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an efficient method for such setting is of interest. Recently, two variants of the extragradient (EG) method are studied in that direction. First, a two-time-scale variant of the EG, named $\mathrm { E G + }$ , was proposed under a smooth structured nonconvexnonconcave setting, with a slow $\mathcal { O } ( \bar { 1 } / k )$ rate on the squared gradient norm, where $k$ denotes the number of iterations. Second, another variant of EG with an anchoring technique, named extra anchored gradient (EAG), was studied under a smooth convex-concave setting, yielding a fast $\mathcal { O } ( 1 / k ^ { 2 } )$ rate on the squared gradient norm. Built upon $\mathrm { E G + }$ and EAG, this paper proposes a two-time-scale EG with anchoring, named fast extragradient (FEG), that has a fast $\mathcal { O } ( 1 / k ^ { 2 } )$ rate on the squared gradient norm for smooth structured nonconvex-nonconcave problems; the corresponding saddle-gradient operator satisfies the negative comonotonicity condition. This paper further develops its backtracking line-search version, named FEG-A, for the case where the problem parameters are not available. The stochastic analysis of FEG is also provided. + +# 1 Introduction + +Recently, nonconvex-nonconcave minimax problems have received an increased attention in the optimization community and the machine learning community due to their applications to generative adversarial network [10] and adversarial training [27]. In this paper, we consider a smooth structured nonconvex-nonconcave minimax problem: + +$$ +\operatorname* { m i n } _ { \pmb { x } \in \mathbb { R } ^ { d _ { \boldsymbol { x } } } } \operatorname* { m a x } _ { \pmb { y } \in \mathbb { R } ^ { d _ { \boldsymbol { y } } } } f ( \pmb { x } , \pmb { y } ) , +$$ + +where $f : \mathbb { R } ^ { d _ { x } } \times \mathbb { R } ^ { d _ { y } } \mathbb { R }$ is smooth and is possibly nonconvex in $_ { \textbf { \em x } }$ for fixed $\textbf { { y } }$ , and possibly nonconcave in $\textbf { { y } }$ for fixed $_ { \textbf { \em x } }$ ; the saddle-gradient operator $\pmb { F } : = ( \nabla _ { x } f , - \nabla _ { y } f )$ satisfies the negative comonotonicity [1]. We construct an efficient (first-order) method, using a saddle gradient operator $\pmb { F }$ for finding a first-order stationary point of the problem (1). + +So far little is known under the nonconvex-nonconcave setting, compared to the convex-concave setting. Recent works [4, 7, 22, 24, 26, 42, 44] studied extragradient-type methods [19, 39] for minimax problems under various structured nonconvex-nonconcave settings. In other words, they consider various non-monotone conditions on $\pmb { F }$ , such as the Minty variational inequality (MVI) condition [4], the weak MVI condition [7], and the negative comonotonicity [1].1 Among them, this paper focuses on the negative comonotonicity condition for a Lipschitz continuous $\pmb { F }$ . To the best of our knowledge, the following two-time-scale variant of the extragradient method, named $\mathrm { E G + }$ : + +$$ +\begin{array} { c } { { z _ { k + 1 / 2 } = z _ { k } - \frac { \alpha _ { k } } { \beta } { \cal F } z _ { k } , } } \\ { { z _ { k + 1 } = z _ { k } - \alpha _ { k } { \cal F } z _ { k + 1 / 2 } , } } \end{array} +$$ + +is the only known (explicit)2 method, using $\pmb { F }$ , that converges under the considered setting3 [7], where $z _ { k } : = ( x _ { k } , y _ { k } )$ . The $\mathrm { E G + }$ , however, has a slow $\mathcal { O } ( 1 / k )$ rate on the squared gradient norm. Note that a similar two-time-scale approach has been found to stabilize the stochastic extragradient method with unbounded noise variance [14]. + +Meanwhile, under the smooth convex-concave setting, recent works [6, 17, 21, 40, 43] suggest that Halpern-type [12] (or anchoring) methods, performing a convex combination of an initial point $z _ { \mathrm { 0 } }$ and the last updated point $z _ { k }$ at each iteration, has a fast $\mathcal { O } ( 1 / k ^ { 2 } )$ rate in terms of the squared gradient norm. In particular, [43] developed the following anchoring variant of the extragradient method, named extra anchored gradient (EAG): + +$$ +\begin{array} { r l } & { z _ { k + 1 / 2 } = z _ { k } + \beta _ { k } ( z _ { 0 } - z _ { k } ) - \alpha _ { k } F z _ { k } , } \\ & { \quad z _ { k + 1 } = z _ { k } + \beta _ { k } ( z _ { 0 } - z _ { k } ) - \alpha _ { k } F z _ { k + 1 / 2 } . } \end{array} +$$ + +This is the first (explicit) method with a fast $\mathcal { O } ( 1 / k ^ { 2 } )$ rate on the squared gradient norm, when $\pmb { F }$ satisfies both the Lipschitz continuity and the monotonicity. [43] also showed that such $\mathcal { O } ( 1 / k ^ { 2 } )$ rate is optimal for first-order methods using a Lipschitz continuous and monotone $\pmb { F }$ . + +Built upon both $\mathrm { E G + }$ and EAG, this paper studies the following class of two-time-scale anchored extragradient methods, named fast extragradient (FEG): + +$$ +\begin{array} { r l } & { z _ { k + 1 / 2 } = z _ { k } + \beta _ { k } ( z _ { 0 } - z _ { k } ) - ( 1 - \beta _ { k } ) ( \alpha _ { k } + 2 \rho _ { k } ) { \cal F } z _ { k } , } \\ & { ~ z _ { k + 1 } = z _ { k } + \beta _ { k } ( z _ { 0 } - z _ { k } ) - \alpha _ { k } { \cal F } z _ { k + 1 / 2 } - ( 1 - \beta _ { k } ) 2 \rho _ { k } { \cal F } z _ { k } . } \end{array} +$$ + +(Class FEG) + +Note that (Class FEG) reuses the $\pmb { F } z _ { k }$ term in the $z _ { k + 1 }$ update, unlike the standard extragradienttype methods, which we found essential for handling the negative comonotonicity condition. We leave further understanding the use of $\pmb { F } z _ { k }$ and the formulation of (Class FEG) as future work. The proposed FEG method (with appropriately chosen step coefficients $\alpha _ { k }$ , $\beta _ { k }$ and $\rho _ { k }$ discussed later) has an $\mathcal { O } ( 1 / k ^ { 2 } )$ rate on the squared gradient norm, under the Lipschitz continuity and the negative comonotonicity conditions on $\pmb { F }$ . To the best of our knowledge, this is the first accelerated method under the nonconvex-nonconcave setting. The FEG also has value under the smooth convex-concave setting. First, when $\pmb { F }$ is Lipschitz continuous and monotone, the rate bound of FEG is about 27/4 times smaller than that of EAG. Also note that the rate bound of FEG is only about four times larger than the $\mathcal { O } ( 1 / k ^ { 2 } )$ lower complexity bound of first-order methods under such setting [43], further closing the gap between the lower and upper complexity bounds. Second, when $\pmb { F }$ is cocoercive, FEG has a rate faster than that of a version of Halpern iteration [12] in [6]. + +We also develop an adaptive variant of FEG, named FEG-A, which updates its parameters, $\alpha _ { k }$ and $\rho _ { k }$ in (Class FEG), adaptively using a backtracking line-search [2, 25, 31]. FEG requires the knowledge of the two problem parameters for the Lipschitz continuity and the comonotonicity of $\pmb { F }$ . However, those global parameters can be conservative, and in practice, they are even usually unknown. For such cases, the FEG-A adaptively and locally estimates the problem parameters, while preserving the fast rate $\mathcal { O } ( 1 / k ^ { 2 } )$ on the squared gradient norm for smooth structured nonconvex-nonconcave minimax problems. + +Lastly, we study a stochastic version of FEG, named S-FEG, which uses an unbiased stochastic estimate of $\pmb { F } z$ , i.e., $\tilde { F } z = F z + \xi$ , instead of $\pmb { F } z$ in FEG, where $\xi$ denotes a stochastic noise. For a Lipschitz continuous and monotone $\pmb { F }$ , we provide a convergence analysis in terms of the expected squared gradient norm. In specific, we show that the S-FEG is stable with a rate $\mathcal { O } ( 1 / k ^ { 2 } ) \overset { \cdot } { + } \mathcal { O } ( \epsilon )$ , when the noise variance decreases in the order of $\mathcal { O } ( \epsilon / k )$ , while being unstable otherwise due to error accumulation. This is similar to the convergence behavior of a stochastic version of Nesterov’s fast gradient method [35, 36], observed in [5], for smooth convex minimization. + +Our main contributions are summarized as follows. + +• We propose the FEG method that has an accelerated convergence rate $\mathcal { O } ( 1 / k ^ { 2 } )$ on the squared gradient norm for smooth structured nonconvex-nonconcave minimax problems. +We present that the FEG method has a rate faster than those of the EAG and the Halpern iteration for smooth convex-concave problems. We construct a backtracking line-search version of FEG, named FEG-A, for the case where the Lipschitz constant and comonotonicity parameters of $\pmb { F }$ are unavailable. +• We analyze a stochastic version of FEG, named S-FEG, for smooth convex-concave problems. + +# 2 Related work + +# 2.1 Methods for convex-concave minimax problems + +The extragradient method [19] is one of the widely used methods for solving smooth convex-concave minimax problems (see, e.g., [4, 7, 22, 24, 26, 42, 44] for its extensions and applications). In terms of the duality gap, $\begin{array} { r } { \operatorname* { m a x } _ { { \pmb y } ^ { \prime } \in \mathcal { V } } f ( { \pmb x } , { \pmb y } ^ { \prime } ) - \operatorname* { m i n } _ { { \pmb x } ^ { \prime } \in \mathcal { X } } f ( { \pmb x } ^ { \prime } , { \pmb y } ) } \end{array}$ , where $\mathcal { X }$ and $\mathcal { V }$ are compact4 domains, the ergodic iterate of the extragradient-type methods [32, 37] have an $\mathcal { O } ( 1 / k )$ rate. Such $\mathcal { O } ( 1 / k )$ rate on the duality gap is order-optimal for the first-order methods [34, 38], leaving no room for√ improvement. On the other hand, the last iterate of the extragradient method has a slower $\mathcal { O } ( 1 / \sqrt { k } )$ rate on the duality gap, under an additional assumption that $\pmb { F }$ has a Lipschitz derivative [9]. In terms of the squared gradient norm, $\| \ b { F z } \| ^ { 2 }$ , the best iterate of the extragradient-type methods [19, 39] have an $\mathcal { O } ( 1 / k )$ rate [40, 41, 43]. The last iterate of the extragradient method also has a rate $\mathcal { O } ( 1 / k )$ , when $\pmb { F }$ is further assumed to have a Lipschitz derivative [9]. Unlike the duality gap, the $\mathcal { O } ( 1 / k )$ rate on the squared gradient norm is not optimal [43]. From now on throughout this paper, we mainly study and compare the convergence rates on the squared gradient norm, which still has room for improvement in convex-concave problems, and has meaning for nonconvex-nonconcave minimax problems, unlike the duality gap. + +Recently, [6, 17, 21, 40, 43] found that Halpern-type [12] (or anchoring) methods yield a fast $\mathcal { O } ( 1 / k ^ { 2 } )$ rate in terms of the squared gradient norm for minimax problems. [17, 21] showed that the (implicit) Halpern iteration [12] with appropriately chosen step coefficients has an $\mathcal { O } ( 1 / k ^ { 2 } )$ rate on the squared norm of a monotone $\pmb { F }$ . Then, for a cocoercive $\pmb { F }$ , an (explicit) version of the Halpern iteration was studied in [6, 17] that has the same fast rate. In addition, [6] constructed a double-loop version of the Halpern iteration for a Lipschitz continuous and monotone $\pmb { F }$ , which has a rate $\tilde { \mathcal { O } } ( 1 / k ^ { 2 } )$ on the squared gradient norm, slower than the rate $\mathcal { O } ( 1 / k ^ { 2 } )$ . While this is promising compared to the $\bar { \mathcal { O } ( 1 / k ) }$ rate of the extragradient methods on the squared gradient norm [40, 41, 43], the computational complexity due to its double-loop nature and a relatively slow rate remained a problem. Very recently, [43] proposed the extra anchored gradient (EAG) method, which is the first (explicit) method with a fast $\bar { \mathcal { O } } ( \bar { 1 } / k ^ { 2 } )$ rate for smooth convex-concave minimax problems, i.e., for Lipschitz continuous and monotone operators. In addition, [43] proved that the EAG is order-optimal by showing that the lower complexity bound of first-order methods is $\Omega ( 1 / k ^ { 2 } )$ . + +$$ +\begin{array} { c c c c c } { { \mathrm { C o c o e r c i v e } } } & { { \subseteq } } & { { \bf M o n o t o n e } } & { { \subseteq } } & { { \mathrm { N e g a t i v e ~ c o m o n o t o n e } } } \\ { { } } & { { } } & { { | \bigcap } } & { { } } & { { | \bigcap } } \\ { { } } & { { } } & { { \mathrm { M V I } } } & { { \subseteq } } & { { \mathrm { W e a k ~ M V I } } } \end{array} +$$ + +Figure 1: Relations between the conditions on $\pmb { F }$ . + +# 2.2 Methods for nonconvex-nonconcave minimax problems + +Some recent literature considered relaxing the monotonicity condition of the saddle gradient operator to tackle modern nonconvex-nonconcave minimax problems. For example, the Minty variational inequality (MVI) condition, i.e., there exists $z _ { \ast } \in Z _ { \ast } ( F )$ satisfying $\langle F z , z - z _ { * } \rangle \ge 0$ for all $z \in \mathbb { R } ^ { d }$ where $Z _ { * } ( F ) : = \{ z _ { * } \in \mathbb { R } ^ { d } : F z _ { * } = \mathbf { 0 } \}$ , is studied in [4, 23, 22, 24]. This condition is also studied under the name, the coherence, in [26, 42, 44]. Moreover, [7] considered a weaker condition, named the weak MVI condition, i.e., for some $\rho < 0$ , there exists $z _ { \ast } \in Z _ { \ast } ( F )$ satisfying $\langle F z , z - z _ { * } \rangle \geq \rho \Vert F z \Vert ^ { 2 }$ for all $z \in \mathbb { R } ^ { d }$ . The weak MVI condition is implied by the negative comonotonicity [1] or, equivalently, the (positive) cohypomonotonicity [3]. The comonotonicity will be further discussed in the upcoming section. + +For $L$ -Lipschitz continuous $\pmb { F }$ , [4, 42] showed that the extragradient-type methods have an $\mathcal { O } ( 1 / k )$ rate on the squared gradient norm under the MVI condition, and [7] developed the $( \mathrm { E G + } )$ method under the weak MVI condition (and thus under the negative comonotonicty), which also has an $\mathcal { O } ( 1 / k )$ rate on the squared gradient norm. To the best of our knowledge, there is no known accelerated method for the nonconvex-nonconcave setting; our proposed FEG method is the first method to have a fast $\mathcal { O } ( 1 / k ^ { 2 } )$ rate under the nonconvex-nonconcave setting. The convergence rates of the existing methods and the FEG on the squared gradient norm are summarized in Table 1. + +Table 1: Comparison of the convergence rates of the existing extragradient-type methods and the FEG, with respect to the squared gradient norm, for smooth structured minimax problems, under various assumptions on the Lipschitz continuous saddle gradient operator $\pmb { F }$ . + +
MethodConvex-concaveNonconvex-nonconcave
Cocoercive MonotoneNegative comonotoneMVIWeak MVI
NormalEG [4, 42]0(1/k)0(1/k）0(1/k）
EG+[7]0(1/k)0(1/k)0(1/k)0(1/k）0(1/k）
AcceleratedHalpern [12, 6]0(1/k²)(1/k²)
EAG [43] FEG (this paper)0(1/k2) 0(1/k2)0(1/k²) 0(1/k2)0(1/k2)
+ +# 3 Preliminaries + +The followings are the two main assumptions for the saddle gradient operator $\pmb { F }$ of the smooth structured nonconvex-nonconcave problem (1). Under such assumptions, we develop efficient methods that find a first-order stationary point $z _ { \ast } \in Z _ { \ast } ( F )$ where $Z _ { * } ( \hat { F } ) : = \{ z _ { * } \in \mathbb { R } ^ { d } : \hat { F } z _ { * } = \mathbf { 0 } \}$ . + +Assumption 1 ( $L$ -Lipschitz continuity). For some $L \in ( 0 , \infty )$ , $\pmb { F }$ satisfies + +$$ +\| F z - F z ^ { \prime } \| \le L \| z - z ^ { \prime } \| , \quad \forall z , z ^ { \prime } \in \mathbb R ^ { d } . +$$ + +Assumption 2 ( $\rho$ -Comonotonicity). For some $\textstyle \rho \in { \bigl ( } - { \frac { 1 } { 2 L } } , \infty { \bigr ) }$ , $\pmb { F }$ satisfies + +$$ +\begin{array} { r } { \langle F z - F z ^ { \prime } , z - z ^ { \prime } \rangle \ge \rho \| F z - F z ^ { \prime } \| ^ { 2 } , \quad \forall z , z ^ { \prime } \in \mathbb { R } ^ { d } . } \end{array} +$$ + +The $\rho$ -comonotonicity consists of three cases depending on the choice of $\rho$ ; the negative comonotonicity when $\rho < 0$ , the monotonicity when $\rho = 0$ , and the cocoercivity when $\rho > 0$ . The negative comonotonicity is weaker than the other two, and is the main focus of this paper. The following is an examplary nonconvex-nonconcave condition that is stronger than the negative comonotonicity [1, 3]. + +Example 1. Let $f$ be twice continuously differentiable and $\gamma$ -weakly-convex-weakly-concave. Further assume that $f$ satisfies + +$$ +\begin{array} { r } { \nabla _ { x x } ^ { 2 } f + \nabla _ { x y } ^ { 2 } f ( \eta I - \nabla _ { y y } ^ { 2 } f ) ^ { - 1 } \nabla _ { y x } ^ { 2 } f \succeq \alpha I , } \\ { - \nabla _ { y y } ^ { 2 } f + \nabla _ { y x } ^ { 2 } f ( \eta I + \nabla _ { x x } ^ { 2 } f ) ^ { - 1 } \nabla _ { x y } ^ { 2 } f \succeq \alpha I , } \end{array} +$$ + +for some $\alpha \geq 0$ and $\eta > \gamma ,$ , named $\alpha \geq 0$ -interaction dominant condition in [11]. Then, the saddle gradient of $f$ satisfies the $- \frac { 1 } { \eta }$ -negative comonotonicity. (See Appendix A.1.) For any $\gamma$ -weaklyconvex-weakly-concave function, the condition (2) holds with $\alpha = - \gamma < 0$ . Its extreme case is + +$\begin{array} { r } { f ( x , y ) = - \frac { \gamma } { 2 } x ^ { 2 } + \frac { \gamma } { 2 } y ^ { 2 } } \end{array}$ , where there is no interaction between x and y. On the other hand, when the +the second terms in the left-hand side of (2) are sufficently positive definite, a nonconvex-nonconave +function satisfies the condition (2) with a nonnegative condition is satisfied when the interaction term of Hess $\alpha$ .n cific, the is domin $\alpha \geq 0$ -interaction dominantany negative curvature $\nabla _ { \substack { x y } } ^ { 2 } f$ +in Hessians $\nabla _ { x x } ^ { 2 } f$ and $- \nabla _ { \boldsymbol { y } \boldsymbol { y } } ^ { 2 } f l l l \boldsymbol { l } \boldsymbol { l } \boldsymbol { l } \boldsymbol { l }$ . + +We next present our proposed FEG, and illustrate that the FEG outperforms existing methods such as $\mathrm { E G + }$ , EAG, and the Halpern iteration, for each three comonoticity case, respectively. + +# 4 Fast extragradient (FEG) method for Lipschitz continuous and comonotone operators + +This section considers an instance of (Class FEG) with $\begin{array} { r } { \alpha _ { k } = \frac { 1 } { L } } \end{array}$ , $\begin{array} { r } { \beta _ { k } = \frac { 1 } { k + 1 } } \end{array}$ , and $\rho _ { k } = \rho$ for all $k \geq 0$ The resulting method, named FEG, is illustrated in Algorithm 1, which has an $\mathcal { O } ( 1 / k ^ { 2 } )$ fast rate with respect to the squared gradient norm, in Theorem 4.1. The proof of Theorem 4.1 is provided in Section 7. + +# Algorithm 1 Fast extragradient (FEG) method + +
Input: z0 ∈ Rd,L ∈ (0,∞o),ρ ∈(- 2,00) for k = 0,1,... do
2+1/=+1(2-（1-1）（+2）F 1
1 2 2k+1= 2k+ k+1
