 , \\bar { 3 } , 4 , 5 , 6 , 7 , 8 , 9 , \\bar { 1 0 } \\} \\times 1 0 ^ { - 1 }$ . We fix a continuous time $T = 2 0$ , and run LMC   \n284 algorithm for $\\left\\lceil { \\frac { T } { h } } \\right\\rceil$ iterations for each $h$ . The procedure is repeated $1 0 ^ { 4 }$ times with different random   \n285 seeds to obtain independent samples. When the corresponding continuous time $t = k h > 1 0$ , we see   \n286 from Eq. (10) that LMC is well converged and the sampling error is saturated by the discretization   \n287 error. Therefore, for each $h$ , we take the last $\\lceil \\frac { 1 0 } { h } \\rceil$ iterates and record the mean and standard deviation   \n288 of their sampling error.   \n289 The experiment results shown in Figure 1 are consistent with our theoretical analysis of the sampling√   \n290 error.Both linear dependence on $\\sqrt { d }$ and $h$ can be identified in and supported by the empirical   \n291 evidence. Note results with smaller $h$ are less accurate because one starts to see the error of empirical   \n292 approximation due to finite samples. Experiments were conducted on a machine with a 2.20GHz   \n293 Intel(R) Xeon(R) E5-2630 v4 CPU and an Nvidia GeForce GTX 1080 GPU. ",
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+            "Figure 1: (a) Dependence of the sampling error of LMC on dimension $d$ and step size $h$ for $f _ { 1 }$ and $f _ { 2 }$ Both axes in Figure 1a and 1b are in log scale. The shaded areas in Figure 1a and 1b represent one standard deviation of the last 10 iterates. The shaded areas in Figure 1c and 1d represent one standard deviation of the last $\\lceil \\frac { 1 0 } { h } \\rceil$ iterations. "
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+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This paper extends the mean-square analysis framework for analyzing the integration error of SDE to analyzing the sampling error in 2-Wasserstein distance. Corresponding mixing time bound unveils how a high-order numerical algorithm can help improve dependence on accuracy tolerance \u000f, and potentially other parameters, such as the dimension. When applied to Langevin Monte Carlo algorithm, it obtains an improved and optimal $\\widetilde { \\mathcal { O } } \\left( \\sqrt { d } / \\epsilon \\right)$ bound, which was previously thought to be obtainable only with the addition of momentum. ",
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+        "text": "01 Here are some possible directions worth further investigations. (i) In data-intensive applications,   \n02 stochastic gradients are typically used for better scalability. It seems natural to apply the mean-square   \n03 analysis framework to study SDE-basd stochastic gradient MCMC methods; (ii) Assumption 2 is   \n04 likely to be an artifact of our analysis; how to establish the optimal mixing time bound in the standard   \n05 log-smooth and log-strongly-convex setup is still an open question; (iii) Motivated by the recent   \n06 result of MALA (Chewi et al., 2020), it would be interesting to know whether the dependence on $\\frac { 1 } { \\epsilon }$   \n07 can be improved to logarithmic, for example if LMC is initialized at a warm start. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M., Kudlur, M., Levenberg, J., Monga, R., Moore, S., Murray, D. G., Steiner, B., Tucker, P., Vasudevan, V., Warden, P., Wicke, M., Yu, Y., and Zheng, X. (2016). Tensorflow: A system for large-scale machine learning. In 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), pages 265–283.   \nAndrieu, C., De Freitas, N., Doucet, A., and Jordan, M. I. (2003). An introduction to mcmc for machine learning. Machine learning, 50(1):5–43.   \nChen, C., Zhang, R., Wang, W., Li, B., and Chen, L. (2018). A unified particle-optimization framework for scalable bayesian sampling. In The Conference on Uncertainty in Artificial Intelligence.   \nCheng, X. and Bartlett, P. L. (2018). Convergence of langevin mcmc in kl-divergence. PMLR 83, (83):186–211.   \nCheng, X., Chatterji, N. S., Abbasi-Yadkori, Y., Bartlett, P. L., and Jordan, M. I. (2018a). 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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] Limitation is discussed in the paragraph below Table 1.   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] To the best of our knowledge, this work does not have any potential negative societal impact.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] Assumptions are fully, clearly stated in theorems.   \n(b) Did you include complete proofs of all theoretical results? [Yes] Complete proofs of all theoretical results are provided in supplementary materials. ",
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