+ +end for + +Theorem 4.1. For the $L$ -Lipschitz continuous and $\rho$ -comonotone operator $\pmb { F }$ with $\rho > - \frac { 1 } { 2 L }$ and for any $z _ { \ast } \in Z _ { \ast } ( F )$ , the sequence $\{ z _ { k } \} _ { k \ge 0 }$ generated by FEG satisfies, for all $k \geq 1$ , + +$$ +\| F z _ { k } \| ^ { 2 } \leq \frac { 4 \| z _ { 0 } - z _ { * } \| ^ { 2 } } { \Big ( \frac { 1 } { L } + 2 \rho \Big ) ^ { 2 } k ^ { 2 } } . +$$ + +The following example shows that the bound (3) of the FEG is exact for $\rho = 0$ and $k = 4 l + 2$ . The bound (3) is not known to be exact in general, and we leave finding the exact bound as future work. + +Example 2. Let $f : \mathbb { R } \times \mathbb { R } \to \mathbb { R }$ be $f ( x , y ) = L x y .$ . Its saddle gradient operator and solution are $\pmb { F } ( x , y ) = ( L y , - L x )$ and $z _ { * } = ( 0 , 0 )$ , respectively. For the initial point $z _ { 0 } = ( x _ { 0 } , y _ { 0 } ) =$ $( 1 , 0 )$ , the sequence $\{ z _ { k } \} _ { k \ge 0 }$ generated by FEG satisfies $\begin{array} { r } { z _ { 4 l + 2 } = \left( 0 , \frac { 1 } { 2 l + 1 } \right) } \end{array}$ for all $l \geq 0$ . Hence, $\begin{array} { r } { \| F z _ { 4 l + 2 } \| ^ { 2 } = \frac { L ^ { 2 } } { ( 2 l + 1 ) ^ { 2 } } = \frac { 4 L ^ { 2 } \| z _ { 0 } - z _ { * } \| ^ { 2 } } { ( 4 l + 2 ) ^ { 2 } } } \end{array}$ for all $l \geq 0 .$ . (See Appendix B.1.) + +We next compare the rate bound (3) with existing analyses for the three cases $\begin{array} { r } { - \frac { 1 } { 2 L } < \rho < 0 , \rho = 0 } \end{array}$ and $\rho > 0$ . + +# 4.1 Comparison to $\mathbf { E G + }$ under the negative comonotonicity $( \rho < 0 )$ ) + +Under the negative comonotonicity with $\begin{array} { r } { - \frac { 1 } { 8 L } < \rho < 0 } \end{array}$ , the $( \mathrm { E G + } )$ method with $\begin{array} { r } { \alpha _ { k } = \frac { 1 } { 2 L } } \end{array}$ and $\begin{array} { r } { \beta = \frac { 1 } { 2 } } \end{array}$ rate on the squared gradient norm. To the best of our knowledge, this is the best known rate, and the FEG has a faster $\mathcal { O } ( 1 \bar { \vert } k ^ { 2 } )$ rate with a wider region of convergence $\begin{array} { r } { - \frac { 1 } { 2 L } < \rho < 0 } \end{array}$ . + +# 4.2 Comparison to EAG under the monotonicity $( \rho = 0$ ) + +For an $L$ -Lipschitz continuous and monotone operator $\pmb { F }$ , [43] proposed two EAG methods, named EAG-C and EAG-V, with same βk = 1k+2 but with different choices of $\alpha _ { k }$ . EAG-C sets $\alpha _ { k }$ to be a constant $\frac { 1 } { 8 L }$ for all $k \geq 0$ in (EAG), and has a large constant 260 in its convergence rate, $\begin{array} { r } { \| \pmb { F } \pmb { z } _ { k } \| ^ { 2 } \le \frac { 2 6 0 L ^ { 2 } \| \pmb { z } _ { 0 } - \pmb { z } _ { * } \| ^ { 2 } } { ( k + 1 ) ^ { 2 } } } \end{array}$ 260L2kz0−z∗k22 for all k ≥ 0. On the other hand, while EAG-V requires a complicated recursive update for {αk}, αk+1 = αk1−α2L2 $\begin{array} { r } { \alpha _ { k + 1 } = \frac { \alpha _ { k } } { 1 - \alpha _ { k } ^ { 2 } L ^ { 2 } } \big ( 1 - \frac { ( k + 2 ) ^ { 2 } } { ( k + 1 ) ( k + 3 ) } \alpha _ { k } ^ { 2 } L ^ { 2 } \big ) } \end{array}$ for all $k \geq 0$ , with $\begin{array} { r } { \alpha _ { 0 } = \frac { 0 . 6 1 8 } { L } } \end{array}$ , its rate has a smaller constant 27. + +![](images/380353bbf2580099fbe770143af658fb4debcb8120afd75852aaa71dc5266deb.jpg) +Figure 2: Numerical result with $\begin{array} { r } { f ( x , y ) = - \frac { 1 } { 6 } x ^ { 2 } + \frac { 2 \sqrt { 2 } } { 3 } x y + \frac { 1 } { 6 } y ^ { 2 } } \end{array}$ . The dashed line represents the theoretical bound (3) of FEG. + +The FEG takes a constant $\begin{array} { r } { \alpha _ { k } \ = \ \frac { 1 } { L } } \end{array}$ , unlike EAG-V, but has an even smaller constant 4 in its convergence rate $\begin{array} { r } { \| \pmb { F } \pmb { z } _ { k } \| ^ { 2 } \le \frac { 4 L ^ { 2 } \| \pmb { z } _ { 0 } - \pmb { z } _ { * } \| ^ { 2 } } { k ^ { 2 } } } \end{array}$ 4L2kz0−z∗k2k2 for ρ = 0. Therefore, the FEG with ρ = 0 has about $2 6 0 / 4$ -times and $2 7 / 4$ -times faster convergence rate compared to those of EAG-C and EAG-V, respectively. Furthermore, the rate bound of FEG with $\rho = 0$ is only about 4-times larger than the lower complexity bound of first-order methods under the considered setting [43], reducing the gap between the lower and upper complexity bounds from 27 to 4. + +# 4.3 Comparison to the Halpern iteration under the cocoercivity $( \rho > 0 )$ ) + +For a $\rho$ -cocoercive operator $\pmb { F }$ , an (explicit) version of Halpern iteration [12], studied in [6], has a fast rate, $\begin{array} { r } { \| \boldsymbol { F } \boldsymbol { z } _ { k } \| ^ { 2 } \le \frac { \| \boldsymbol { z } _ { 0 } - \boldsymbol { z } _ { * } \| ^ { 2 } } { \rho ^ { 2 } k ^ { 2 } } } \end{array}$ kz0−z∗k2ρ2k2 . Note that while the ρ-cocoercivity implies the 1ρ -Lipschitz continuity, there is case where the $\rho$ -cocoercive (and thus Lipschitz continuous) operator has a Lipschitz constant $L$ smaller than $\frac { 1 } { \rho }$ . Since $\begin{array} { r } { L \le \frac { 1 } { \rho } } \end{array}$ , the FEG has a rate $\begin{array} { r } { \| \boldsymbol { F } \boldsymbol { z } _ { k } \| ^ { 2 } \le \frac { 4 \| \boldsymbol { z } _ { 0 } - \boldsymbol { z } _ { * } \| ^ { 2 } } { ( 1 / L + 2 \rho ) ^ { 2 } k ^ { 2 } } = \frac { 4 \| \boldsymbol { z } _ { 0 } - \boldsymbol { z } _ { * } \| ^ { 2 } } { 9 \rho ^ { 2 } k ^ { 2 } } } \end{array}$ that is faster than that of Halpern iteration. However, if we take into account that the FEG requires computing the saddle gradient twice per iteration, unlike Halpern iteration studied in [6], the FEG method has a slower rate in terms of the number of gradient computations. If we narrow down to the case $\begin{array} { r } { L < \frac { 1 } { 2 \rho } } \end{array}$ , the FEG has a faster rate, $\begin{array} { r } { \| \ b { F } \ b { z } _ { k } \| ^ { 2 } \leq \frac { 4 \| \ b { z } _ { 0 } - \ b { z } _ { * } \| ^ { 2 } } { ( 1 / L + 2 \rho ) ^ { 2 } k ^ { 2 } } < \frac { \| \ b { z } _ { 0 } - \ b { z } _ { * } \| ^ { 2 } } { 4 \rho ^ { 2 } k ^ { 2 } } } \end{array}$ . For such case, the FEG has a rate faster than that of the Halpern iteration, even in terms of the number of gradient computations. + +# 4.4 Toy example + +rformed a toy exp, which has an iment on a simple quadratic-Lipschitz continuous and nction, -comon $\begin{array} { r } { f ( x , y ) = \frac { \rho L ^ { 2 } } { 2 } x ^ { 2 } + L \sqrt { 1 - \rho ^ { 2 } L ^ { 2 } } x y - } \end{array}$ ${ \frac { \rho L ^ { 2 } } { 2 } } y ^ { 2 }$ $L$ $\rho$ $\begin{array} { r } { \rho = - \frac { 1 } { 3 L } } \end{array}$ and $L = 1$ , Figure 2 illustrates that the FEG converges with an accelerated rate whereas $\mathrm { E G + }$ , EAG-C, EAG-V, and the (explicit) version of Halpern iteration [6] diverge. This example presents that the existing guarantees on convergence and acceleration of the aforementioned methods under the convex-concave setting do not generalize to the nonconvex-nonconcave setting. + +# 5 FEG with backtracking line-search + +The FEG requires the knowledge of the two global parameters $L$ and $\rho$ for Lipschitz continuity and comonotonicity, respectively. Those global parameters are often difficult to compute in practice and can be locally conservative. To handle these two disadvantages, we employ the backtracking line-search technique [2, 25, 31] in FEG. We adaptively decrease the two step size parameters, $\tau$ and $\eta$ , to satisfy the both conditions, the local $\frac { 1 } { \tau _ { - } }$ -Lipschitz continuity and the $\frac { \eta - \tau } { 2 }$ -comonotonicity.5 A pseudocode of the resulting method, named FEG-A, is illustrated in Algorithm 2. For a detailed description of the FEG-A, see Algorithm 4 in Appendix C.1. + +# Algorithm 2 Fast extragradient method with adaptive step size (FEG-A) + +Input: $\boldsymbol { z } _ { 0 } \in \mathbb { R } ^ { d }$ , $\tau _ { - 1 } \in ( \operatorname* { m a x } \{ 0 , - 2 \rho \} , \infty )$ , $\eta _ { 0 } \in ( 0 , \infty )$ , $\delta \in ( 0 , 1 )$ +Find the smallest nonnegative integer $i _ { 0 }$ such that $\hat { z } = { z _ { 0 } - \tau _ { - 1 } ( 1 - \delta ) ^ { i _ { 0 } } F z _ { 0 } }$ satisfies $\Vert \pmb { F } \hat { z } -$ +$\begin{array} { r } { \pmb { F } z _ { 0 } \| \leq \frac { 1 } { \tau _ { - 1 } ( 1 - \delta ) ^ { i _ { 0 } } } \| \hat { \pmb { z } } - z _ { 0 } \| } \end{array}$ . +$\tau _ { 0 } = \tau _ { - 1 } ( 1 - \delta ) ^ { i _ { 0 } }$ , $z _ { \mathrm { 1 } } = z _ { \mathrm { 0 } } - \tau _ { 0 } F z _ { \mathrm { 0 } }$ . +for $k = 1 , 2 , \ldots$ do $i _ { k } = j _ { k } = 0$ . Increase each $i _ { k }$ and $j _ { k }$ one by one until + +$$ +\begin{array} { r l } & { \hat { z } _ { k + 1 / 2 } = z _ { k } + \cfrac { 1 } { k + 1 } ( z _ { 0 } - z _ { k } ) - \bigg ( 1 - \cfrac { 1 } { k + 1 } \bigg ) \eta _ { k - 1 } ( 1 - \delta ) ^ { j _ { k } } F z _ { k } \qquad \mathrm { a n d } } \\ & { \hat { z } _ { k + 1 } = z _ { k } + \cfrac { 1 } { k + 1 } ( z _ { 0 } - z _ { k } ) - \tau _ { k - 1 } ( 1 - \delta ) ^ { i _ { k } } F z _ { k + 1 / 2 } } \\ & { \qquad - \bigg ( 1 - \cfrac { 1 } { k + 1 } \bigg ) ( \eta _ { k - 1 } ( 1 - \delta ) ^ { j _ { k } } - \tau _ { k - 1 } ( 1 - \delta ) ^ { i _ { k } } ) F z _ { k } } \end{array} +$$ + +satisfy both conditions, + +$$ +\begin{array} { r l } & { \qquad \| F \hat { z } _ { k + 1 } - F \hat { z } _ { k + 1 / 2 } \| \le \frac { 1 } { \tau _ { k - 1 } ( 1 - \delta ) ^ { i _ { k } } } \| \hat { z } _ { k + 1 } - \hat { z } _ { k + 1 / 2 } \| \qquad \mathrm { a n d } } \\ & { \qquad \langle F \hat { z } _ { k + 1 } - F z _ { k } , \hat { z } _ { k + 1 } - z _ { k } \rangle \ge \frac { \eta _ { k - 1 } ( 1 - \delta ) ^ { j _ { k } } - \tau _ { k - 1 } ( 1 - \delta ) ^ { i _ { k } } } { 2 } \| F \hat { z } _ { k + 1 } - F z _ { k } \| ^ { 2 } . } \\ & { \qquad \frac { \tilde { \mathbf { \phi } } _ { k } } { \mathrm { \Delta } r } = \tau _ { k - 1 } ( 1 - \delta ) ^ { i _ { k } } , \eta _ { k } = \eta _ { k - 1 } ( 1 - \delta ) ^ { j _ { k } } , z _ { k + 1 } = \hat { z } _ { k + 1 } . } \end{array} +$$ + +The following lemma shows that each of the nonincreasing sequences $\{ \tau _ { k } \} _ { k \ge 0 }$ and $\{ \eta _ { k } \} _ { k \ge 0 }$ of the FEG-A has a positive lower bound, and thus FEG-A is well-defined6, under the condition $\rho > - \frac { \tau _ { k } } { 2 }$ . This condition for ρ can be weaker than the condition ρ > − 12L of FEG, since the local Lipschitz parameter $\scriptstyle { \frac { 1 } { \tau _ { k } } }$ can be smaller than $L$ . This is another benefit of using a backtracking line-search in FEG, over the standard FEG. + +Lemma 5.1. For the $L$ -Lipschitz and $\rho$ -comonotone operator $\pmb { F }$ and a given constant $\delta \in ( 0 , 1 )$ , the step size $\tau _ { k }$ of FEG-A is lower bounded by a positive value $\begin{array} { r } { \underline { { \tau } } : = \operatorname* { m i n } \left\{ \tau _ { - 1 } , { \frac { 1 - \delta } { L } } \right\} } \end{array}$ for all $k \geq 0$ , and $\begin{array} { r } { { \mathfrak { j } } f \rho > - { \frac { \tau _ { k } } { 2 } } } \end{array}$ , the step size $\eta _ { k }$ is lower bounded by a positive value $\operatorname* { m i n } \left\{ \eta _ { 0 } , ( 1 - \delta ) \bigl ( \tau _ { k } + 2 \rho \bigr ) \right\}$ for all $k \geq 1$ . + +The FEG-A method also has the following $\mathcal { O } ( 1 / k ^ { 2 } )$ rate with respect to the squared gradient norm in Theorem 5.1, when $\rho > - \frac { \tau _ { k } } { 2 }$ . The proof is provided in Section 7 and Appendix C.3. + +Theorem 5.1. For the $L$ -Lipschitz and $\rho$ -comonotone operator $\pmb { F }$ and for any $z _ { \ast } \in Z _ { \ast } ( F )$ , the sequence $\{ z _ { k } \} _ { k \ge 0 }$ generated by FEG-A satisfies + +$$ +\| F z _ { k } \| ^ { 2 } \leq \frac { 4 \| z _ { 0 } - z _ { * } \| ^ { 2 } } { ( ( k - 1 ) \eta _ { k } + \tau _ { k } + 2 \rho ) ^ { 2 } } +$$ + +for all $k \geq 1$ , if $\rho > - \frac { \tau _ { k } } { 2 }$ + +This rate bound of FEG-A reduces to that of FEG in Theorem 4.1, when we choose $\begin{array} { r } { \tau _ { - 1 } = \frac { 1 } { L } } \end{array}$ and $\begin{array} { r } { \eta _ { 0 } = \frac { 1 } { L } + 2 \rho } \end{array}$ for FEG-A. + +# 6 FEG under stochastic setting + +When exactly computing $\pmb { F } z$ is expensive in practice, one usually instead consider its stochastic estimate for computational efficiency (see, e.g., [13, 16, 26, 33, 40, 42, 44]). This section also considers using a stochastic oracle in FEG for smooth convex-concave problems. In specific, this section assumes that we only have access to a noisy saddle gradient oracle, $\tilde { F } z _ { k / 2 } = F \bar { z } _ { k / 2 } + \xi _ { k / 2 }$ , where $\{ \xi _ { k / 2 } \} _ { k \geq 0 }$ are independent random variables satisfying $\mathbb { E } [ \xi _ { k / 2 } ] = 0$ and $\mathbb { E } [ \| \xi _ { k / 2 } \| ^ { 2 } ] = \sigma _ { k / 2 } ^ { 2 }$ for all $k \geq 0$ . Under this setting, we study a stochastic first-order method, named stochastic fast extragradient (S-FEG) method, illustrated in Algorithm 3. + +# Algorithm 3 Stochastic fast extragradient (S-FEG) method + +Input: $\boldsymbol { z } _ { 0 } \in \mathbb { R } ^ { d }$ , $L \in ( 0 , \infty )$ . for $k = 0 , 1 , \ldots$ do + +$$ +\begin{array} { l } { { z _ { k + 1 / 2 } = z _ { k } + \displaystyle \frac { 1 } { k + 1 } ( z _ { 0 } - z _ { k } ) - \left( 1 - \displaystyle \frac { 1 } { k + 1 } \right) \displaystyle \frac { 1 } { L } \tilde { F } z _ { k } } } \\ { { z _ { k + 1 } = z _ { k } + \displaystyle \frac { 1 } { k + 1 } ( z _ { 0 } - z _ { k } ) - \displaystyle \frac { 1 } { L } \tilde { F } z _ { k + 1 / 2 } } } \end{array} +$$ + +# end for + +The following theorem provides an upper bound of the expected squared gradient norm for the S-FEG. (See Appendix D.3 for the proof.) + +Theorem 6.1. Let $\tilde { F } z _ { k / 2 } = F z _ { k / 2 } + \xi _ { k / 2 }$ , where $\{ \xi _ { k / 2 } \} _ { k \geq 0 }$ are independent random variables satisfying $\mathbb { E } [ \xi _ { k / 2 } ] = 0$ and $\mathbb { E } [ \| \xi _ { k / 2 } \| ^ { 2 } ] = \sigma _ { k / 2 } ^ { 2 }$ for all $k \geq 0$ . Then, for the $L$ -Lipschitz continuous and monotone operator $F$ and for any $z _ { * } \in Z _ { * } ( F )$ , the sequence $\{ z _ { k } \} _ { k \ge 0 }$ generated by S-FEG satisfies + +$$ +\mathbb { E } [ \| F z _ { k } \| ^ { 2 } ] \le \frac { 4 L ^ { 2 } \| z _ { 0 } - z _ { * } \| ^ { 2 } } { k ^ { 2 } } + \frac { 6 } { k ^ { 2 } } \left[ \sigma _ { 0 } ^ { 2 } + \sum _ { l = 1 } ^ { k - 1 } ( l ^ { 2 } \sigma _ { l } ^ { 2 } + ( l + 1 ) ^ { 2 } \sigma _ { l + 1 / 2 } ^ { 2 } ) \right] +$$ + +for all $k \geq 1$ . Furthermore, if $\sigma _ { 0 } ^ { 2 } \le \frac { \epsilon } { 6 }$ , $\sigma _ { k } ^ { 2 } \le \frac { \epsilon } { 6 k }$ and $\textstyle \sigma _ { k + 1 / 2 } ^ { 2 } \leq \frac { \epsilon } { 6 ( k + 1 ) }$ for all $k \geq 1$ , then the bound (4) reduces to + +$$ +\mathbb { E } [ \| F z _ { k } \| ^ { 2 } ] \le \frac { 4 L ^ { 2 } \| z _ { 0 } - z _ { * } \| ^ { 2 } } { k ^ { 2 } } + \epsilon +$$ + +for all $k \geq 1$ + +Here, we needed the noise variance $\sigma _ { k / 2 } ^ { 2 }$ to decrease in the order of $\mathcal { O } ( 1 / k )$ so that the stochastic error of the S-FEG does not accumulate. Otherwise, if $\sigma _ { k / 2 } ^ { 2 }$ is a constant for all $k$ , the error accumulates with rate ${ \mathcal { O } } ( k )$ . In short, the S-FEG will suffer from error accumulation, unless the stochastic error decreases with rate $\mathcal { O } ( 1 / k )$ . Such error accumulation behavior also appears in a stochastic version of Nesterov’s fast gradient method [35, 36] for smooth convex minimization [5, 8]. Similar to [5], we believe that adjusting the step coefficients of the S-FEG can make the S-FEG become relatively stable even with a constant noise, which we leave as future work. + +# 7 Convergence analysis with nonincreasing potential lemma + +We analyze FEG and FEG-A by finding a nonincreasing potential function in a form $V _ { k } ~ =$ $a _ { k } \| F z _ { k } \| ^ { 2 } - b _ { k } \left. F z _ { k } , z _ { 0 } - z _ { k } \right.$ in the lemma below. We provide a similar potential lemma for S-FEG in Appendix D.2. The convergence analyses of EAG and Halpern iteration are also based on such potential function [6, 43]. + +Lemma 7.1. Let $\{ z _ { k } \} _ { k \ge 0 }$ be the sequence generated by (Class FEG) with $\{ \alpha _ { k } \} _ { k \geq 0 } , \{ \beta _ { k } \} _ { k \geq 0 } ,$ $\{ L _ { k } \} _ { k \ge 0 } \subset ( 0 , \infty )$ and $\{ \rho _ { k } \} _ { k \ge 0 } \subset \mathbb { R } ,$ , satisfying $\alpha _ { 0 } \in ( 0 , \infty )$ , $\textstyle \alpha _ { k } \in { \bigl ( } 0 , { \frac { 1 } { L _ { k } } } { \bigr ] }$ , $\beta _ { 0 } = 1$ , $\{ \beta _ { k } \} _ { k \ge 1 } \subseteq$ $( 0 , 1 )$ for all $k \geq 1$ , and + +$$ +\frac { ( 1 - \beta _ { k + 1 } ) } { 2 \beta _ { k + 1 } } ( \alpha _ { k + 1 } + 2 \rho _ { k + 1 } ) - \rho _ { k + 1 } \leq \frac { 1 } { 2 \beta _ { k } } ( \alpha _ { k } + 2 \rho _ { k } ) - \rho _ { k } +$$ + +for all $k \geq 0$ . Assume that the following conditions are satisfied. + +$$ +\begin{array} { c } { { \| { \pmb F } { \boldsymbol z } _ { 1 } - { \pmb F } { \boldsymbol z } _ { 0 } \| \leq L _ { 0 } \| { \boldsymbol z } _ { 1 } - { \boldsymbol z } _ { 0 } \| } } \\ { { \| { \pmb F } { \boldsymbol z } _ { k + 1 } - { \pmb F } { \boldsymbol z } _ { k + 1 / 2 } \| \leq L _ { k } \| { \boldsymbol z } _ { k + 1 } - { \boldsymbol z } _ { k + 1 / 2 } \| } } \\ { { \langle { \pmb F } { \boldsymbol z } _ { k + 1 } - { \pmb F } { \boldsymbol z } _ { k } , { \boldsymbol z } _ { k + 1 } - { \pmb z } _ { k } \rangle \geq \rho _ { k } \| { \pmb F } { \boldsymbol z } _ { k + 1 } - { \pmb F } { \boldsymbol z } _ { k } \| ^ { 2 } } } \end{array} +$$ + +Then the potential function + +$$ +V _ { k } = a _ { k } \| F z _ { k } \| ^ { 2 } - b _ { k } \left. F z _ { k } , z _ { 0 } - z _ { k } \right. +$$ + +with $\begin{array} { r } { a _ { 0 } = \frac { \alpha _ { 0 } ( L _ { 0 } ^ { 2 } \alpha _ { 0 } ^ { 2 } - 1 ) } { 2 } } \end{array}$ , $b _ { 0 } = 0 , b _ { 1 } = 1 ,$ + +$$ +a _ { k } = { \frac { b _ { k } ( 1 - \beta _ { k } ) } { 2 \beta _ { k } } } ( \alpha _ { k } + 2 \rho _ { k } ) - b _ { k } \rho _ { k } a n d b _ { k + 1 } = { \frac { b _ { k } } { 1 - \beta _ { k } } } +$$ + +for all $k \geq 1$ satisfies $V _ { k } \le V _ { k - 1 }$ for all $k \geq 1$ + +Based on the above potential lemma, we next provide a convergence analysis of FEG. The analyses for the convergence rate of FEG-A and S-FEG, i.e., the proofs of Theorem 5.1 and Theorem 6.1, are similar to that of FEG and are provided in Appendix C.3 and Appendix D.3. + +# 7.1 Convergence analysis for FEG + +Proof of Theorem 4.1. Recall that FEG is equivalent to (Class FEG) with $\begin{array} { r } { \alpha _ { k } = \frac { 1 } { L } , \beta _ { k } = \frac { 1 } { k + 1 } } \end{array}$ , and $\rho _ { k } = \rho$ . It is straightforward to verify that the given $\{ \alpha _ { k } \} _ { k \ge 0 }$ and $\{ \beta _ { k } \} _ { k \ge 0 }$ satisfy the conditions in Lemma 7.1 with $L _ { k } = L$ for all $k \geq 0$ . Since + +$$ +\begin{array} { c } { { a _ { k } = \displaystyle \frac { b _ { k } ( 1 - \beta _ { k } ) } { 2 \beta _ { k } } ( \alpha _ { k } + 2 \rho _ { k } ) - b _ { k } \rho _ { k } = \displaystyle \frac { k ^ { 2 } } { 2 } \Big ( \frac 1 L + 2 \rho \Big ) - k \rho \qquad \mathrm { a n d } } } \\ { { b _ { k } = \displaystyle \frac { 1 } { 1 - \beta _ { k - 1 } } b _ { k - 1 } = \Big ( \prod _ { i = 1 } ^ { k - 1 } \frac 1 { 1 - \beta _ { i } } \Big ) b _ { 1 } = k , } } \end{array} +$$ + +Lemma 7.1 implies that + +$$ +0 = V _ { 0 } \geq V _ { k } = \left( \frac { k ^ { 2 } } { 2 } \Big ( \frac { 1 } { L } + 2 \rho \Big ) - k \rho \right) \| F z _ { k } \| ^ { 2 } - k \left. F z _ { k } , z _ { 0 } - z _ { k } \right. . +$$ + +Therefore, + +$$ +\begin{array} { r l } & { \displaystyle \frac { k ^ { 2 } } { 2 } \Big ( \frac { 1 } { L } + 2 \rho \Big ) \| F z _ { k } \| ^ { 2 } \leq k \langle F z _ { k } , z _ { 0 } - z _ { k } \rangle + k \rho \| F z _ { k } \| ^ { 2 } } \\ & { \quad \quad \quad \quad = k \langle F z _ { k } , z _ { 0 } - z _ { * } \rangle + k \langle F z _ { k } , z _ { * } - z _ { k } \rangle + k \rho \| F z _ { k } \| ^ { 2 } } \\ & { \quad \quad \quad \leq k \langle F z _ { k } , z _ { 0 } - z _ { * } \rangle \quad \quad \quad \quad ( \because \rho \mathrm { - c o m o n o t o n i c i t y ~ o f ~ } F ) } \\ & { \quad \quad \quad \quad \leq k \| F z _ { k } \| \| z _ { 0 } - z _ { * } \| . } \end{array} +$$ + +The desired result follows directly by dividing both sides by $\begin{array} { r } { \frac { k ^ { 2 } } { 2 } \left( \frac { 1 } { L } + 2 \rho \right) \| F z _ { k } \| } \end{array}$ . + +# 8 Discussion: first-order methods for Lipschitz continuous operators + +Throughout this paper, we studied and constructed efficient methods in a class of first-order methods: + +$$ +z _ { k } \in z _ { 0 } + \operatorname { s p a n } \{ F z _ { 0 } , \cdot \cdot \cdot , F z _ { k } \} +$$ + +denoted by $\mathcal { A }$ , for smooth structured nonconvex-nonconcave problems. We observed that all existing first-order methods, including the FEG, required an additional condition, such as the negative comonoticity, on a Lipschitz continuous $\pmb { F }$ to guarantee convergence. One would then be curious whether or not there exists an (efficient) method in class $\mathcal { A }$ that guarantees convergence without any additional condition on a Lipschitz continuous $\pmb { F }$ . Unfortunately, the following lemma states that there exists a worst-case7 smooth example that none of the methods in $\mathcal { A }$ can find its stationary point. The corresponding smooth function is illustrated in Figure 3. + +![](images/417af2708fada253c811f0d36e8f36eb36b53d148b42635dd18fbe25dc8f1b31.jpg) +Figure 3: A smooth worst-case example $f ( x , y )$ (5) with $L = R = 1$ for first-order methods. any sequence $\{ z _ { k } \} _ { k \ge 0 }$ generated by a first-order method in class $\mathcal { A }$ starting from $( 0 , 0 )$ is contained in the line $x = y$ . + +Lemma 8.1. Let us consider the following function $f : \mathbb { R } ^ { 2 } \to \mathbb { R }$ for some $L , R > 0$ : + +$$ +\begin{array} { r } { f ( x , y ) = \left\{ \begin{array} { l l } { \frac { R } { 2 } } & { f o r x < y - \sqrt { \frac { R } { L } } } \\ { - \frac { L } { 2 } ( x - y ) ^ { 2 } - \sqrt { L R } ( x - y ) } & { f o r y - \sqrt { \frac { R } { L } } \leq x < y } \\ { \frac { L } { 2 } ( x - y ) ^ { 2 } - \sqrt { L R } ( x - y ) } & { f o r y \leq x < y + \sqrt { \frac { R } { L } } } \\ { - \frac { R } { 2 } } & { f o r y + \sqrt { \frac { R } { L } } < x . } \end{array} \right. } \end{array} +$$ + +Its saddle-gradient operator $\pmb { F }$ is $L$ -Lipschitz continuous but not comonotone.8 Then, the sequence $\{ z _ { k } \} _ { k \ge 0 }$ generated by any first-order method in class $\mathcal { A }$ with $z _ { 0 } = ( 0 , 0 )$ satisfies $\| { \pmb { F } } { \pmb { z } } _ { k } \| ^ { 2 } = 2 L R$ for all $k \geq 0$ . + +Proof. $\pmb { F }$ satisfies $F ( x , y ) = ( - \sqrt { L R } , - \sqrt { L R } )$ whenever $x = y$ . Hence, for all sequences $\{ z _ { k } \} _ { k \ge 0 }$ satisfying $z _ { 0 } = ( 0 , 0 )$ and $z _ { k } \in z _ { 0 } + \operatorname { s p a n } \{ F z _ { 0 } , \cdot \cdot \cdot , F z _ { k } \}$ for all $k \geq 0$ , we have that $\{ z _ { k } \} _ { k \ge 0 } \subseteq$ $\{ z = ( x , y ) \in \mathbb { R } ^ { 2 } | x = y \}$ ; thus, $\| { \pmb { F } } z _ { k } \| ^ { 2 } = 2 L R$ for all $k \geq 0$ . + +The lemma implies that one should consider a class of methods, other than the class $\mathcal { A }$ , to guarantee finding a stationary point of any smooth problem, which we leave as future work. We also leave finding additional conditions for a Lipschitz continuous $\pmb { F }$ , weaker than the weak MVI condition and the negative comonotonicity (with $\begin{array} { r } { { \Dot { \rho } } > - \frac { 1 } { 2 L } ) } \end{array}$ , which guarantee convergence or its accelerated rate, respectively, as future work. + +# 9 Conclusion + +This paper proposed a two-time-scale and anchored extragradient method, named FEG, for smooth structured nonconvex-nonconcave problems. The proposed FEG has an accelerated $\mathcal { O } ( 1 / k ^ { 2 } )$ rate, with respect to the squared gradient norm, for the Lipschitz continuous and negative comonotone operators for the first time. The FEG also has value for smooth convex-concave problems, compared to existing works. We further studied its backtracking line-search version, named FEG-A, for the smooth structured nonconvex-nonconcave problems and studied its stochastic version, named S-FEG, for smooth convex-concave problems. We leave extending this work to stochastic, composite, or more general nonconvex-nonconcave setting and applying to more realistic problems as future work. + +# Acknowledgments and Disclosure of Funding + +This work was supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2019R1A5A1028324), the POSCO Science Fellowship of POSCO TJ Park Foundation, and the Samsung Science and Technology Foundation (No. SSTFBA2101-02). + +# References + +[1] H. H. Bauschke, W. M. Moursi, and X. Wang. Generalized monotone operators and their averaged resolvents. Mathematical Programming, 189(1):55–74, 2021. +[2] A. Beck and M. Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci., 2(1):183–202, 2009. +[3] P. L. Combettes and T. Pennanen. Proximal methods for cohypomonotone operators. SIAM J. Control Optim., 43(2):731–42, 2004. +[4] C. D. Dang and G. Lan. 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Accelerated algorithms for smooth convex-concave minimax problems with $\mathcal { O } ( 1 / k ^ { 2 } )$ rate on squared gradient norm. In Proc. Intl. Conf. Mach. Learn, 2021. +[44] Z. Zhou, P. Mertikopoulos, N. Bambos, S. Boyd, and P. Glynn. Stochastic mirror descent in variationally coherent optimization problems. In Neural Info. Proc. Sys., 2017. \ No newline at end of file diff --git a/parse/train/AYAgKFl78z/AYAgKFl78z_content_list.json b/parse/train/AYAgKFl78z/AYAgKFl78z_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..2cbf82aef9f34c646d1cab3129822e5d20fd2edb --- /dev/null +++ b/parse/train/AYAgKFl78z/AYAgKFl78z_content_list.json @@ -0,0 +1,1562 @@ +[ + { + "type": "text", + "text": "Fast Extra Gradient Methods for Smooth Structured Nonconvex-Nonconcave Minimax Problems ", + "text_level": 1, + "bbox": [ + 181, + 122, + 818, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sucheol Lee Department of Mathematical Sciences KAIST Daejeon, Republic of Korea csfh1379@kaist.ac.kr ", + "bbox": [ + 220, + 222, + 472, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Donghwan Kim Department of Mathematical Sciences KAIST Daejeon, Republic of Korea donghwankim@kaist.ac.kr ", + "bbox": [ + 526, + 222, + 777, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 327, + 535, + 343 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Modern minimax problems, such as generative adversarial network and adversarial training, are often under a nonconvex-nonconcave setting, and developing an efficient method for such setting is of interest. Recently, two variants of the extragradient (EG) method are studied in that direction. First, a two-time-scale variant of the EG, named $\\mathrm { E G + }$ , was proposed under a smooth structured nonconvexnonconcave setting, with a slow $\\mathcal { O } ( \\bar { 1 } / k )$ rate on the squared gradient norm, where $k$ denotes the number of iterations. Second, another variant of EG with an anchoring technique, named extra anchored gradient (EAG), was studied under a smooth convex-concave setting, yielding a fast $\\mathcal { O } ( 1 / k ^ { 2 } )$ rate on the squared gradient norm. Built upon $\\mathrm { E G + }$ and EAG, this paper proposes a two-time-scale EG with anchoring, named fast extragradient (FEG), that has a fast $\\mathcal { O } ( 1 / k ^ { 2 } )$ rate on the squared gradient norm for smooth structured nonconvex-nonconcave problems; the corresponding saddle-gradient operator satisfies the negative comonotonicity condition. This paper further develops its backtracking line-search version, named FEG-A, for the case where the problem parameters are not available. The stochastic analysis of FEG is also provided. ", + "bbox": [ + 232, + 358, + 766, + 579 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 604, + 312, + 622 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recently, nonconvex-nonconcave minimax problems have received an increased attention in the optimization community and the machine learning community due to their applications to generative adversarial network [10] and adversarial training [27]. In this paper, we consider a smooth structured nonconvex-nonconcave minimax problem: ", + "bbox": [ + 174, + 637, + 823, + 693 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/34f3ac3a9e2482f1dbde9278f3df5217bf4d4b22aceb6ca5cce31ac1970b7f1b.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\pmb { x } \\in \\mathbb { R } ^ { d _ { \\boldsymbol { x } } } } \\operatorname* { m a x } _ { \\pmb { y } \\in \\mathbb { R } ^ { d _ { \\boldsymbol { y } } } } f ( \\pmb { x } , \\pmb { y } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 423, + 699, + 573, + 724 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "where $f : \\mathbb { R } ^ { d _ { x } } \\times \\mathbb { R } ^ { d _ { y } } \\mathbb { R }$ is smooth and is possibly nonconvex in $_ { \\textbf { \\em x } }$ for fixed $\\textbf { { y } }$ , and possibly nonconcave in $\\textbf { { y } }$ for fixed $_ { \\textbf { \\em x } }$ ; the saddle-gradient operator $\\pmb { F } : = ( \\nabla _ { x } f , - \\nabla _ { y } f )$ satisfies the negative comonotonicity [1]. We construct an efficient (first-order) method, using a saddle gradient operator $\\pmb { F }$ for finding a first-order stationary point of the problem (1). ", + "bbox": [ + 174, + 733, + 823, + 790 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "So far little is known under the nonconvex-nonconcave setting, compared to the convex-concave setting. Recent works [4, 7, 22, 24, 26, 42, 44] studied extragradient-type methods [19, 39] for minimax problems under various structured nonconvex-nonconcave settings. In other words, they consider various non-monotone conditions on $\\pmb { F }$ , such as the Minty variational inequality (MVI) condition [4], the weak MVI condition [7], and the negative comonotonicity [1].1 Among them, this paper focuses on the negative comonotonicity condition for a Lipschitz continuous $\\pmb { F }$ . To the best of our knowledge, the following two-time-scale variant of the extragradient method, named $\\mathrm { E G + }$ : ", + "bbox": [ + 174, + 796, + 825, + 866 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 90, + 825, + 119 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/be87972bb07ea37d3b1242f4c17bf83af8e769a3fe7fd27010661ac3ba8045a2.jpg", + "text": "$$\n\\begin{array} { c } { { z _ { k + 1 / 2 } = z _ { k } - \\frac { \\alpha _ { k } } { \\beta } { \\cal F } z _ { k } , } } \\\\ { { z _ { k + 1 } = z _ { k } - \\alpha _ { k } { \\cal F } z _ { k + 1 / 2 } , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 401, + 126, + 596, + 176 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "is the only known (explicit)2 method, using $\\pmb { F }$ , that converges under the considered setting3 [7], where $z _ { k } : = ( x _ { k } , y _ { k } )$ . The $\\mathrm { E G + }$ , however, has a slow $\\mathcal { O } ( 1 / k )$ rate on the squared gradient norm. Note that a similar two-time-scale approach has been found to stabilize the stochastic extragradient method with unbounded noise variance [14]. ", + "bbox": [ + 173, + 181, + 825, + 238 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Meanwhile, under the smooth convex-concave setting, recent works [6, 17, 21, 40, 43] suggest that Halpern-type [12] (or anchoring) methods, performing a convex combination of an initial point $z _ { \\mathrm { 0 } }$ and the last updated point $z _ { k }$ at each iteration, has a fast $\\mathcal { O } ( 1 / k ^ { 2 } )$ rate in terms of the squared gradient norm. In particular, [43] developed the following anchoring variant of the extragradient method, named extra anchored gradient (EAG): ", + "bbox": [ + 173, + 244, + 825, + 314 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/fd4b411901765ff54f504100c3b3b692556308262a78daf7d60cabe8196e5b11.jpg", + "text": "$$\n\\begin{array} { r l } & { z _ { k + 1 / 2 } = z _ { k } + \\beta _ { k } ( z _ { 0 } - z _ { k } ) - \\alpha _ { k } F z _ { k } , } \\\\ & { \\quad z _ { k + 1 } = z _ { k } + \\beta _ { k } ( z _ { 0 } - z _ { k } ) - \\alpha _ { k } F z _ { k + 1 / 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 319, + 648, + 361 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This is the first (explicit) method with a fast $\\mathcal { O } ( 1 / k ^ { 2 } )$ rate on the squared gradient norm, when $\\pmb { F }$ satisfies both the Lipschitz continuity and the monotonicity. [43] also showed that such $\\mathcal { O } ( 1 / k ^ { 2 } )$ rate is optimal for first-order methods using a Lipschitz continuous and monotone $\\pmb { F }$ . ", + "bbox": [ + 174, + 364, + 823, + 409 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Built upon both $\\mathrm { E G + }$ and EAG, this paper studies the following class of two-time-scale anchored extragradient methods, named fast extragradient (FEG): ", + "bbox": [ + 171, + 414, + 823, + 443 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/edd564677a00f9214a567ae965295ad19386acff4b2dc3661561eff2f334dcf9.jpg", + "text": "$$\n\\begin{array} { r l } & { z _ { k + 1 / 2 } = z _ { k } + \\beta _ { k } ( z _ { 0 } - z _ { k } ) - ( 1 - \\beta _ { k } ) ( \\alpha _ { k } + 2 \\rho _ { k } ) { \\cal F } z _ { k } , } \\\\ & { ~ z _ { k + 1 } = z _ { k } + \\beta _ { k } ( z _ { 0 } - z _ { k } ) - \\alpha _ { k } { \\cal F } z _ { k + 1 / 2 } - ( 1 - \\beta _ { k } ) 2 \\rho _ { k } { \\cal F } z _ { k } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 448, + 673, + 488 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(Class FEG) ", + "bbox": [ + 740, + 460, + 823, + 474 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Note that (Class FEG) reuses the $\\pmb { F } z _ { k }$ term in the $z _ { k + 1 }$ update, unlike the standard extragradienttype methods, which we found essential for handling the negative comonotonicity condition. We leave further understanding the use of $\\pmb { F } z _ { k }$ and the formulation of (Class FEG) as future work. The proposed FEG method (with appropriately chosen step coefficients $\\alpha _ { k }$ , $\\beta _ { k }$ and $\\rho _ { k }$ discussed later) has an $\\mathcal { O } ( 1 / k ^ { 2 } )$ rate on the squared gradient norm, under the Lipschitz continuity and the negative comonotonicity conditions on $\\pmb { F }$ . To the best of our knowledge, this is the first accelerated method under the nonconvex-nonconcave setting. The FEG also has value under the smooth convex-concave setting. First, when $\\pmb { F }$ is Lipschitz continuous and monotone, the rate bound of FEG is about 27/4 times smaller than that of EAG. Also note that the rate bound of FEG is only about four times larger than the $\\mathcal { O } ( 1 / k ^ { 2 } )$ lower complexity bound of first-order methods under such setting [43], further closing the gap between the lower and upper complexity bounds. Second, when $\\pmb { F }$ is cocoercive, FEG has a rate faster than that of a version of Halpern iteration [12] in [6]. ", + "bbox": [ + 173, + 492, + 826, + 659 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We also develop an adaptive variant of FEG, named FEG-A, which updates its parameters, $\\alpha _ { k }$ and $\\rho _ { k }$ in (Class FEG), adaptively using a backtracking line-search [2, 25, 31]. FEG requires the knowledge of the two problem parameters for the Lipschitz continuity and the comonotonicity of $\\pmb { F }$ . However, those global parameters can be conservative, and in practice, they are even usually unknown. For such cases, the FEG-A adaptively and locally estimates the problem parameters, while preserving the fast rate $\\mathcal { O } ( 1 / k ^ { 2 } )$ on the squared gradient norm for smooth structured nonconvex-nonconcave minimax problems. ", + "bbox": [ + 173, + 665, + 825, + 762 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Lastly, we study a stochastic version of FEG, named S-FEG, which uses an unbiased stochastic estimate of $\\pmb { F } z$ , i.e., $\\tilde { F } z = F z + \\xi$ , instead of $\\pmb { F } z$ in FEG, where $\\xi$ denotes a stochastic noise. For a Lipschitz continuous and monotone $\\pmb { F }$ , we provide a convergence analysis in terms of the expected squared gradient norm. In specific, we show that the S-FEG is stable with a rate $\\mathcal { O } ( 1 / k ^ { 2 } ) \\overset { \\cdot } { + } \\mathcal { O } ( \\epsilon )$ , when the noise variance decreases in the order of $\\mathcal { O } ( \\epsilon / k )$ , while being unstable otherwise due to error accumulation. This is similar to the convergence behavior of a stochastic version of Nesterov’s fast gradient method [35, 36], observed in [5], for smooth convex minimization. ", + "bbox": [ + 173, + 767, + 825, + 840 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 92, + 823, + 119 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our main contributions are summarized as follows. ", + "bbox": [ + 174, + 126, + 509, + 140 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• We propose the FEG method that has an accelerated convergence rate $\\mathcal { O } ( 1 / k ^ { 2 } )$ on the squared gradient norm for smooth structured nonconvex-nonconcave minimax problems. \nWe present that the FEG method has a rate faster than those of the EAG and the Halpern iteration for smooth convex-concave problems. We construct a backtracking line-search version of FEG, named FEG-A, for the case where the Lipschitz constant and comonotonicity parameters of $\\pmb { F }$ are unavailable. \n• We analyze a stochastic version of FEG, named S-FEG, for smooth convex-concave problems. ", + "bbox": [ + 215, + 151, + 825, + 277 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Related work ", + "text_level": 1, + "bbox": [ + 174, + 301, + 318, + 318 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 Methods for convex-concave minimax problems ", + "text_level": 1, + "bbox": [ + 174, + 333, + 544, + 348 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The extragradient method [19] is one of the widely used methods for solving smooth convex-concave minimax problems (see, e.g., [4, 7, 22, 24, 26, 42, 44] for its extensions and applications). In terms of the duality gap, $\\begin{array} { r } { \\operatorname* { m a x } _ { { \\pmb y } ^ { \\prime } \\in \\mathcal { V } } f ( { \\pmb x } , { \\pmb y } ^ { \\prime } ) - \\operatorname* { m i n } _ { { \\pmb x } ^ { \\prime } \\in \\mathcal { X } } f ( { \\pmb x } ^ { \\prime } , { \\pmb y } ) } \\end{array}$ , where $\\mathcal { X }$ and $\\mathcal { V }$ are compact4 domains, the ergodic iterate of the extragradient-type methods [32, 37] have an $\\mathcal { O } ( 1 / k )$ rate. Such $\\mathcal { O } ( 1 / k )$ rate on the duality gap is order-optimal for the first-order methods [34, 38], leaving no room for√ improvement. On the other hand, the last iterate of the extragradient method has a slower $\\mathcal { O } ( 1 / \\sqrt { k } )$ rate on the duality gap, under an additional assumption that $\\pmb { F }$ has a Lipschitz derivative [9]. In terms of the squared gradient norm, $\\| \\ b { F z } \\| ^ { 2 }$ , the best iterate of the extragradient-type methods [19, 39] have an $\\mathcal { O } ( 1 / k )$ rate [40, 41, 43]. The last iterate of the extragradient method also has a rate $\\mathcal { O } ( 1 / k )$ , when $\\pmb { F }$ is further assumed to have a Lipschitz derivative [9]. Unlike the duality gap, the $\\mathcal { O } ( 1 / k )$ rate on the squared gradient norm is not optimal [43]. From now on throughout this paper, we mainly study and compare the convergence rates on the squared gradient norm, which still has room for improvement in convex-concave problems, and has meaning for nonconvex-nonconcave minimax problems, unlike the duality gap. ", + "bbox": [ + 173, + 358, + 826, + 554 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Recently, [6, 17, 21, 40, 43] found that Halpern-type [12] (or anchoring) methods yield a fast $\\mathcal { O } ( 1 / k ^ { 2 } )$ rate in terms of the squared gradient norm for minimax problems. [17, 21] showed that the (implicit) Halpern iteration [12] with appropriately chosen step coefficients has an $\\mathcal { O } ( 1 / k ^ { 2 } )$ rate on the squared norm of a monotone $\\pmb { F }$ . Then, for a cocoercive $\\pmb { F }$ , an (explicit) version of the Halpern iteration was studied in [6, 17] that has the same fast rate. In addition, [6] constructed a double-loop version of the Halpern iteration for a Lipschitz continuous and monotone $\\pmb { F }$ , which has a rate $\\tilde { \\mathcal { O } } ( 1 / k ^ { 2 } )$ on the squared gradient norm, slower than the rate $\\mathcal { O } ( 1 / k ^ { 2 } )$ . While this is promising compared to the $\\bar { \\mathcal { O } ( 1 / k ) }$ rate of the extragradient methods on the squared gradient norm [40, 41, 43], the computational complexity due to its double-loop nature and a relatively slow rate remained a problem. Very recently, [43] proposed the extra anchored gradient (EAG) method, which is the first (explicit) method with a fast $\\bar { \\mathcal { O } } ( \\bar { 1 } / k ^ { 2 } )$ rate for smooth convex-concave minimax problems, i.e., for Lipschitz continuous and monotone operators. In addition, [43] proved that the EAG is order-optimal by showing that the lower complexity bound of first-order methods is $\\Omega ( 1 / k ^ { 2 } )$ . ", + "bbox": [ + 173, + 560, + 825, + 741 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/809493f0af3a49bfb3fe1ede2a779bf7b12bcbbd732f36c6179ba488ea52db84.jpg", + "text": "$$\n\\begin{array} { c c c c c } { { \\mathrm { C o c o e r c i v e } } } & { { \\subseteq } } & { { \\bf M o n o t o n e } } & { { \\subseteq } } & { { \\mathrm { N e g a t i v e ~ c o m o n o t o n e } } } \\\\ { { } } & { { } } & { { | \\bigcap } } & { { } } & { { | \\bigcap } } \\\\ { { } } & { { } } & { { \\mathrm { M V I } } } & { { \\subseteq } } & { { \\mathrm { W e a k ~ M V I } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 762, + 694, + 810 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "", + "image_caption": [ + "Figure 1: Relations between the conditions on $\\pmb { F }$ . " + ], + "image_footnote": [], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Methods for nonconvex-nonconcave minimax problems ", + "text_level": 1, + "bbox": [ + 173, + 90, + 598, + 106 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Some recent literature considered relaxing the monotonicity condition of the saddle gradient operator to tackle modern nonconvex-nonconcave minimax problems. For example, the Minty variational inequality (MVI) condition, i.e., there exists $z _ { \\ast } \\in Z _ { \\ast } ( F )$ satisfying $\\langle F z , z - z _ { * } \\rangle \\ge 0$ for all $z \\in \\mathbb { R } ^ { d }$ where $Z _ { * } ( F ) : = \\{ z _ { * } \\in \\mathbb { R } ^ { d } : F z _ { * } = \\mathbf { 0 } \\}$ , is studied in [4, 23, 22, 24]. This condition is also studied under the name, the coherence, in [26, 42, 44]. Moreover, [7] considered a weaker condition, named the weak MVI condition, i.e., for some $\\rho < 0$ , there exists $z _ { \\ast } \\in Z _ { \\ast } ( F )$ satisfying $\\langle F z , z - z _ { * } \\rangle \\geq \\rho \\Vert F z \\Vert ^ { 2 }$ for all $z \\in \\mathbb { R } ^ { d }$ . The weak MVI condition is implied by the negative comonotonicity [1] or, equivalently, the (positive) cohypomonotonicity [3]. The comonotonicity will be further discussed in the upcoming section. ", + "bbox": [ + 173, + 116, + 825, + 244 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For $L$ -Lipschitz continuous $\\pmb { F }$ , [4, 42] showed that the extragradient-type methods have an $\\mathcal { O } ( 1 / k )$ rate on the squared gradient norm under the MVI condition, and [7] developed the $( \\mathrm { E G + } )$ method under the weak MVI condition (and thus under the negative comonotonicty), which also has an $\\mathcal { O } ( 1 / k )$ rate on the squared gradient norm. To the best of our knowledge, there is no known accelerated method for the nonconvex-nonconcave setting; our proposed FEG method is the first method to have a fast $\\mathcal { O } ( 1 / k ^ { 2 } )$ rate under the nonconvex-nonconcave setting. The convergence rates of the existing methods and the FEG on the squared gradient norm are summarized in Table 1. ", + "bbox": [ + 173, + 250, + 825, + 348 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/43e976f98ce2cace0267d5b149ecb9eb71a4e7aa783f17d781c0a7c59c073923.jpg", + "table_caption": [ + "Table 1: Comparison of the convergence rates of the existing extragradient-type methods and the FEG, with respect to the squared gradient norm, for smooth structured minimax problems, under various assumptions on the Lipschitz continuous saddle gradient operator $\\pmb { F }$ . " + ], + "table_footnote": [], + "table_body": "
MethodConvex-concaveNonconvex-nonconcave
Cocoercive MonotoneNegative comonotoneMVIWeak MVI
NormalEG [4, 42]0(1/k)0(1/k）0(1/k）
EG+[7]0(1/k)0(1/k)0(1/k)0(1/k）0(1/k）
AcceleratedHalpern [12, 6]0(1/k²)(1/k²)
EAG [43] FEG (this paper)0(1/k2) 0(1/k2)0(1/k²) 0(1/k2)0(1/k2)
", + "bbox": [ + 176, + 407, + 820, + 529 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 Preliminaries ", + "text_level": 1, + "bbox": [ + 174, + 554, + 318, + 571 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The followings are the two main assumptions for the saddle gradient operator $\\pmb { F }$ of the smooth structured nonconvex-nonconcave problem (1). Under such assumptions, we develop efficient methods that find a first-order stationary point $z _ { \\ast } \\in Z _ { \\ast } ( F )$ where $Z _ { * } ( \\hat { F } ) : = \\{ z _ { * } \\in \\mathbb { R } ^ { d } : \\hat { F } z _ { * } = \\mathbf { 0 } \\}$ . ", + "bbox": [ + 174, + 585, + 826, + 627 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Assumption 1 ( $L$ -Lipschitz continuity). For some $L \\in ( 0 , \\infty )$ , $\\pmb { F }$ satisfies ", + "bbox": [ + 174, + 631, + 665, + 646 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/58b398d57fdabf5eae781a8cad620b97441318815530f3e4a6e3d730f59592d0.jpg", + "text": "$$\n\\| F z - F z ^ { \\prime } \\| \\le L \\| z - z ^ { \\prime } \\| , \\quad \\forall z , z ^ { \\prime } \\in \\mathbb R ^ { d } .\n$$", + "text_format": "latex", + "bbox": [ + 351, + 651, + 645, + 670 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Assumption 2 ( $\\rho$ -Comonotonicity). For some $\\textstyle \\rho \\in { \\bigl ( } - { \\frac { 1 } { 2 L } } , \\infty { \\bigr ) }$ , $\\pmb { F }$ satisfies ", + "bbox": [ + 173, + 676, + 669, + 694 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f517a07e39a41fd0b3b884dac93732d30c287f839e2f96809371b781868b805c.jpg", + "text": "$$\n\\begin{array} { r } { \\langle F z - F z ^ { \\prime } , z - z ^ { \\prime } \\rangle \\ge \\rho \\| F z - F z ^ { \\prime } \\| ^ { 2 } , \\quad \\forall z , z ^ { \\prime } \\in \\mathbb { R } ^ { d } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 310, + 699, + 687, + 718 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The $\\rho$ -comonotonicity consists of three cases depending on the choice of $\\rho$ ; the negative comonotonicity when $\\rho < 0$ , the monotonicity when $\\rho = 0$ , and the cocoercivity when $\\rho > 0$ . The negative comonotonicity is weaker than the other two, and is the main focus of this paper. The following is an examplary nonconvex-nonconcave condition that is stronger than the negative comonotonicity [1, 3]. ", + "bbox": [ + 174, + 729, + 826, + 786 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Example 1. Let $f$ be twice continuously differentiable and $\\gamma$ -weakly-convex-weakly-concave. Further assume that $f$ satisfies ", + "bbox": [ + 174, + 790, + 825, + 819 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a80262cf18f822172cf8e321a77ba19488f8a7472e73f345fcb48c608c0dd382.jpg", + "text": "$$\n\\begin{array} { r } { \\nabla _ { x x } ^ { 2 } f + \\nabla _ { x y } ^ { 2 } f ( \\eta I - \\nabla _ { y y } ^ { 2 } f ) ^ { - 1 } \\nabla _ { y x } ^ { 2 } f \\succeq \\alpha I , } \\\\ { - \\nabla _ { y y } ^ { 2 } f + \\nabla _ { y x } ^ { 2 } f ( \\eta I + \\nabla _ { x x } ^ { 2 } f ) ^ { - 1 } \\nabla _ { x y } ^ { 2 } f \\succeq \\alpha I , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 339, + 821, + 655, + 866 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "for some $\\alpha \\geq 0$ and $\\eta > \\gamma ,$ , named $\\alpha \\geq 0$ -interaction dominant condition in [11]. Then, the saddle gradient of $f$ satisfies the $- \\frac { 1 } { \\eta }$ -negative comonotonicity. (See Appendix A.1.) For any $\\gamma$ -weaklyconvex-weakly-concave function, the condition (2) holds with $\\alpha = - \\gamma < 0$ . Its extreme case is ", + "bbox": [ + 173, + 867, + 825, + 912 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { f ( x , y ) = - \\frac { \\gamma } { 2 } x ^ { 2 } + \\frac { \\gamma } { 2 } y ^ { 2 } } \\end{array}$ , where there is no interaction between x and y. On the other hand, when the \nthe second terms in the left-hand side of (2) are sufficently positive definite, a nonconvex-nonconave \nfunction satisfies the condition (2) with a nonnegative condition is satisfied when the interaction term of Hess $\\alpha$ .n cific, the is domin $\\alpha \\geq 0$ -interaction dominantany negative curvature $\\nabla _ { \\substack { x y } } ^ { 2 } f$ \nin Hessians $\\nabla _ { x x } ^ { 2 } f$ and $- \\nabla _ { \\boldsymbol { y } \\boldsymbol { y } } ^ { 2 } f l l l \\boldsymbol { l } \\boldsymbol { l } \\boldsymbol { l } \\boldsymbol { l }$ . ", + "bbox": [ + 173, + 89, + 825, + 166 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We next present our proposed FEG, and illustrate that the FEG outperforms existing methods such as $\\mathrm { E G + }$ , EAG, and the Halpern iteration, for each three comonoticity case, respectively. ", + "bbox": [ + 173, + 174, + 823, + 204 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 Fast extragradient (FEG) method for Lipschitz continuous and comonotone operators ", + "text_level": 1, + "bbox": [ + 174, + 223, + 730, + 258 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This section considers an instance of (Class FEG) with $\\begin{array} { r } { \\alpha _ { k } = \\frac { 1 } { L } } \\end{array}$ , $\\begin{array} { r } { \\beta _ { k } = \\frac { 1 } { k + 1 } } \\end{array}$ , and $\\rho _ { k } = \\rho$ for all $k \\geq 0$ The resulting method, named FEG, is illustrated in Algorithm 1, which has an $\\mathcal { O } ( 1 / k ^ { 2 } )$ fast rate with respect to the squared gradient norm, in Theorem 4.1. The proof of Theorem 4.1 is provided in Section 7. ", + "bbox": [ + 173, + 271, + 826, + 332 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1 Fast extragradient (FEG) method ", + "text_level": 1, + "bbox": [ + 174, + 344, + 480, + 359 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/e6f770a71e1f68a45474304fcaeed89a31d829b4be358d3bc85afcb0ac396edf.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Input: z0 ∈ Rd,L ∈ (0,∞o),ρ ∈(- 2,00) for k = 0,1,... do
2+1/=+1(2-（1-1）（+2）F 1
1 2 2k+1= 2k+ k+1
", + "bbox": [ + 184, + 361, + 745, + 460 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "end for ", + "bbox": [ + 189, + 459, + 243, + 473 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 4.1. For the $L$ -Lipschitz continuous and $\\rho$ -comonotone operator $\\pmb { F }$ with $\\rho > - \\frac { 1 } { 2 L }$ and for any $z _ { \\ast } \\in Z _ { \\ast } ( F )$ , the sequence $\\{ z _ { k } \\} _ { k \\ge 0 }$ generated by FEG satisfies, for all $k \\geq 1$ , ", + "bbox": [ + 171, + 500, + 823, + 529 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e2f6cd273461c9b92d8707e1f757dd91d03d1770236f8581afadd094ba3dcb75.jpg", + "text": "$$\n\\| F z _ { k } \\| ^ { 2 } \\leq \\frac { 4 \\| z _ { 0 } - z _ { * } \\| ^ { 2 } } { \\Big ( \\frac { 1 } { L } + 2 \\rho \\Big ) ^ { 2 } k ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 408, + 536, + 589, + 583 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The following example shows that the bound (3) of the FEG is exact for $\\rho = 0$ and $k = 4 l + 2$ . The bound (3) is not known to be exact in general, and we leave finding the exact bound as future work. ", + "bbox": [ + 173, + 595, + 825, + 626 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Example 2. Let $f : \\mathbb { R } \\times \\mathbb { R } \\to \\mathbb { R }$ be $f ( x , y ) = L x y .$ . Its saddle gradient operator and solution are $\\pmb { F } ( x , y ) = ( L y , - L x )$ and $z _ { * } = ( 0 , 0 )$ , respectively. For the initial point $z _ { 0 } = ( x _ { 0 } , y _ { 0 } ) =$ $( 1 , 0 )$ , the sequence $\\{ z _ { k } \\} _ { k \\ge 0 }$ generated by FEG satisfies $\\begin{array} { r } { z _ { 4 l + 2 } = \\left( 0 , \\frac { 1 } { 2 l + 1 } \\right) } \\end{array}$ for all $l \\geq 0$ . Hence, $\\begin{array} { r } { \\| F z _ { 4 l + 2 } \\| ^ { 2 } = \\frac { L ^ { 2 } } { ( 2 l + 1 ) ^ { 2 } } = \\frac { 4 L ^ { 2 } \\| z _ { 0 } - z _ { * } \\| ^ { 2 } } { ( 4 l + 2 ) ^ { 2 } } } \\end{array}$ for all $l \\geq 0 .$ . (See Appendix B.1.) ", + "bbox": [ + 173, + 628, + 826, + 699 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We next compare the rate bound (3) with existing analyses for the three cases $\\begin{array} { r } { - \\frac { 1 } { 2 L } < \\rho < 0 , \\rho = 0 } \\end{array}$ and $\\rho > 0$ . ", + "bbox": [ + 173, + 709, + 823, + 739 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 Comparison to $\\mathbf { E G + }$ under the negative comonotonicity $( \\rho < 0 )$ ) ", + "text_level": 1, + "bbox": [ + 174, + 755, + 651, + 770 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Under the negative comonotonicity with $\\begin{array} { r } { - \\frac { 1 } { 8 L } < \\rho < 0 } \\end{array}$ , the $( \\mathrm { E G + } )$ method with $\\begin{array} { r } { \\alpha _ { k } = \\frac { 1 } { 2 L } } \\end{array}$ and $\\begin{array} { r } { \\beta = \\frac { 1 } { 2 } } \\end{array}$ rate on the squared gradient norm. To the best of our knowledge, this is the best known rate, and the FEG has a faster $\\mathcal { O } ( 1 \\bar { \\vert } k ^ { 2 } )$ rate with a wider region of convergence $\\begin{array} { r } { - \\frac { 1 } { 2 L } < \\rho < 0 } \\end{array}$ . ", + "bbox": [ + 174, + 780, + 825, + 824 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 Comparison to EAG under the monotonicity $( \\rho = 0$ ) ", + "text_level": 1, + "bbox": [ + 173, + 839, + 576, + 856 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For an $L$ -Lipschitz continuous and monotone operator $\\pmb { F }$ , [43] proposed two EAG methods, named EAG-C and EAG-V, with same βk = 1k+2 but with different choices of $\\alpha _ { k }$ . EAG-C sets $\\alpha _ { k }$ to be a constant $\\frac { 1 } { 8 L }$ for all $k \\geq 0$ in (EAG), and has a large constant 260 in its convergence rate, $\\begin{array} { r } { \\| \\pmb { F } \\pmb { z } _ { k } \\| ^ { 2 } \\le \\frac { 2 6 0 L ^ { 2 } \\| \\pmb { z } _ { 0 } - \\pmb { z } _ { * } \\| ^ { 2 } } { ( k + 1 ) ^ { 2 } } } \\end{array}$ 260L2kz0−z∗k22 for all k ≥ 0. On the other hand, while EAG-V requires a complicated recursive update for {αk}, αk+1 = αk1−α2L2 \u0000 $\\begin{array} { r } { \\alpha _ { k + 1 } = \\frac { \\alpha _ { k } } { 1 - \\alpha _ { k } ^ { 2 } L ^ { 2 } } \\big ( 1 - \\frac { ( k + 2 ) ^ { 2 } } { ( k + 1 ) ( k + 3 ) } \\alpha _ { k } ^ { 2 } L ^ { 2 } \\big ) } \\end{array}$ for all $k \\geq 0$ , with $\\begin{array} { r } { \\alpha _ { 0 } = \\frac { 0 . 6 1 8 } { L } } \\end{array}$ , its rate has a smaller constant 27. ", + "bbox": [ + 174, + 866, + 825, + 912 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/380353bbf2580099fbe770143af658fb4debcb8120afd75852aaa71dc5266deb.jpg", + "image_caption": [ + "Figure 2: Numerical result with $\\begin{array} { r } { f ( x , y ) = - \\frac { 1 } { 6 } x ^ { 2 } + \\frac { 2 \\sqrt { 2 } } { 3 } x y + \\frac { 1 } { 6 } y ^ { 2 } } \\end{array}$ . The dashed line represents the theoretical bound (3) of FEG. " + ], + "image_footnote": [], + "bbox": [ + 333, + 95, + 643, + 277 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 339, + 826, + 398 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The FEG takes a constant $\\begin{array} { r } { \\alpha _ { k } \\ = \\ \\frac { 1 } { L } } \\end{array}$ , unlike EAG-V, but has an even smaller constant 4 in its convergence rate $\\begin{array} { r } { \\| \\pmb { F } \\pmb { z } _ { k } \\| ^ { 2 } \\le \\frac { 4 L ^ { 2 } \\| \\pmb { z } _ { 0 } - \\pmb { z } _ { * } \\| ^ { 2 } } { k ^ { 2 } } } \\end{array}$ 4L2kz0−z∗k2k2 for ρ = 0. Therefore, the FEG with ρ = 0 has about $2 6 0 / 4$ -times and $2 7 / 4$ -times faster convergence rate compared to those of EAG-C and EAG-V, respectively. Furthermore, the rate bound of FEG with $\\rho = 0$ is only about 4-times larger than the lower complexity bound of first-order methods under the considered setting [43], reducing the gap between the lower and upper complexity bounds from 27 to 4. ", + "bbox": [ + 173, + 404, + 826, + 494 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 Comparison to the Halpern iteration under the cocoercivity $( \\rho > 0 )$ ) ", + "text_level": 1, + "bbox": [ + 174, + 510, + 679, + 525 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For a $\\rho$ -cocoercive operator $\\pmb { F }$ , an (explicit) version of Halpern iteration [12], studied in [6], has a fast rate, $\\begin{array} { r } { \\| \\boldsymbol { F } \\boldsymbol { z } _ { k } \\| ^ { 2 } \\le \\frac { \\| \\boldsymbol { z } _ { 0 } - \\boldsymbol { z } _ { * } \\| ^ { 2 } } { \\rho ^ { 2 } k ^ { 2 } } } \\end{array}$ kz0−z∗k2ρ2k2 . Note that while the ρ-cocoercivity implies the 1ρ -Lipschitz continuity, there is case where the $\\rho$ -cocoercive (and thus Lipschitz continuous) operator has a Lipschitz constant $L$ smaller than $\\frac { 1 } { \\rho }$ . Since $\\begin{array} { r } { L \\le \\frac { 1 } { \\rho } } \\end{array}$ , the FEG has a rate $\\begin{array} { r } { \\| \\boldsymbol { F } \\boldsymbol { z } _ { k } \\| ^ { 2 } \\le \\frac { 4 \\| \\boldsymbol { z } _ { 0 } - \\boldsymbol { z } _ { * } \\| ^ { 2 } } { ( 1 / L + 2 \\rho ) ^ { 2 } k ^ { 2 } } = \\frac { 4 \\| \\boldsymbol { z } _ { 0 } - \\boldsymbol { z } _ { * } \\| ^ { 2 } } { 9 \\rho ^ { 2 } k ^ { 2 } } } \\end{array}$ that is faster than that of Halpern iteration. However, if we take into account that the FEG requires computing the saddle gradient twice per iteration, unlike Halpern iteration studied in [6], the FEG method has a slower rate in terms of the number of gradient computations. If we narrow down to the case $\\begin{array} { r } { L < \\frac { 1 } { 2 \\rho } } \\end{array}$ , the FEG has a faster rate, $\\begin{array} { r } { \\| \\ b { F } \\ b { z } _ { k } \\| ^ { 2 } \\leq \\frac { 4 \\| \\ b { z } _ { 0 } - \\ b { z } _ { * } \\| ^ { 2 } } { ( 1 / L + 2 \\rho ) ^ { 2 } k ^ { 2 } } < \\frac { \\| \\ b { z } _ { 0 } - \\ b { z } _ { * } \\| ^ { 2 } } { 4 \\rho ^ { 2 } k ^ { 2 } } } \\end{array}$ . For such case, the FEG has a rate faster than that of the Halpern iteration, even in terms of the number of gradient computations. ", + "bbox": [ + 173, + 535, + 826, + 694 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.4 Toy example ", + "text_level": 1, + "bbox": [ + 174, + 710, + 300, + 727 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "rformed a toy exp, which has an iment on a simple quadratic-Lipschitz continuous and nction, -comon $\\begin{array} { r } { f ( x , y ) = \\frac { \\rho L ^ { 2 } } { 2 } x ^ { 2 } + L \\sqrt { 1 - \\rho ^ { 2 } L ^ { 2 } } x y - } \\end{array}$ ${ \\frac { \\rho L ^ { 2 } } { 2 } } y ^ { 2 }$ $L$ $\\rho$ $\\begin{array} { r } { \\rho = - \\frac { 1 } { 3 L } } \\end{array}$ and $L = 1$ , Figure 2 illustrates that the FEG converges with an accelerated rate whereas $\\mathrm { E G + }$ , EAG-C, EAG-V, and the (explicit) version of Halpern iteration [6] diverge. This example presents that the existing guarantees on convergence and acceleration of the aforementioned methods under the convex-concave setting do not generalize to the nonconvex-nonconcave setting. ", + "bbox": [ + 173, + 736, + 825, + 832 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 FEG with backtracking line-search ", + "text_level": 1, + "bbox": [ + 173, + 852, + 500, + 869 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The FEG requires the knowledge of the two global parameters $L$ and $\\rho$ for Lipschitz continuity and comonotonicity, respectively. Those global parameters are often difficult to compute in practice and can be locally conservative. To handle these two disadvantages, we employ the backtracking line-search technique [2, 25, 31] in FEG. We adaptively decrease the two step size parameters, $\\tau$ and $\\eta$ , to satisfy the both conditions, the local $\\frac { 1 } { \\tau _ { - } }$ -Lipschitz continuity and the $\\frac { \\eta - \\tau } { 2 }$ -comonotonicity.5 A pseudocode of the resulting method, named FEG-A, is illustrated in Algorithm 2. For a detailed description of the FEG-A, see Algorithm 4 in Appendix C.1. ", + "bbox": [ + 174, + 882, + 823, + 911 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 825, + 162 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Algorithm 2 Fast extragradient method with adaptive step size (FEG-A) ", + "text_level": 1, + "bbox": [ + 174, + 175, + 650, + 190 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Input: $\\boldsymbol { z } _ { 0 } \\in \\mathbb { R } ^ { d }$ , $\\tau _ { - 1 } \\in ( \\operatorname* { m a x } \\{ 0 , - 2 \\rho \\} , \\infty )$ , $\\eta _ { 0 } \\in ( 0 , \\infty )$ , $\\delta \\in ( 0 , 1 )$ \nFind the smallest nonnegative integer $i _ { 0 }$ such that $\\hat { z } = { z _ { 0 } - \\tau _ { - 1 } ( 1 - \\delta ) ^ { i _ { 0 } } F z _ { 0 } }$ satisfies $\\Vert \\pmb { F } \\hat { z } -$ \n$\\begin{array} { r } { \\pmb { F } z _ { 0 } \\| \\leq \\frac { 1 } { \\tau _ { - 1 } ( 1 - \\delta ) ^ { i _ { 0 } } } \\| \\hat { \\pmb { z } } - z _ { 0 } \\| } \\end{array}$ . \n$\\tau _ { 0 } = \\tau _ { - 1 } ( 1 - \\delta ) ^ { i _ { 0 } }$ , $z _ { \\mathrm { 1 } } = z _ { \\mathrm { 0 } } - \\tau _ { 0 } F z _ { \\mathrm { 0 } }$ . \nfor $k = 1 , 2 , \\ldots$ do $i _ { k } = j _ { k } = 0$ . Increase each $i _ { k }$ and $j _ { k }$ one by one until ", + "bbox": [ + 184, + 194, + 823, + 297 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/74a178cf962bc3b725f9e2bc1d72f3bdb7d7506cd2287b60cdddc3505b5ab96b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\hat { z } _ { k + 1 / 2 } = z _ { k } + \\cfrac { 1 } { k + 1 } ( z _ { 0 } - z _ { k } ) - \\bigg ( 1 - \\cfrac { 1 } { k + 1 } \\bigg ) \\eta _ { k - 1 } ( 1 - \\delta ) ^ { j _ { k } } F z _ { k } \\qquad \\mathrm { a n d } } \\\\ & { \\hat { z } _ { k + 1 } = z _ { k } + \\cfrac { 1 } { k + 1 } ( z _ { 0 } - z _ { k } ) - \\tau _ { k - 1 } ( 1 - \\delta ) ^ { i _ { k } } F z _ { k + 1 / 2 } } \\\\ & { \\qquad - \\bigg ( 1 - \\cfrac { 1 } { k + 1 } \\bigg ) ( \\eta _ { k - 1 } ( 1 - \\delta ) ^ { j _ { k } } - \\tau _ { k - 1 } ( 1 - \\delta ) ^ { i _ { k } } ) F z _ { k } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 303, + 761, + 398 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "satisfy both conditions, ", + "bbox": [ + 214, + 402, + 369, + 416 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/c660f47f968af31a63209b2ff8c54203fc5b0e6e1e84c18f83944ce1e9138c84.jpg", + "text": "$$\n\\begin{array} { r l } & { \\qquad \\| F \\hat { z } _ { k + 1 } - F \\hat { z } _ { k + 1 / 2 } \\| \\le \\frac { 1 } { \\tau _ { k - 1 } ( 1 - \\delta ) ^ { i _ { k } } } \\| \\hat { z } _ { k + 1 } - \\hat { z } _ { k + 1 / 2 } \\| \\qquad \\mathrm { a n d } } \\\\ & { \\qquad \\langle F \\hat { z } _ { k + 1 } - F z _ { k } , \\hat { z } _ { k + 1 } - z _ { k } \\rangle \\ge \\frac { \\eta _ { k - 1 } ( 1 - \\delta ) ^ { j _ { k } } - \\tau _ { k - 1 } ( 1 - \\delta ) ^ { i _ { k } } } { 2 } \\| F \\hat { z } _ { k + 1 } - F z _ { k } \\| ^ { 2 } . } \\\\ & { \\qquad \\frac { \\tilde { \\mathbf { \\phi } } _ { k } } { \\mathrm { \\Delta } r } = \\tau _ { k - 1 } ( 1 - \\delta ) ^ { i _ { k } } , \\eta _ { k } = \\eta _ { k - 1 } ( 1 - \\delta ) ^ { j _ { k } } , z _ { k + 1 } = \\hat { z } _ { k + 1 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 220, + 422, + 784, + 516 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The following lemma shows that each of the nonincreasing sequences $\\{ \\tau _ { k } \\} _ { k \\ge 0 }$ and $\\{ \\eta _ { k } \\} _ { k \\ge 0 }$ of the FEG-A has a positive lower bound, and thus FEG-A is well-defined6, under the condition $\\rho > - \\frac { \\tau _ { k } } { 2 }$ . This condition for ρ can be weaker than the condition ρ > − 12L of FEG, since the local Lipschitz parameter $\\scriptstyle { \\frac { 1 } { \\tau _ { k } } }$ can be smaller than $L$ . This is another benefit of using a backtracking line-search in FEG, over the standard FEG. ", + "bbox": [ + 173, + 540, + 826, + 617 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Lemma 5.1. For the $L$ -Lipschitz and $\\rho$ -comonotone operator $\\pmb { F }$ and a given constant $\\delta \\in ( 0 , 1 )$ , the step size $\\tau _ { k }$ of FEG-A is lower bounded by a positive value $\\begin{array} { r } { \\underline { { \\tau } } : = \\operatorname* { m i n } \\left\\{ \\tau _ { - 1 } , { \\frac { 1 - \\delta } { L } } \\right\\} } \\end{array}$ for all $k \\geq 0$ , and $\\begin{array} { r } { { \\mathfrak { j } } f \\rho > - { \\frac { \\tau _ { k } } { 2 } } } \\end{array}$ , the step size $\\eta _ { k }$ is lower bounded by a positive value $\\operatorname* { m i n } \\left\\{ \\eta _ { 0 } , ( 1 - \\delta ) \\bigl ( \\tau _ { k } + 2 \\rho \\bigr ) \\right\\}$ for all $k \\geq 1$ . ", + "bbox": [ + 173, + 621, + 826, + 681 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The FEG-A method also has the following $\\mathcal { O } ( 1 / k ^ { 2 } )$ rate with respect to the squared gradient norm in Theorem 5.1, when $\\rho > - \\frac { \\tau _ { k } } { 2 }$ . The proof is provided in Section 7 and Appendix C.3. ", + "bbox": [ + 171, + 689, + 823, + 719 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 5.1. For the $L$ -Lipschitz and $\\rho$ -comonotone operator $\\pmb { F }$ and for any $z _ { \\ast } \\in Z _ { \\ast } ( F )$ , the sequence $\\{ z _ { k } \\} _ { k \\ge 0 }$ generated by FEG-A satisfies ", + "bbox": [ + 173, + 720, + 823, + 750 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/5164e93d7f1b4359c7f7f6dbd2162ce4c0504c4aa4073a8adaa11f4154f686bc.jpg", + "text": "$$\n\\| F z _ { k } \\| ^ { 2 } \\leq \\frac { 4 \\| z _ { 0 } - z _ { * } \\| ^ { 2 } } { ( ( k - 1 ) \\eta _ { k } + \\tau _ { k } + 2 \\rho ) ^ { 2 } }\n$$", + "text_format": "latex", + "bbox": [ + 377, + 751, + 619, + 789 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "for all $k \\geq 1$ , if $\\rho > - \\frac { \\tau _ { k } } { 2 }$ ", + "bbox": [ + 173, + 789, + 344, + 806 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "This rate bound of FEG-A reduces to that of FEG in Theorem 4.1, when we choose $\\begin{array} { r } { \\tau _ { - 1 } = \\frac { 1 } { L } } \\end{array}$ and $\\begin{array} { r } { \\eta _ { 0 } = \\frac { 1 } { L } + 2 \\rho } \\end{array}$ for FEG-A. ", + "bbox": [ + 173, + 815, + 825, + 847 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 FEG under stochastic setting ", + "text_level": 1, + "bbox": [ + 174, + 88, + 447, + 107 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "When exactly computing $\\pmb { F } z$ is expensive in practice, one usually instead consider its stochastic estimate for computational efficiency (see, e.g., [13, 16, 26, 33, 40, 42, 44]). This section also considers using a stochastic oracle in FEG for smooth convex-concave problems. In specific, this section assumes that we only have access to a noisy saddle gradient oracle, $\\tilde { F } z _ { k / 2 } = F \\bar { z } _ { k / 2 } + \\xi _ { k / 2 }$ , where $\\{ \\xi _ { k / 2 } \\} _ { k \\geq 0 }$ are independent random variables satisfying $\\mathbb { E } [ \\xi _ { k / 2 } ] = 0$ and $\\mathbb { E } [ \\| \\xi _ { k / 2 } \\| ^ { 2 } ] = \\sigma _ { k / 2 } ^ { 2 }$ for all $k \\geq 0$ . Under this setting, we study a stochastic first-order method, named stochastic fast extragradient (S-FEG) method, illustrated in Algorithm 3. ", + "bbox": [ + 173, + 118, + 826, + 224 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Algorithm 3 Stochastic fast extragradient (S-FEG) method ", + "text_level": 1, + "bbox": [ + 173, + 237, + 562, + 252 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Input: $\\boldsymbol { z } _ { 0 } \\in \\mathbb { R } ^ { d }$ , $L \\in ( 0 , \\infty )$ . for $k = 0 , 1 , \\ldots$ do ", + "bbox": [ + 186, + 255, + 387, + 285 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/382b2e06eef0b6657f3ff684e9fe2da5e9d7a5a736d6b52f6b28cc4a4ba49a8c.jpg", + "text": "$$\n\\begin{array} { l } { { z _ { k + 1 / 2 } = z _ { k } + \\displaystyle \\frac { 1 } { k + 1 } ( z _ { 0 } - z _ { k } ) - \\left( 1 - \\displaystyle \\frac { 1 } { k + 1 } \\right) \\displaystyle \\frac { 1 } { L } \\tilde { F } z _ { k } } } \\\\ { { z _ { k + 1 } = z _ { k } + \\displaystyle \\frac { 1 } { k + 1 } ( z _ { 0 } - z _ { k } ) - \\displaystyle \\frac { 1 } { L } \\tilde { F } z _ { k + 1 / 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 282, + 692, + 348 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "end for ", + "text_level": 1, + "bbox": [ + 189, + 352, + 243, + 364 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The following theorem provides an upper bound of the expected squared gradient norm for the S-FEG. (See Appendix D.3 for the proof.) ", + "bbox": [ + 173, + 381, + 823, + 410 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Theorem 6.1. Let $\\tilde { F } z _ { k / 2 } = F z _ { k / 2 } + \\xi _ { k / 2 }$ , where $\\{ \\xi _ { k / 2 } \\} _ { k \\geq 0 }$ are independent random variables satisfying $\\mathbb { E } [ \\xi _ { k / 2 } ] = 0$ and $\\mathbb { E } [ \\| \\xi _ { k / 2 } \\| ^ { 2 } ] = \\sigma _ { k / 2 } ^ { 2 }$ for all $k \\geq 0$ . Then, for the $L$ -Lipschitz continuous and monotone operator $F$ and for any $z _ { * } \\in Z _ { * } ( F )$ , the sequence $\\{ z _ { k } \\} _ { k \\ge 0 }$ generated by S-FEG satisfies ", + "bbox": [ + 173, + 412, + 825, + 476 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/7f9606efa3d23c5eceeaa2b97aa883b0f48834a7c9f80110e6314d9c3d452b63.jpg", + "text": "$$\n\\mathbb { E } [ \\| F z _ { k } \\| ^ { 2 } ] \\le \\frac { 4 L ^ { 2 } \\| z _ { 0 } - z _ { * } \\| ^ { 2 } } { k ^ { 2 } } + \\frac { 6 } { k ^ { 2 } } \\left[ \\sigma _ { 0 } ^ { 2 } + \\sum _ { l = 1 } ^ { k - 1 } ( l ^ { 2 } \\sigma _ { l } ^ { 2 } + ( l + 1 ) ^ { 2 } \\sigma _ { l + 1 / 2 } ^ { 2 } ) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 253, + 477, + 743, + 521 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "for all $k \\geq 1$ . Furthermore, if $\\sigma _ { 0 } ^ { 2 } \\le \\frac { \\epsilon } { 6 }$ , $\\sigma _ { k } ^ { 2 } \\le \\frac { \\epsilon } { 6 k }$ and $\\textstyle \\sigma _ { k + 1 / 2 } ^ { 2 } \\leq \\frac { \\epsilon } { 6 ( k + 1 ) }$ for all $k \\geq 1$ , then the bound (4) reduces to ", + "bbox": [ + 171, + 523, + 826, + 554 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/673a740db6b460eeffefa1d512531991080b20378946835845d2546c335a2530.jpg", + "text": "$$\n\\mathbb { E } [ \\| F z _ { k } \\| ^ { 2 } ] \\le \\frac { 4 L ^ { 2 } \\| z _ { 0 } - z _ { * } \\| ^ { 2 } } { k ^ { 2 } } + \\epsilon\n$$", + "text_format": "latex", + "bbox": [ + 380, + 556, + 617, + 589 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "for all $k \\geq 1$ ", + "bbox": [ + 171, + 590, + 263, + 606 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Here, we needed the noise variance $\\sigma _ { k / 2 } ^ { 2 }$ to decrease in the order of $\\mathcal { O } ( 1 / k )$ so that the stochastic error of the S-FEG does not accumulate. Otherwise, if $\\sigma _ { k / 2 } ^ { 2 }$ is a constant for all $k$ , the error accumulates with rate ${ \\mathcal { O } } ( k )$ . In short, the S-FEG will suffer from error accumulation, unless the stochastic error decreases with rate $\\mathcal { O } ( 1 / k )$ . Such error accumulation behavior also appears in a stochastic version of Nesterov’s fast gradient method [35, 36] for smooth convex minimization [5, 8]. Similar to [5], we believe that adjusting the step coefficients of the S-FEG can make the S-FEG become relatively stable even with a constant noise, which we leave as future work. ", + "bbox": [ + 173, + 613, + 826, + 719 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7 Convergence analysis with nonincreasing potential lemma ", + "text_level": 1, + "bbox": [ + 173, + 737, + 687, + 756 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We analyze FEG and FEG-A by finding a nonincreasing potential function in a form $V _ { k } ~ =$ $a _ { k } \\| F z _ { k } \\| ^ { 2 } - b _ { k } \\left. F z _ { k } , z _ { 0 } - z _ { k } \\right.$ in the lemma below. We provide a similar potential lemma for S-FEG in Appendix D.2. The convergence analyses of EAG and Halpern iteration are also based on such potential function [6, 43]. ", + "bbox": [ + 173, + 767, + 826, + 824 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Lemma 7.1. Let $\\{ z _ { k } \\} _ { k \\ge 0 }$ be the sequence generated by (Class FEG) with $\\{ \\alpha _ { k } \\} _ { k \\geq 0 } , \\{ \\beta _ { k } \\} _ { k \\geq 0 } ,$ $\\{ L _ { k } \\} _ { k \\ge 0 } \\subset ( 0 , \\infty )$ and $\\{ \\rho _ { k } \\} _ { k \\ge 0 } \\subset \\mathbb { R } ,$ , satisfying $\\alpha _ { 0 } \\in ( 0 , \\infty )$ , $\\textstyle \\alpha _ { k } \\in { \\bigl ( } 0 , { \\frac { 1 } { L _ { k } } } { \\bigr ] }$ , $\\beta _ { 0 } = 1$ , $\\{ \\beta _ { k } \\} _ { k \\ge 1 } \\subseteq$ $( 0 , 1 )$ for all $k \\geq 1$ , and ", + "bbox": [ + 173, + 825, + 825, + 873 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/e295973a7169ca644968b92ff5d1a1fa708621d1fbf16782684725f666275069.jpg", + "text": "$$\n\\frac { ( 1 - \\beta _ { k + 1 } ) } { 2 \\beta _ { k + 1 } } ( \\alpha _ { k + 1 } + 2 \\rho _ { k + 1 } ) - \\rho _ { k + 1 } \\leq \\frac { 1 } { 2 \\beta _ { k } } ( \\alpha _ { k } + 2 \\rho _ { k } ) - \\rho _ { k }\n$$", + "text_format": "latex", + "bbox": [ + 295, + 876, + 700, + 910 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "for all $k \\geq 0$ . Assume that the following conditions are satisfied. ", + "bbox": [ + 171, + 90, + 596, + 106 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/49b2b519ba42b6693179bf8ae5dd8a3bd4b5c97be3b8c358b5251c46eff12ccb.jpg", + "text": "$$\n\\begin{array} { c } { { \\| { \\pmb F } { \\boldsymbol z } _ { 1 } - { \\pmb F } { \\boldsymbol z } _ { 0 } \\| \\leq L _ { 0 } \\| { \\boldsymbol z } _ { 1 } - { \\boldsymbol z } _ { 0 } \\| } } \\\\ { { \\| { \\pmb F } { \\boldsymbol z } _ { k + 1 } - { \\pmb F } { \\boldsymbol z } _ { k + 1 / 2 } \\| \\leq L _ { k } \\| { \\boldsymbol z } _ { k + 1 } - { \\boldsymbol z } _ { k + 1 / 2 } \\| } } \\\\ { { \\langle { \\pmb F } { \\boldsymbol z } _ { k + 1 } - { \\pmb F } { \\boldsymbol z } _ { k } , { \\boldsymbol z } _ { k + 1 } - { \\pmb z } _ { k } \\rangle \\geq \\rho _ { k } \\| { \\pmb F } { \\boldsymbol z } _ { k + 1 } - { \\pmb F } { \\boldsymbol z } _ { k } \\| ^ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 267, + 108, + 630, + 165 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Then the potential function ", + "bbox": [ + 173, + 167, + 352, + 181 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/1d43dcacaaf4f5dd94eb7f8ec44f9db705f678335179477f50b44f6a1ddb97a2.jpg", + "text": "$$\nV _ { k } = a _ { k } \\| F z _ { k } \\| ^ { 2 } - b _ { k } \\left. F z _ { k } , z _ { 0 } - z _ { k } \\right.\n$$", + "text_format": "latex", + "bbox": [ + 369, + 184, + 629, + 202 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "with $\\begin{array} { r } { a _ { 0 } = \\frac { \\alpha _ { 0 } ( L _ { 0 } ^ { 2 } \\alpha _ { 0 } ^ { 2 } - 1 ) } { 2 } } \\end{array}$ , $b _ { 0 } = 0 , b _ { 1 } = 1 ,$ ", + "bbox": [ + 174, + 205, + 431, + 227 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/4be93e42dfda0c79f0664cc896405edf4cc225f1e62124c32060169ffdeba45f.jpg", + "text": "$$\na _ { k } = { \\frac { b _ { k } ( 1 - \\beta _ { k } ) } { 2 \\beta _ { k } } } ( \\alpha _ { k } + 2 \\rho _ { k } ) - b _ { k } \\rho _ { k } a n d b _ { k + 1 } = { \\frac { b _ { k } } { 1 - \\beta _ { k } } }\n$$", + "text_format": "latex", + "bbox": [ + 295, + 229, + 702, + 263 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "for all $k \\geq 1$ satisfies $V _ { k } \\le V _ { k - 1 }$ for all $k \\geq 1$ ", + "bbox": [ + 173, + 265, + 482, + 281 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Based on the above potential lemma, we next provide a convergence analysis of FEG. The analyses for the convergence rate of FEG-A and S-FEG, i.e., the proofs of Theorem 5.1 and Theorem 6.1, are similar to that of FEG and are provided in Appendix C.3 and Appendix D.3. ", + "bbox": [ + 174, + 290, + 825, + 333 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7.1 Convergence analysis for FEG ", + "text_level": 1, + "bbox": [ + 174, + 347, + 423, + 363 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Proof of Theorem 4.1. Recall that FEG is equivalent to (Class FEG) with $\\begin{array} { r } { \\alpha _ { k } = \\frac { 1 } { L } , \\beta _ { k } = \\frac { 1 } { k + 1 } } \\end{array}$ , and $\\rho _ { k } = \\rho$ . It is straightforward to verify that the given $\\{ \\alpha _ { k } \\} _ { k \\ge 0 }$ and $\\{ \\beta _ { k } \\} _ { k \\ge 0 }$ satisfy the conditions in Lemma 7.1 with $L _ { k } = L$ for all $k \\geq 0$ . Since ", + "bbox": [ + 176, + 372, + 825, + 416 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/36e62d72f9634c54f042ed2fe814f1d388d8ee2c8e91ac2076dd3ce1bef1823a.jpg", + "text": "$$\n\\begin{array} { c } { { a _ { k } = \\displaystyle \\frac { b _ { k } ( 1 - \\beta _ { k } ) } { 2 \\beta _ { k } } ( \\alpha _ { k } + 2 \\rho _ { k } ) - b _ { k } \\rho _ { k } = \\displaystyle \\frac { k ^ { 2 } } { 2 } \\Big ( \\frac 1 L + 2 \\rho \\Big ) - k \\rho \\qquad \\mathrm { a n d } } } \\\\ { { b _ { k } = \\displaystyle \\frac { 1 } { 1 - \\beta _ { k - 1 } } b _ { k - 1 } = \\Big ( \\prod _ { i = 1 } ^ { k - 1 } \\frac 1 { 1 - \\beta _ { i } } \\Big ) b _ { 1 } = k , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 271, + 420, + 728, + 498 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Lemma 7.1 implies that ", + "bbox": [ + 173, + 501, + 331, + 515 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/98caa60be890fc6b4faf7b664b065c56e97985f31c6546a7f24b54a8a26e579e.jpg", + "text": "$$\n0 = V _ { 0 } \\geq V _ { k } = \\left( \\frac { k ^ { 2 } } { 2 } \\Big ( \\frac { 1 } { L } + 2 \\rho \\Big ) - k \\rho \\right) \\| F z _ { k } \\| ^ { 2 } - k \\left. F z _ { k } , z _ { 0 } - z _ { k } \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 264, + 517, + 732, + 553 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Therefore, ", + "bbox": [ + 173, + 555, + 245, + 569 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/1dfdd9ea91944e95ca9d03700e1fc5e2d6630d84ba1c74aedd4c0e3604a09ecd.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\frac { k ^ { 2 } } { 2 } \\Big ( \\frac { 1 } { L } + 2 \\rho \\Big ) \\| F z _ { k } \\| ^ { 2 } \\leq k \\langle F z _ { k } , z _ { 0 } - z _ { k } \\rangle + k \\rho \\| F z _ { k } \\| ^ { 2 } } \\\\ & { \\quad \\quad \\quad \\quad = k \\langle F z _ { k } , z _ { 0 } - z _ { * } \\rangle + k \\langle F z _ { k } , z _ { * } - z _ { k } \\rangle + k \\rho \\| F z _ { k } \\| ^ { 2 } } \\\\ & { \\quad \\quad \\quad \\leq k \\langle F z _ { k } , z _ { 0 } - z _ { * } \\rangle \\quad \\quad \\quad \\quad ( \\because \\rho \\mathrm { - c o m o n o t o n i c i t y ~ o f ~ } F ) } \\\\ & { \\quad \\quad \\quad \\quad \\leq k \\| F z _ { k } \\| \\| z _ { 0 } - z _ { * } \\| . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 571, + 754, + 660 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The desired result follows directly by dividing both sides by $\\begin{array} { r } { \\frac { k ^ { 2 } } { 2 } \\left( \\frac { 1 } { L } + 2 \\rho \\right) \\| F z _ { k } \\| } \\end{array}$ . ", + "bbox": [ + 171, + 661, + 704, + 683 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "8 Discussion: first-order methods for Lipschitz continuous operators ", + "text_level": 1, + "bbox": [ + 169, + 698, + 761, + 715 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Throughout this paper, we studied and constructed efficient methods in a class of first-order methods: ", + "bbox": [ + 169, + 728, + 823, + 743 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/03be5922e9969d2018e6ac91d4ae72346b3eff2a43120b876c721ba8237810a6.jpg", + "text": "$$\nz _ { k } \\in z _ { 0 } + \\operatorname { s p a n } \\{ F z _ { 0 } , \\cdot \\cdot \\cdot , F z _ { k } \\}\n$$", + "text_format": "latex", + "bbox": [ + 387, + 747, + 611, + 763 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "denoted by $\\mathcal { A }$ , for smooth structured nonconvex-nonconcave problems. We observed that all existing first-order methods, including the FEG, required an additional condition, such as the negative comonoticity, on a Lipschitz continuous $\\pmb { F }$ to guarantee convergence. One would then be curious whether or not there exists an (efficient) method in class $\\mathcal { A }$ that guarantees convergence without any additional condition on a Lipschitz continuous $\\pmb { F }$ . Unfortunately, the following lemma states that there exists a worst-case7 smooth example that none of the methods in $\\mathcal { A }$ can find its stationary point. The corresponding smooth function is illustrated in Figure 3. ", + "bbox": [ + 173, + 766, + 826, + 864 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/417af2708fada253c811f0d36e8f36eb36b53d148b42635dd18fbe25dc8f1b31.jpg", + "image_caption": [ + "Figure 3: A smooth worst-case example $f ( x , y )$ (5) with $L = R = 1$ for first-order methods. any sequence $\\{ z _ { k } \\} _ { k \\ge 0 }$ generated by a first-order method in class $\\mathcal { A }$ starting from $( 0 , 0 )$ is contained in the line $x = y$ . " + ], + "image_footnote": [], + "bbox": [ + 333, + 114, + 643, + 275 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Lemma 8.1. Let us consider the following function $f : \\mathbb { R } ^ { 2 } \\to \\mathbb { R }$ for some $L , R > 0$ : ", + "bbox": [ + 171, + 354, + 730, + 372 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/337423b721db43d80d764145e8e0607d246394769080daa8296b7c981a4a7770.jpg", + "text": "$$\n\\begin{array} { r } { f ( x , y ) = \\left\\{ \\begin{array} { l l } { \\frac { R } { 2 } } & { f o r x < y - \\sqrt { \\frac { R } { L } } } \\\\ { - \\frac { L } { 2 } ( x - y ) ^ { 2 } - \\sqrt { L R } ( x - y ) } & { f o r y - \\sqrt { \\frac { R } { L } } \\leq x < y } \\\\ { \\frac { L } { 2 } ( x - y ) ^ { 2 } - \\sqrt { L R } ( x - y ) } & { f o r y \\leq x < y + \\sqrt { \\frac { R } { L } } } \\\\ { - \\frac { R } { 2 } } & { f o r y + \\sqrt { \\frac { R } { L } } < x . } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 277, + 377, + 720, + 474 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Its saddle-gradient operator $\\pmb { F }$ is $L$ -Lipschitz continuous but not comonotone.8 Then, the sequence $\\{ z _ { k } \\} _ { k \\ge 0 }$ generated by any first-order method in class $\\mathcal { A }$ with $z _ { 0 } = ( 0 , 0 )$ satisfies $\\| { \\pmb { F } } { \\pmb { z } } _ { k } \\| ^ { 2 } = 2 L R$ for all $k \\geq 0$ . ", + "bbox": [ + 174, + 482, + 823, + 525 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Proof. $\\pmb { F }$ satisfies $F ( x , y ) = ( - \\sqrt { L R } , - \\sqrt { L R } )$ whenever $x = y$ . Hence, for all sequences $\\{ z _ { k } \\} _ { k \\ge 0 }$ satisfying $z _ { 0 } = ( 0 , 0 )$ and $z _ { k } \\in z _ { 0 } + \\operatorname { s p a n } \\{ F z _ { 0 } , \\cdot \\cdot \\cdot , F z _ { k } \\}$ for all $k \\geq 0$ , we have that $\\{ z _ { k } \\} _ { k \\ge 0 } \\subseteq$ $\\{ z = ( x , y ) \\in \\mathbb { R } ^ { 2 } | x = y \\}$ ; thus, $\\| { \\pmb { F } } z _ { k } \\| ^ { 2 } = 2 L R$ for all $k \\geq 0$ . ", + "bbox": [ + 173, + 539, + 825, + 584 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The lemma implies that one should consider a class of methods, other than the class $\\mathcal { A }$ , to guarantee finding a stationary point of any smooth problem, which we leave as future work. We also leave finding additional conditions for a Lipschitz continuous $\\pmb { F }$ , weaker than the weak MVI condition and the negative comonotonicity (with $\\begin{array} { r } { { \\Dot { \\rho } } > - \\frac { 1 } { 2 L } ) } \\end{array}$ , which guarantee convergence or its accelerated rate, respectively, as future work. ", + "bbox": [ + 173, + 597, + 825, + 667 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "9 Conclusion ", + "text_level": 1, + "bbox": [ + 173, + 686, + 299, + 704 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This paper proposed a two-time-scale and anchored extragradient method, named FEG, for smooth structured nonconvex-nonconcave problems. The proposed FEG has an accelerated $\\mathcal { O } ( 1 / k ^ { 2 } )$ rate, with respect to the squared gradient norm, for the Lipschitz continuous and negative comonotone operators for the first time. The FEG also has value for smooth convex-concave problems, compared to existing works. We further studied its backtracking line-search version, named FEG-A, for the smooth structured nonconvex-nonconcave problems and studied its stochastic version, named S-FEG, for smooth convex-concave problems. We leave extending this work to stochastic, composite, or more general nonconvex-nonconcave setting and applying to more realistic problems as future work. ", + "bbox": [ + 173, + 718, + 826, + 830 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments and Disclosure of Funding ", + "text_level": 1, + "bbox": [ + 174, + 88, + 553, + 107 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "This work was supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2019R1A5A1028324), the POSCO Science Fellowship of POSCO TJ Park Foundation, and the Samsung Science and Technology Foundation (No. SSTFBA2101-02). 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We", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 413, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 259, + 424 + ], + "score": 1.0, + "content": "leave further understanding the use of", + "type": "text" + }, + { + "bbox": [ + 259, + 413, + 279, + 424 + ], + "score": 0.91, + "content": "\\pmb { F } z _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 413, + 505, + 424 + ], + "score": 1.0, + "content": "and the formulation of (Class FEG) as future work. The", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 423, + 380, + 437 + ], + "score": 1.0, + "content": "proposed FEG method (with appropriately chosen step coefficients", + "type": "text" + }, + { + "bbox": [ + 381, + 424, + 393, + 435 + ], + "score": 0.82, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 423, + 397, + 437 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 397, + 424, + 409, + 435 + ], + "score": 0.86, + "content": "\\beta _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 423, + 427, + 437 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 428, + 425, + 439, + 435 + ], + "score": 0.86, + "content": "\\rho _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "discussed later)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 135, + 447 + ], + "score": 1.0, + "content": "has an", + "type": "text" + }, + { + "bbox": [ + 135, + 434, + 171, + 446 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "rate on the squared gradient norm, under the Lipschitz continuity and the negative", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 446, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 229, + 457 + ], + "score": 1.0, + "content": "comonotonicity conditions on", + "type": "text" + }, + { + "bbox": [ + 230, + 446, + 239, + 455 + ], + "score": 0.82, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 446, + 506, + 457 + ], + "score": 1.0, + "content": ". To the best of our knowledge, this is the first accelerated method", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "under the nonconvex-nonconcave setting. The FEG also has value under the smooth convex-concave", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 187, + 479 + ], + "score": 1.0, + "content": "setting. First, when", + "type": "text" + }, + { + "bbox": [ + 187, + 468, + 197, + 477 + ], + "score": 0.82, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "is Lipschitz continuous and monotone, the rate bound of FEG is about 27/4", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 476, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 506, + 491 + ], + "score": 1.0, + "content": "times smaller than that of EAG. Also note that the rate bound of FEG is only about four times larger", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 142, + 501 + ], + "score": 1.0, + "content": "than the", + "type": "text" + }, + { + "bbox": [ + 142, + 488, + 179, + 501 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "lower complexity bound of first-order methods under such setting [43], further", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 499, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 437, + 513 + ], + "score": 1.0, + "content": "closing the gap between the lower and upper complexity bounds. Second, when", + "type": "text" + }, + { + "bbox": [ + 437, + 500, + 447, + 510 + ], + "score": 0.8, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 499, + 507, + 513 + ], + "score": 1.0, + "content": "is cocoercive,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 510, + 406, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 406, + 523 + ], + "score": 1.0, + "content": "FEG has a rate faster than that of a version of Halpern iteration [12] in [6].", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 526, + 504, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 462, + 540 + ], + "score": 1.0, + "content": "We also develop an adaptive variant of FEG, named FEG-A, which updates its parameters,", + "type": "text" + }, + { + "bbox": [ + 462, + 529, + 475, + 538 + ], + "score": 0.86, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 526, + 492, + 540 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 492, + 529, + 504, + 539 + ], + "score": 0.83, + "content": "\\rho _ { k }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 536, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 505, + 551 + ], + "score": 1.0, + "content": "in (Class FEG), adaptively using a backtracking line-search [2, 25, 31]. FEG requires the knowledge", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 547, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 451, + 562 + ], + "score": 1.0, + "content": "of the two problem parameters for the Lipschitz continuity and the comonotonicity of", + "type": "text" + }, + { + "bbox": [ + 452, + 549, + 461, + 559 + ], + "score": 0.76, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 547, + 506, + 562 + ], + "score": 1.0, + "content": ". However,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 560, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 506, + 572 + ], + "score": 1.0, + "content": "those global parameters can be conservative, and in practice, they are even usually unknown. For", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 569, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 505, + 584 + ], + "score": 1.0, + "content": "such cases, the FEG-A adaptively and locally estimates the problem parameters, while preserving", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 156, + 595 + ], + "score": 1.0, + "content": "the fast rate", + "type": "text" + }, + { + "bbox": [ + 157, + 581, + 193, + 594 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "on the squared gradient norm for smooth structured nonconvex-nonconcave", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 593, + 187, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 187, + 605 + ], + "score": 1.0, + "content": "minimax problems.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 608, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "Lastly, we study a stochastic version of FEG, named S-FEG, which uses an unbiased stochastic", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 152, + 633 + ], + "score": 1.0, + "content": "estimate of", + "type": "text" + }, + { + "bbox": [ + 152, + 621, + 168, + 632 + ], + "score": 0.85, + "content": "\\pmb { F } z", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 621, + 188, + 633 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 189, + 620, + 248, + 633 + ], + "score": 0.92, + "content": "\\tilde { F } z = F z + \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 621, + 293, + 633 + ], + "score": 1.0, + "content": ", instead of", + "type": "text" + }, + { + "bbox": [ + 293, + 622, + 308, + 632 + ], + "score": 0.87, + "content": "\\pmb { F } z", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 621, + 369, + 633 + ], + "score": 1.0, + "content": "in FEG, where", + "type": "text" + }, + { + "bbox": [ + 369, + 622, + 375, + 633 + ], + "score": 0.84, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 621, + 506, + 633 + ], + "score": 1.0, + "content": "denotes a stochastic noise. For a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 252, + 644 + ], + "score": 1.0, + "content": "Lipschitz continuous and monotone", + "type": "text" + }, + { + "bbox": [ + 252, + 633, + 262, + 642 + ], + "score": 0.79, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 632, + 505, + 644 + ], + "score": 1.0, + "content": ", we provide a convergence analysis in terms of the expected", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 433, + 657 + ], + "score": 1.0, + "content": "squared gradient norm. In specific, we show that the S-FEG is stable with a rate", + "type": "text" + }, + { + "bbox": [ + 434, + 642, + 502, + 655 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } ) \\overset { \\cdot } { + } \\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 642, + 506, + 657 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 311, + 666 + ], + "score": 1.0, + "content": "when the noise variance decreases in the order of", + "type": "text" + }, + { + "bbox": [ + 312, + 654, + 342, + 666 + ], + "score": 0.93, + "content": "\\mathcal { O } ( \\epsilon / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 654, + 506, + 666 + ], + "score": 1.0, + "content": ", while being unstable otherwise due to", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 681, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 120, + 680, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 120, + 680, + 505, + 693 + ], + "score": 1.0, + "content": "2A proximal point method converges under the negative comonotonicity [1, 18], but such implicit method is", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 690, + 370, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 370, + 702 + ], + "score": 1.0, + "content": "not preferable over explicit methods in practice due to its implicit nature.", + "type": "text" + } + ] + }, + { + "bbox": [ + 119, + 698, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 119, + 698, + 141, + 714 + ], + "score": 1.0, + "content": "3The", + "type": "text" + }, + { + "bbox": [ + 141, + 701, + 160, + 711 + ], + "score": 0.81, + "content": "\\mathrm { E G + }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 698, + 398, + 714 + ], + "score": 1.0, + "content": "was originally shown to work under the weak MVI condition of", + "type": "text" + }, + { + "bbox": [ + 399, + 702, + 407, + 711 + ], + "score": 0.82, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 698, + 505, + 714 + ], + "score": 1.0, + "content": ", which is weaker than the", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 712, + 198, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 198, + 723 + ], + "score": 1.0, + "content": "negative comonotonicity.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 72, + 505, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 72, + 506, + 96 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 100, + 365, + 140 + ], + "lines": [ + { + "bbox": [ + 246, + 100, + 365, + 140 + ], + "spans": [ + { + "bbox": [ + 246, + 100, + 365, + 140 + ], + "score": 0.91, + "content": "\\begin{array} { c } { { z _ { k + 1 / 2 } = z _ { k } - \\frac { \\alpha _ { k } } { \\beta } { \\cal F } z _ { k } , } } \\\\ { { z _ { k + 1 } = z _ { k } - \\alpha _ { k } { \\cal F } z _ { k + 1 / 2 } , } } \\end{array}", + "type": "interline_equation", + "image_path": "be87972bb07ea37d3b1242f4c17bf83af8e769a3fe7fd27010661ac3ba8045a2.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 100, + 365, + 120.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 246, + 120.0, + 365, + 140.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 144, + 505, + 189 + ], + "lines": [ + { + "bbox": [ + 104, + 144, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 104, + 144, + 289, + 158 + ], + "score": 1.0, + "content": "is the only known (explicit)2 method, using", + "type": "text" + }, + { + "bbox": [ + 290, + 146, + 299, + 155 + ], + "score": 0.84, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 144, + 506, + 158 + ], + "score": 1.0, + "content": ", that converges under the considered setting3 [7],", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 156, + 507, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 133, + 169 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 156, + 196, + 168 + ], + "score": 0.93, + "content": "z _ { k } : = ( x _ { k } , y _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 156, + 220, + 169 + ], + "score": 1.0, + "content": ". The", + "type": "text" + }, + { + "bbox": [ + 221, + 156, + 241, + 167 + ], + "score": 0.83, + "content": "\\mathrm { E G + }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 156, + 330, + 169 + ], + "score": 1.0, + "content": ", however, has a slow", + "type": "text" + }, + { + "bbox": [ + 330, + 156, + 362, + 168 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 156, + 507, + 169 + ], + "score": 1.0, + "content": "rate on the squared gradient norm.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 166, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 506, + 179 + ], + "score": 1.0, + "content": "Note that a similar two-time-scale approach has been found to stabilize the stochastic extragradient", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 177, + 287, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 287, + 190 + ], + "score": 1.0, + "content": "method with unbounded noise variance [14].", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 144, + 507, + 190 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 194, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 105, + 193, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 506, + 208 + ], + "score": 1.0, + "content": "Meanwhile, under the smooth convex-concave setting, recent works [6, 17, 21, 40, 43] suggest that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 504, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 492, + 219 + ], + "score": 1.0, + "content": "Halpern-type [12] (or anchoring) methods, performing a convex combination of an initial point", + "type": "text" + }, + { + "bbox": [ + 493, + 207, + 504, + 216 + ], + "score": 0.82, + "content": "z _ { \\mathrm { 0 } }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 208, + 228 + ], + "score": 1.0, + "content": "and the last updated point", + "type": "text" + }, + { + "bbox": [ + 208, + 217, + 219, + 227 + ], + "score": 0.86, + "content": "z _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 215, + 325, + 228 + ], + "score": 1.0, + "content": "at each iteration, has a fast", + "type": "text" + }, + { + "bbox": [ + 325, + 216, + 362, + 228 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "rate in terms of the squared gradient", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "norm. In particular, [43] developed the following anchoring variant of the extragradient method,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 263, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 263, + 250 + ], + "score": 1.0, + "content": "named extra anchored gradient (EAG):", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 193, + 506, + 250 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 253, + 397, + 286 + ], + "lines": [ + { + "bbox": [ + 214, + 253, + 397, + 286 + ], + "spans": [ + { + "bbox": [ + 214, + 253, + 397, + 286 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { z _ { k + 1 / 2 } = z _ { k } + \\beta _ { k } ( z _ { 0 } - z _ { k } ) - \\alpha _ { k } F z _ { k } , } \\\\ & { \\quad z _ { k + 1 } = z _ { k } + \\beta _ { k } ( z _ { 0 } - z _ { k } ) - \\alpha _ { k } F z _ { k + 1 / 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "fd4b411901765ff54f504100c3b3b692556308262a78daf7d60cabe8196e5b11.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 214, + 253, + 397, + 269.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 214, + 269.5, + 397, + 286.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 504, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 504, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 288, + 303 + ], + "score": 1.0, + "content": "This is the first (explicit) method with a fast", + "type": "text" + }, + { + "bbox": [ + 288, + 289, + 325, + 302 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 289, + 494, + 303 + ], + "score": 1.0, + "content": "rate on the squared gradient norm, when", + "type": "text" + }, + { + "bbox": [ + 495, + 291, + 504, + 300 + ], + "score": 0.8, + "content": "\\pmb { F }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 450, + 313 + ], + "score": 1.0, + "content": "satisfies both the Lipschitz continuity and the monotonicity. 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We", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 413, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 259, + 424 + ], + "score": 1.0, + "content": "leave further understanding the use of", + "type": "text" + }, + { + "bbox": [ + 259, + 413, + 279, + 424 + ], + "score": 0.91, + "content": "\\pmb { F } z _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 413, + 505, + 424 + ], + "score": 1.0, + "content": "and the formulation of (Class FEG) as future work. The", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 423, + 380, + 437 + ], + "score": 1.0, + "content": "proposed FEG method (with appropriately chosen step coefficients", + "type": "text" + }, + { + "bbox": [ + 381, + 424, + 393, + 435 + ], + "score": 0.82, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 423, + 397, + 437 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 397, + 424, + 409, + 435 + ], + "score": 0.86, + "content": "\\beta _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 423, + 427, + 437 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 428, + 425, + 439, + 435 + ], + "score": 0.86, + "content": "\\rho _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "discussed later)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 135, + 447 + ], + "score": 1.0, + "content": "has an", + "type": "text" + }, + { + "bbox": [ + 135, + 434, + 171, + 446 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "rate on the squared gradient norm, under the Lipschitz continuity and the negative", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 446, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 229, + 457 + ], + "score": 1.0, + "content": "comonotonicity conditions on", + "type": "text" + }, + { + "bbox": [ + 230, + 446, + 239, + 455 + ], + "score": 0.82, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 446, + 506, + 457 + ], + "score": 1.0, + "content": ". To the best of our knowledge, this is the first accelerated method", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "under the nonconvex-nonconcave setting. The FEG also has value under the smooth convex-concave", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 187, + 479 + ], + "score": 1.0, + "content": "setting. First, when", + "type": "text" + }, + { + "bbox": [ + 187, + 468, + 197, + 477 + ], + "score": 0.82, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "is Lipschitz continuous and monotone, the rate bound of FEG is about 27/4", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 476, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 506, + 491 + ], + "score": 1.0, + "content": "times smaller than that of EAG. Also note that the rate bound of FEG is only about four times larger", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 142, + 501 + ], + "score": 1.0, + "content": "than the", + "type": "text" + }, + { + "bbox": [ + 142, + 488, + 179, + 501 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "lower complexity bound of first-order methods under such setting [43], further", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 499, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 437, + 513 + ], + "score": 1.0, + "content": "closing the gap between the lower and upper complexity bounds. Second, when", + "type": "text" + }, + { + "bbox": [ + 437, + 500, + 447, + 510 + ], + "score": 0.8, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 499, + 507, + 513 + ], + "score": 1.0, + "content": "is cocoercive,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 510, + 406, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 406, + 523 + ], + "score": 1.0, + "content": "FEG has a rate faster than that of a version of Halpern iteration [12] in [6].", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 390, + 507, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 526, + 504, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 462, + 540 + ], + "score": 1.0, + "content": "We also develop an adaptive variant of FEG, named FEG-A, which updates its parameters,", + "type": "text" + }, + { + "bbox": [ + 462, + 529, + 475, + 538 + ], + "score": 0.86, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 526, + 492, + 540 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 492, + 529, + 504, + 539 + ], + "score": 0.83, + "content": "\\rho _ { k }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 536, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 505, + 551 + ], + "score": 1.0, + "content": "in (Class FEG), adaptively using a backtracking line-search [2, 25, 31]. FEG requires the knowledge", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 547, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 451, + 562 + ], + "score": 1.0, + "content": "of the two problem parameters for the Lipschitz continuity and the comonotonicity of", + "type": "text" + }, + { + "bbox": [ + 452, + 549, + 461, + 559 + ], + "score": 0.76, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 547, + 506, + 562 + ], + "score": 1.0, + "content": ". However,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 560, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 506, + 572 + ], + "score": 1.0, + "content": "those global parameters can be conservative, and in practice, they are even usually unknown. For", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 569, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 505, + 584 + ], + "score": 1.0, + "content": "such cases, the FEG-A adaptively and locally estimates the problem parameters, while preserving", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 156, + 595 + ], + "score": 1.0, + "content": "the fast rate", + "type": "text" + }, + { + "bbox": [ + 157, + 581, + 193, + 594 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "on the squared gradient norm for smooth structured nonconvex-nonconcave", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 593, + 187, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 187, + 605 + ], + "score": 1.0, + "content": "minimax problems.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 526, + 506, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 608, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "Lastly, we study a stochastic version of FEG, named S-FEG, which uses an unbiased stochastic", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 152, + 633 + ], + "score": 1.0, + "content": "estimate of", + "type": "text" + }, + { + "bbox": [ + 152, + 621, + 168, + 632 + ], + "score": 0.85, + "content": "\\pmb { F } z", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 621, + 188, + 633 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 189, + 620, + 248, + 633 + ], + "score": 0.92, + "content": "\\tilde { F } z = F z + \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 621, + 293, + 633 + ], + "score": 1.0, + "content": ", instead of", + "type": "text" + }, + { + "bbox": [ + 293, + 622, + 308, + 632 + ], + "score": 0.87, + "content": "\\pmb { F } z", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 621, + 369, + 633 + ], + "score": 1.0, + "content": "in FEG, where", + "type": "text" + }, + { + "bbox": [ + 369, + 622, + 375, + 633 + ], + "score": 0.84, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 621, + 506, + 633 + ], + "score": 1.0, + "content": "denotes a stochastic noise. For a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 252, + 644 + ], + "score": 1.0, + "content": "Lipschitz continuous and monotone", + "type": "text" + }, + { + "bbox": [ + 252, + 633, + 262, + 642 + ], + "score": 0.79, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 632, + 505, + 644 + ], + "score": 1.0, + "content": ", we provide a convergence analysis in terms of the expected", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 433, + 657 + ], + "score": 1.0, + "content": "squared gradient norm. 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This is similar to the convergence behavior of a stochastic version of Nesterov’s", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 427, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 427, + 96 + ], + "score": 1.0, + "content": "fast gradient method [35, 36], observed in [5], for smooth convex minimization.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 609, + 506, + 666 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "error accumulation. This is similar to the convergence behavior of a stochastic version of Nesterov’s", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 427, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 427, + 96 + ], + "score": 1.0, + "content": "fast gradient method [35, 36], observed in [5], for smooth convex minimization.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 312, + 111 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 313, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 313, + 113 + ], + "score": 1.0, + "content": "Our main contributions are summarized as follows.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 132, + 120, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 132, + 120, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 132, + 120, + 437, + 133 + ], + "score": 1.0, + "content": "• We propose the FEG method that has an accelerated convergence rate", + "type": "text" + }, + { + "bbox": [ + 438, + 120, + 474, + 132 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 120, + 505, + 133 + ], + "score": 1.0, + "content": "on the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 131, + 500, + 144 + ], + "spans": [ + { + "bbox": [ + 141, + 131, + 500, + 144 + ], + "score": 1.0, + "content": "squared gradient norm for smooth structured nonconvex-nonconcave minimax problems.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 137, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 137, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "We present that the FEG method has a rate faster than those of the EAG and the Halpern", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 156, + 332, + 169 + ], + "spans": [ + { + "bbox": [ + 141, + 156, + 332, + 169 + ], + "score": 1.0, + "content": "iteration for smooth convex-concave problems.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 140, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 140, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "We construct a backtracking line-search version of FEG, named FEG-A, for the case where", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 142, + 183, + 447, + 195 + ], + "spans": [ + { + "bbox": [ + 142, + 183, + 371, + 195 + ], + "score": 1.0, + "content": "the Lipschitz constant and comonotonicity parameters of", + "type": "text" + }, + { + "bbox": [ + 371, + 183, + 381, + 193 + ], + "score": 0.82, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 183, + 447, + 195 + ], + "score": 1.0, + "content": "are unavailable.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 131, + 196, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 131, + 196, + 506, + 210 + ], + "score": 1.0, + "content": "• We analyze a stochastic version of FEG, named S-FEG, for smooth convex-concave prob-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 208, + 167, + 221 + ], + "spans": [ + { + "bbox": [ + 141, + 208, + 167, + 221 + ], + "score": 1.0, + "content": "lems.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 107, + 239, + 195, + 252 + ], + "lines": [ + { + "bbox": [ + 105, + 238, + 196, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 196, + 254 + ], + "score": 1.0, + "content": "2 Related work", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 107, + 264, + 333, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 262, + 334, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 334, + 278 + ], + "score": 1.0, + "content": "2.1 Methods for convex-concave minimax problems", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 506, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 283, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 297 + ], + "score": 1.0, + "content": "The extragradient method [19] is one of the widely used methods for solving smooth convex-concave", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 294, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 308 + ], + "score": 1.0, + "content": "minimax problems (see, e.g., [4, 7, 22, 24, 26, 42, 44] for its extensions and applications). In terms", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 304, + 507, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 304, + 183, + 320 + ], + "score": 1.0, + "content": "of the duality gap,", + "type": "text" + }, + { + "bbox": [ + 183, + 306, + 342, + 318 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { { \\pmb y } ^ { \\prime } \\in \\mathcal { V } } f ( { \\pmb x } , { \\pmb y } ^ { \\prime } ) - \\operatorname* { m i n } _ { { \\pmb x } ^ { \\prime } \\in \\mathcal { X } } f ( { \\pmb x } ^ { \\prime } , { \\pmb y } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 304, + 373, + 320 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 373, + 307, + 383, + 316 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 304, + 401, + 320 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 401, + 306, + 411, + 317 + ], + "score": 0.85, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 304, + 507, + 320 + ], + "score": 1.0, + "content": "are compact4 domains,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 393, + 330 + ], + "score": 1.0, + "content": "the ergodic iterate of the extragradient-type methods [32, 37] have an", + "type": "text" + }, + { + "bbox": [ + 393, + 317, + 425, + 329 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 316, + 472, + 330 + ], + "score": 1.0, + "content": "rate. Such", + "type": "text" + }, + { + "bbox": [ + 473, + 317, + 505, + 329 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 328, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 506, + 341 + ], + "score": 1.0, + "content": "rate on the duality gap is order-optimal for the first-order methods [34, 38], leaving no room for√", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 338, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 464, + 354 + ], + "score": 1.0, + "content": "improvement. On the other hand, the last iterate of the extragradient method has a slower", + "type": "text" + }, + { + "bbox": [ + 464, + 339, + 505, + 352 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / \\sqrt { k } )", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 340, + 363 + ], + "score": 1.0, + "content": "rate on the duality gap, under an additional assumption that", + "type": "text" + }, + { + "bbox": [ + 341, + 352, + 351, + 361 + ], + "score": 0.81, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "has a Lipschitz derivative [9]. In terms", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 232, + 375 + ], + "score": 1.0, + "content": "of the squared gradient norm,", + "type": "text" + }, + { + "bbox": [ + 232, + 361, + 262, + 374 + ], + "score": 0.95, + "content": "\\| \\ b { F z } \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 361, + 506, + 375 + ], + "score": 1.0, + "content": ", the best iterate of the extragradient-type methods [19, 39]", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 139, + 387 + ], + "score": 1.0, + "content": "have an", + "type": "text" + }, + { + "bbox": [ + 139, + 373, + 171, + 385 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 372, + 470, + 387 + ], + "score": 1.0, + "content": "rate [40, 41, 43]. The last iterate of the extragradient method also has a rate", + "type": "text" + }, + { + "bbox": [ + 470, + 373, + 502, + 385 + ], + "score": 0.9, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 372, + 506, + 387 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 130, + 397 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 384, + 140, + 394 + ], + "score": 0.82, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 384, + 454, + 397 + ], + "score": 1.0, + "content": "is further assumed to have a Lipschitz derivative [9]. Unlike the duality gap, the", + "type": "text" + }, + { + "bbox": [ + 454, + 384, + 486, + 396 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "rate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "on the squared gradient norm is not optimal [43]. From now on throughout this paper, we mainly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "score": 1.0, + "content": "study and compare the convergence rates on the squared gradient norm, which still has room for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "score": 1.0, + "content": "improvement in convex-concave problems, and has meaning for nonconvex-nonconcave minimax", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 425, + 241, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 241, + 442 + ], + "score": 1.0, + "content": "problems, unlike the duality gap.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 468, + 456 + ], + "score": 1.0, + "content": "Recently, [6, 17, 21, 40, 43] found that Halpern-type [12] (or anchoring) methods yield a fast", + "type": "text" + }, + { + "bbox": [ + 468, + 443, + 505, + 456 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 455, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 506, + 467 + ], + "score": 1.0, + "content": "rate in terms of the squared gradient norm for minimax problems. [17, 21] showed that the (implicit)", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 390, + 479 + ], + "score": 1.0, + "content": "Halpern iteration [12] with appropriately chosen step coefficients has an", + "type": "text" + }, + { + "bbox": [ + 390, + 465, + 427, + 478 + ], + "score": 0.94, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 465, + 506, + 479 + ], + "score": 1.0, + "content": "rate on the squared", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 194, + 489 + ], + "score": 1.0, + "content": "norm of a monotone", + "type": "text" + }, + { + "bbox": [ + 194, + 477, + 204, + 487 + ], + "score": 0.78, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 477, + 306, + 489 + ], + "score": 1.0, + "content": ". Then, for a cocoercive", + "type": "text" + }, + { + "bbox": [ + 306, + 477, + 315, + 487 + ], + "score": 0.79, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 477, + 505, + 489 + ], + "score": 1.0, + "content": ", an (explicit) version of the Halpern iteration", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "was studied in [6, 17] that has the same fast rate. In addition, [6] constructed a double-loop version", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 498, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 381, + 513 + ], + "score": 1.0, + "content": "of the Halpern iteration for a Lipschitz continuous and monotone", + "type": "text" + }, + { + "bbox": [ + 382, + 500, + 391, + 510 + ], + "score": 0.77, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 498, + 468, + 513 + ], + "score": 1.0, + "content": ", which has a rate", + "type": "text" + }, + { + "bbox": [ + 468, + 498, + 505, + 512 + ], + "score": 0.94, + "content": "\\tilde { \\mathcal { O } } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 318, + 523 + ], + "score": 1.0, + "content": "on the squared gradient norm, slower than the rate", + "type": "text" + }, + { + "bbox": [ + 319, + 510, + 355, + 523 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 510, + 505, + 523 + ], + "score": 1.0, + "content": ". While this is promising compared", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 133, + 535 + ], + "score": 1.0, + "content": "to the", + "type": "text" + }, + { + "bbox": [ + 134, + 522, + 166, + 534 + ], + "score": 0.93, + "content": "\\bar { \\mathcal { O } ( 1 / k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "rate of the extragradient methods on the squared gradient norm [40, 41, 43], the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "computational complexity due to its double-loop nature and a relatively slow rate remained a problem.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "Very recently, [43] proposed the extra anchored gradient (EAG) method, which is the first (explicit)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 183, + 567 + ], + "score": 1.0, + "content": "method with a fast", + "type": "text" + }, + { + "bbox": [ + 184, + 554, + 220, + 567 + ], + "score": 0.93, + "content": "\\bar { \\mathcal { O } } ( \\bar { 1 } / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "rate for smooth convex-concave minimax problems, i.e., for Lipschitz", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 563, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 579 + ], + "score": 1.0, + "content": "continuous and monotone operators. 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"• We propose the FEG method that has an accelerated convergence rate", + "type": "text" + }, + { + "bbox": [ + 438, + 120, + 474, + 132 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 120, + 505, + 133 + ], + "score": 1.0, + "content": "on the", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 131, + 500, + 144 + ], + "spans": [ + { + "bbox": [ + 141, + 131, + 500, + 144 + ], + "score": 1.0, + "content": "squared gradient norm for smooth structured nonconvex-nonconcave minimax problems.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 137, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 137, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "We present that the FEG method has a rate faster than those of the EAG and the Halpern", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 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In terms", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 304, + 507, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 304, + 183, + 320 + ], + "score": 1.0, + "content": "of the duality gap,", + "type": "text" + }, + { + "bbox": [ + 183, + 306, + 342, + 318 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { { \\pmb y } ^ { \\prime } \\in \\mathcal { V } } f ( { \\pmb x } , { \\pmb y } ^ { \\prime } ) - \\operatorname* { m i n } _ { { \\pmb x } ^ { \\prime } \\in \\mathcal { X } } f ( { \\pmb x } ^ { \\prime } , { \\pmb y } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 304, + 373, + 320 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 373, + 307, + 383, + 316 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 304, + 401, + 320 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 401, + 306, + 411, + 317 + ], + "score": 0.85, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 304, + 507, + 320 + ], + "score": 1.0, + "content": "are compact4 domains,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 393, + 330 + ], + "score": 1.0, + "content": "the ergodic iterate of the extragradient-type methods [32, 37] have an", + "type": "text" + }, + { + "bbox": [ + 393, + 317, + 425, + 329 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 316, + 472, + 330 + ], + "score": 1.0, + "content": "rate. Such", + "type": "text" + }, + { + "bbox": [ + 473, + 317, + 505, + 329 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 328, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 506, + 341 + ], + "score": 1.0, + "content": "rate on the duality gap is order-optimal for the first-order methods [34, 38], leaving no room for√", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 338, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 464, + 354 + ], + "score": 1.0, + "content": "improvement. On the other hand, the last iterate of the extragradient method has a slower", + "type": "text" + }, + { + "bbox": [ + 464, + 339, + 505, + 352 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / \\sqrt { k } )", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 340, + 363 + ], + "score": 1.0, + "content": "rate on the duality gap, under an additional assumption that", + "type": "text" + }, + { + "bbox": [ + 341, + 352, + 351, + 361 + ], + "score": 0.81, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "has a Lipschitz derivative [9]. In terms", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 232, + 375 + ], + "score": 1.0, + "content": "of the squared gradient norm,", + "type": "text" + }, + { + "bbox": [ + 232, + 361, + 262, + 374 + ], + "score": 0.95, + "content": "\\| \\ b { F z } \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 361, + 506, + 375 + ], + "score": 1.0, + "content": ", the best iterate of the extragradient-type methods [19, 39]", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 139, + 387 + ], + "score": 1.0, + "content": "have an", + "type": "text" + }, + { + "bbox": [ + 139, + 373, + 171, + 385 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 372, + 470, + 387 + ], + "score": 1.0, + "content": "rate [40, 41, 43]. The last iterate of the extragradient method also has a rate", + "type": "text" + }, + { + "bbox": [ + 470, + 373, + 502, + 385 + ], + "score": 0.9, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 372, + 506, + 387 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 130, + 397 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 384, + 140, + 394 + ], + "score": 0.82, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 384, + 454, + 397 + ], + "score": 1.0, + "content": "is further assumed to have a Lipschitz derivative [9]. Unlike the duality gap, the", + "type": "text" + }, + { + "bbox": [ + 454, + 384, + 486, + 396 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "rate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "on the squared gradient norm is not optimal [43]. From now on throughout this paper, we mainly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "score": 1.0, + "content": "study and compare the convergence rates on the squared gradient norm, which still has room for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "score": 1.0, + "content": "improvement in convex-concave problems, and has meaning for nonconvex-nonconcave minimax", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 425, + 241, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 241, + 442 + ], + "score": 1.0, + "content": "problems, unlike the duality gap.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 283, + 507, + 442 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 468, + 456 + ], + "score": 1.0, + "content": "Recently, [6, 17, 21, 40, 43] found that Halpern-type [12] (or anchoring) methods yield a fast", + "type": "text" + }, + { + "bbox": [ + 468, + 443, + 505, + 456 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 455, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 506, + 467 + ], + "score": 1.0, + "content": "rate in terms of the squared gradient norm for minimax problems. [17, 21] showed that the (implicit)", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 390, + 479 + ], + "score": 1.0, + "content": "Halpern iteration [12] with appropriately chosen step coefficients has an", + "type": "text" + }, + { + "bbox": [ + 390, + 465, + 427, + 478 + ], + "score": 0.94, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 465, + 506, + 479 + ], + "score": 1.0, + "content": "rate on the squared", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 194, + 489 + ], + "score": 1.0, + "content": "norm of a monotone", + "type": "text" + }, + { + "bbox": [ + 194, + 477, + 204, + 487 + ], + "score": 0.78, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 477, + 306, + 489 + ], + "score": 1.0, + "content": ". Then, for a cocoercive", + "type": "text" + }, + { + "bbox": [ + 306, + 477, + 315, + 487 + ], + "score": 0.79, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 477, + 505, + 489 + ], + "score": 1.0, + "content": ", an (explicit) version of the Halpern iteration", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "was studied in [6, 17] that has the same fast rate. 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While this is promising compared", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 133, + 535 + ], + "score": 1.0, + "content": "to the", + "type": "text" + }, + { + "bbox": [ + 134, + 522, + 166, + 534 + ], + "score": 0.93, + "content": "\\bar { \\mathcal { O } ( 1 / k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "rate of the extragradient methods on the squared gradient norm [40, 41, 43], the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "computational complexity due to its double-loop nature and a relatively slow rate remained a problem.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "Very recently, [43] proposed the extra anchored gradient (EAG) method, which is the first (explicit)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 183, + 567 + ], + "score": 1.0, + "content": "method with a fast", + "type": "text" + }, + { + "bbox": [ + 184, + 554, + 220, + 567 + ], + "score": 0.93, + "content": "\\bar { \\mathcal { O } } ( \\bar { 1 } / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "rate for smooth convex-concave minimax problems, i.e., for Lipschitz", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 563, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 579 + ], + "score": 1.0, + "content": "continuous and monotone operators. 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MethodConvex-concaveNonconvex-nonconcave
Cocoercive MonotoneNegative comonotoneMVIWeak MVI
NormalEG [4, 42]0(1/k)0(1/k）0(1/k）
EG+[7]0(1/k)0(1/k)0(1/k)0(1/k）0(1/k）
AcceleratedHalpern [12, 6]0(1/k²)(1/k²)
EAG [43] FEG (this paper)0(1/k2) 0(1/k2)0(1/k²) 0(1/k2)0(1/k2)
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For some", + "type": "text" + }, + { + "bbox": [ + 294, + 536, + 359, + 551 + ], + "score": 0.92, + "content": "\\textstyle \\rho \\in { \\bigl ( } - { \\frac { 1 } { 2 L } } , \\infty { \\bigr ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 534, + 363, + 552 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 364, + 537, + 373, + 547 + ], + "score": 0.7, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 534, + 410, + 552 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 554, + 421, + 569 + ], + "lines": [ + { + "bbox": [ + 190, + 554, + 421, + 569 + ], + "spans": [ + { + "bbox": [ + 190, + 554, + 421, + 569 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\langle F z - F z ^ { \\prime } , z - z ^ { \\prime } \\rangle \\ge \\rho \\| F z - F z ^ { \\prime } \\| ^ { 2 } , \\quad \\forall z , z ^ { \\prime } \\in \\mathbb { R } ^ { d } . } \\end{array}", + "type": "interline_equation", + "image_path": "f517a07e39a41fd0b3b884dac93732d30c287f839e2f96809371b781868b805c.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 190, + 554, + 421, + 569 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 506, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 124, + 591 + ], + "score": 1.0, + "content": "The", + "type": "text" + }, + { + "bbox": [ + 125, + 581, + 131, + 591 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 578, + 405, + 591 + ], + "score": 1.0, + "content": "-comonotonicity consists of three cases depending on the choice of", + "type": "text" + }, + { + "bbox": [ + 406, + 581, + 412, + 591 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "; the negative comono-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 163, + 603 + ], + "score": 1.0, + "content": "tonicity when", + "type": "text" + }, + { + "bbox": [ + 164, + 591, + 188, + 601 + ], + "score": 0.9, + "content": "\\rho < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 591, + 286, + 603 + ], + "score": 1.0, + "content": ", the monotonicity when", + "type": "text" + }, + { + "bbox": [ + 287, + 591, + 311, + 601 + ], + "score": 0.91, + "content": "\\rho = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 591, + 422, + 603 + ], + "score": 1.0, + "content": ", and the cocoercivity when", + "type": "text" + }, + { + "bbox": [ + 423, + 591, + 447, + 601 + ], + "score": 0.9, + "content": "\\rho > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 591, + 505, + 603 + ], + "score": 1.0, + "content": ". The negative", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "comonotonicity is weaker than the other two, and is the main focus of this paper. The following is an", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "score": 1.0, + "content": "examplary nonconvex-nonconcave condition that is stronger than the negative comonotonicity [1, 3].", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 174, + 639 + ], + "score": 1.0, + "content": "Example 1. Let", + "type": "text" + }, + { + "bbox": [ + 174, + 627, + 182, + 638 + ], + "score": 0.77, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 624, + 345, + 639 + ], + "score": 1.0, + "content": "be twice continuously differentiable and", + "type": "text" + }, + { + "bbox": [ + 346, + 628, + 353, + 638 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 624, + 506, + 639 + ], + "score": 1.0, + "content": "-weakly-convex-weakly-concave. 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Then, the saddle", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 696, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 696, + 155, + 712 + ], + "score": 1.0, + "content": "gradient of", + "type": "text" + }, + { + "bbox": [ + 155, + 699, + 162, + 710 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 696, + 215, + 712 + ], + "score": 1.0, + "content": "satisfies the", + "type": "text" + }, + { + "bbox": [ + 216, + 698, + 231, + 713 + ], + "score": 0.91, + "content": "- \\frac { 1 } { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 696, + 463, + 712 + ], + "score": 1.0, + "content": "-negative comonotonicity. (See Appendix A.1.) 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Its extreme case is", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "4", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 72, + 366, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 366, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 366, + 86 + ], + "score": 1.0, + "content": "2.2 Methods for nonconvex-nonconcave minimax problems", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 92, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 106, + 92, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 505, + 105 + ], + "score": 1.0, + "content": "Some recent literature considered relaxing the monotonicity condition of the saddle gradient operator", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 115 + ], + "score": 1.0, + "content": "to tackle modern nonconvex-nonconcave minimax problems. 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This condition", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "score": 1.0, + "content": "is also studied under the name, the coherence, in [26, 42, 44]. Moreover, [7] considered a weaker", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 335, + 162 + ], + "score": 1.0, + "content": "condition, named the weak MVI condition, i.e., for some", + "type": "text" + }, + { + "bbox": [ + 335, + 149, + 360, + 160 + ], + "score": 0.9, + "content": "\\rho < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 147, + 410, + 162 + ], + "score": 1.0, + "content": ", there exists", + "type": "text" + }, + { + "bbox": [ + 411, + 148, + 462, + 160 + ], + "score": 0.93, + "content": "z _ { \\ast } \\in Z _ { \\ast } ( F )", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 159, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 107, + 159, + 214, + 172 + ], + "score": 0.91, + "content": "\\langle F z , z - z _ { * } \\rangle \\geq \\rho \\Vert F z \\Vert ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 159, + 244, + 174 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 245, + 159, + 280, + 171 + ], + "score": 0.91, + "content": "z \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 159, + 506, + 174 + ], + "score": 1.0, + "content": ". The weak MVI condition is implied by the negative", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 104, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "comonotonicity [1] or, equivalently, the (positive) cohypomonotonicity [3]. 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The convergence rates", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 485, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 485, + 277 + ], + "score": 1.0, + "content": "of the existing methods and the FEG on the squared gradient norm are summarized in Table 1.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 198, + 506, + 277 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 323, + 502, + 419 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 284, + 505, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 283, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 298 + ], + "score": 1.0, + "content": "Table 1: Comparison of the convergence rates of the existing extragradient-type methods and the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "FEG, with respect to the squared gradient norm, for smooth structured minimax problems, under", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 306, + 415, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 401, + 318 + ], + "score": 1.0, + "content": "various assumptions on the Lipschitz continuous saddle gradient operator", + "type": "text" + }, + { + "bbox": [ + 402, + 307, + 411, + 316 + ], + "score": 0.77, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 306, + 415, + 318 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "table_body", + "bbox": [ + 108, + 323, + 502, + 419 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 323, + 502, + 419 + ], + "spans": [ + { + "bbox": [ + 108, + 323, + 502, + 419 + ], + "score": 0.981, + "html": "
MethodConvex-concaveNonconvex-nonconcave
Cocoercive MonotoneNegative comonotoneMVIWeak MVI
NormalEG [4, 42]0(1/k)0(1/k）0(1/k）
EG+[7]0(1/k)0(1/k)0(1/k)0(1/k）0(1/k）
AcceleratedHalpern [12, 6]0(1/k²)(1/k²)
EAG [43] FEG (this paper)0(1/k2) 0(1/k2)0(1/k²) 0(1/k2)0(1/k2)
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iteration, for each three comonoticity case, respectively.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 107, + 177, + 447, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 447, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 447, + 192 + ], + "score": 1.0, + "content": "4 Fast extragradient (FEG) method for Lipschitz continuous and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 123, + 192, + 241, + 206 + ], + "spans": [ + { + "bbox": [ + 123, + 192, + 241, + 206 + ], + "score": 1.0, + "content": "comonotone operators", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 215, + 506, + 263 + ], + "lines": [ + { + "bbox": [ + 104, + 213, + 503, + 233 + ], + "spans": [ + { + "bbox": [ + 104, + 213, + 322, + 233 + ], + "score": 1.0, + "content": "This section considers an instance of (Class FEG) with", + "type": "text" + }, + { + "bbox": [ 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506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 431, + 243 + ], + "score": 1.0, + "content": "The resulting method, named FEG, is illustrated in Algorithm 1, which has an", + "type": "text" + }, + { + "bbox": [ + 431, + 229, + 468, + 242 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 228, + 506, + 243 + ], + "score": 1.0, + "content": "fast rate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "with respect to the squared gradient norm, in Theorem 4.1. The proof of Theorem 4.1 is provided in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 251, + 150, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 150, + 263 + ], + "score": 1.0, + "content": "Section 7.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 107, + 273, + 294, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 295, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 295, + 288 + ], + "score": 1.0, + "content": "Algorithm 1 Fast extragradient (FEG) method", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "table", + "bbox": [ + 113, + 286, + 456, + 365 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 113, + 286, + 456, + 365 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 286, + 455, + 365 + ], + "spans": [ + { + "bbox": [ + 113, + 286, + 455, + 365 + ], + "score": 0.26, + "html": "
Input: z0 ∈ Rd,L ∈ (0,∞o),ρ ∈(- 2,00) for k = 0,1,... do
2+1/=+1(2-（1-1）（+2）F 1
1 2 2k+1= 2k+ k+1
", + "type": "table", + "image_path": "e6f770a71e1f68a45474304fcaeed89a31d829b4be358d3bc85afcb0ac396edf.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 113, + 286, + 456, + 312.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 113, + 312.3333333333333, + 456, + 338.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 113, + 338.66666666666663, + 456, + 364.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 116, + 364, + 149, + 375 + ], + "lines": [ + { + "bbox": [ + 115, + 363, + 150, + 376 + ], + "spans": [ + { + "bbox": [ + 115, + 363, + 150, + 376 + ], + "score": 1.0, + "content": "end for", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 105, + 396, + 504, + 419 + ], + "lines": [ + { + "bbox": [ + 104, + 393, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 104, + 393, + 197, + 412 + ], + "score": 1.0, + "content": "Theorem 4.1. For the", + "type": "text" + }, + { + "bbox": [ + 198, + 397, + 205, + 406 + ], + "score": 0.64, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 393, + 308, + 412 + ], + "score": 1.0, + "content": "-Lipschitz continuous and", + "type": "text" + }, + { + "bbox": [ + 308, + 398, + 314, + 408 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 393, + 403, + 412 + ], + "score": 1.0, + "content": "-comonotone operator", + "type": "text" + }, + { + "bbox": [ + 403, + 397, + 413, + 406 + ], + "score": 0.78, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 393, + 433, + 412 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 433, + 395, + 472, + 409 + ], + "score": 0.93, + "content": "\\rho > - \\frac { 1 } { 2 L }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 393, + 506, + 412 + ], + "score": 1.0, + "content": "and for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 406, + 439, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 123, + 421 + ], + "score": 1.0, + "content": "any", + "type": "text" + }, + { + "bbox": [ + 124, + 407, + 175, + 419 + ], + "score": 0.93, + "content": "z _ { \\ast } \\in Z _ { \\ast } ( F )", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 406, + 233, + 421 + ], + "score": 1.0, + "content": ", the sequence", + "type": "text" + }, + { + "bbox": [ + 233, + 407, + 269, + 420 + ], + "score": 0.93, + "content": "\\{ z _ { k } \\} _ { k \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 406, + 410, + 421 + ], + "score": 1.0, + "content": "generated by FEG satisfies, for all", + "type": "text" + }, + { + "bbox": [ + 410, + 408, + 435, + 418 + ], + "score": 0.93, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 406, + 439, + 421 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 425, + 361, + 462 + ], + "lines": [ + { + "bbox": [ + 250, + 425, + 361, + 462 + ], + "spans": [ + { + "bbox": [ + 250, + 425, + 361, + 462 + ], + "score": 0.94, + "content": "\\| F z _ { k } \\| ^ { 2 } \\leq \\frac { 4 \\| z _ { 0 } - z _ { * } \\| ^ { 2 } } { \\Big ( \\frac { 1 } { L } + 2 \\rho \\Big ) ^ { 2 } k ^ { 2 } } .", + "type": "interline_equation", + "image_path": "e2f6cd273461c9b92d8707e1f757dd91d03d1770236f8581afadd094ba3dcb75.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 425, + 361, + 443.5 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 250, + 443.5, + 361, + 462.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 472, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 394, + 485 + ], + "score": 1.0, + "content": "The following example shows that the bound (3) of the FEG is exact for", + "type": "text" + }, + { + "bbox": [ + 395, + 474, + 419, + 484 + ], + "score": 0.91, + "content": "\\rho = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 473, + 437, + 485 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 438, + 473, + 483, + 484 + ], + "score": 0.91, + "content": "k = 4 l + 2", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 473, + 505, + 485 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 483, + 504, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 504, + 496 + ], + "score": 1.0, + "content": "bound (3) is not known to be exact in general, and we leave finding the exact bound as future work.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 506, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 176, + 511 + ], + "score": 1.0, + "content": "Example 2. Let", + "type": "text" + }, + { + "bbox": [ + 176, + 499, + 251, + 510 + ], + "score": 0.86, + "content": "f : \\mathbb { R } \\times \\mathbb { R } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 497, + 264, + 511 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 264, + 499, + 327, + 511 + ], + "score": 0.87, + "content": "f ( x , y ) = L x y .", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 497, + 506, + 511 + ], + "score": 1.0, + "content": ". Its saddle gradient operator and solution", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 509, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 123, + 524 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 123, + 510, + 216, + 522 + ], + "score": 0.9, + "content": "\\pmb { F } ( x , y ) = ( L y , - L x )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 509, + 237, + 524 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 237, + 511, + 286, + 522 + ], + "score": 0.92, + "content": "z _ { * } = ( 0 , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 509, + 433, + 524 + ], + "score": 1.0, + "content": ", respectively. For the initial point", + "type": "text" + }, + { + "bbox": [ + 433, + 510, + 505, + 522 + ], + "score": 0.91, + "content": "z _ { 0 } = ( x _ { 0 } , y _ { 0 } ) =", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 520, + 508, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 128, + 534 + ], + "score": 0.9, + "content": "( 1 , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 520, + 189, + 538 + ], + "score": 1.0, + "content": ", the sequence", + "type": "text" + }, + { + "bbox": [ + 189, + 522, + 225, + 534 + ], + "score": 0.91, + "content": "\\{ z _ { k } \\} _ { k \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 520, + 340, + 538 + ], + "score": 1.0, + "content": "generated by FEG satisfies", + "type": "text" + }, + { + "bbox": [ + 340, + 521, + 416, + 536 + ], + "score": 0.92, + "content": "\\begin{array} { r } { z _ { 4 l + 2 } = \\left( 0 , \\frac { 1 } { 2 l + 1 } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 520, + 446, + 538 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 446, + 523, + 470, + 533 + ], + "score": 0.87, + "content": "l \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 520, + 508, + 538 + ], + "score": 1.0, + "content": ". 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Input: z0 ∈ Rd,L ∈ (0,∞o),ρ ∈(- 2,00) for k = 0,1,... do
2+1/=+1(2-（1-1）（+2）F 1
1 2 2k+1= 2k+ k+1
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For the", + "type": "text" + }, + { + "bbox": [ + 198, + 397, + 205, + 406 + ], + "score": 0.64, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 393, + 308, + 412 + ], + "score": 1.0, + "content": "-Lipschitz continuous and", + "type": "text" + }, + { + "bbox": [ + 308, + 398, + 314, + 408 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 393, + 403, + 412 + ], + "score": 1.0, + "content": "-comonotone operator", + "type": "text" + }, + { + "bbox": [ + 403, + 397, + 413, + 406 + ], + "score": 0.78, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 393, + 433, + 412 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 433, + 395, + 472, + 409 + ], + "score": 0.93, + "content": "\\rho > - \\frac { 1 } { 2 L }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 393, + 506, + 412 + ], + "score": 1.0, + "content": "and for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 406, + 439, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 123, + 421 + ], + "score": 1.0, + "content": "any", + "type": "text" + }, + { + "bbox": [ + 124, + 407, + 175, + 419 + ], + "score": 0.93, + "content": "z _ { \\ast } \\in Z _ { \\ast } ( F )", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 406, + 233, + 421 + ], + "score": 1.0, + "content": ", the sequence", + "type": "text" + }, + { + "bbox": [ + 233, + 407, + 269, + 420 + ], + "score": 0.93, + "content": "\\{ z _ { k } \\} _ { k \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 406, + 410, + 421 + ], + "score": 1.0, + "content": "generated by FEG satisfies, for all", + "type": "text" + }, + { + "bbox": [ + 410, + 408, + 435, + 418 + ], + "score": 0.93, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 406, + 439, + 421 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 393, + 506, + 421 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 425, + 361, + 462 + ], + "lines": [ + { + "bbox": [ + 250, + 425, + 361, + 462 + ], + "spans": [ + { + "bbox": [ + 250, + 425, + 361, + 462 + ], + "score": 0.94, + "content": "\\| F z _ { k } \\| ^ { 2 } \\leq \\frac { 4 \\| z _ { 0 } - z _ { * } \\| ^ { 2 } } { \\Big ( \\frac { 1 } { L } + 2 \\rho \\Big ) ^ { 2 } k ^ { 2 } } .", + "type": "interline_equation", + "image_path": "e2f6cd273461c9b92d8707e1f757dd91d03d1770236f8581afadd094ba3dcb75.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 425, + 361, + 443.5 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 250, + 443.5, + 361, + 462.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 472, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 394, + 485 + ], + "score": 1.0, + "content": "The following example shows that the bound (3) of the FEG is exact for", + "type": "text" + }, + { + "bbox": [ + 395, + 474, + 419, + 484 + ], + "score": 0.91, + "content": "\\rho = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 473, + 437, + 485 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 438, + 473, + 483, + 484 + ], + "score": 0.91, + "content": "k = 4 l + 2", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 473, + 505, + 485 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 483, + 504, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 504, + 496 + ], + "score": 1.0, + "content": "bound (3) is not known to be exact in general, and we leave finding the exact bound as future work.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 106, + 473, + 505, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 506, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 176, + 511 + ], + "score": 1.0, + "content": "Example 2. Let", + "type": "text" + }, + { + "bbox": [ + 176, + 499, + 251, + 510 + ], + "score": 0.86, + "content": "f : \\mathbb { R } \\times \\mathbb { R } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 497, + 264, + 511 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 264, + 499, + 327, + 511 + ], + "score": 0.87, + "content": "f ( x , y ) = L x y .", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 497, + 506, + 511 + ], + "score": 1.0, + "content": ". Its saddle gradient operator and solution", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 509, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 123, + 524 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 123, + 510, + 216, + 522 + ], + "score": 0.9, + "content": "\\pmb { F } ( x , y ) = ( L y , - L x )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 509, + 237, + 524 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 237, + 511, + 286, + 522 + ], + "score": 0.92, + "content": "z _ { * } = ( 0 , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 509, + 433, + 524 + ], + "score": 1.0, + "content": ", respectively. 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Hence,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 532, + 398, + 554 + ], + "spans": [ + { + "bbox": [ + 107, + 535, + 260, + 553 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\| F z _ { 4 l + 2 } \\| ^ { 2 } = \\frac { L ^ { 2 } } { ( 2 l + 1 ) ^ { 2 } } = \\frac { 4 L ^ { 2 } \\| z _ { 0 } - z _ { * } \\| ^ { 2 } } { ( 4 l + 2 ) ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 532, + 288, + 554 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 288, + 538, + 311, + 550 + ], + "score": 0.87, + "content": "l \\geq 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 532, + 398, + 554 + ], + "score": 1.0, + "content": ". (See Appendix B.1.)", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 497, + 508, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 562, + 504, + 586 + ], + "lines": [ + { + "bbox": [ + 104, + 558, + 501, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 558, + 416, + 578 + ], + "score": 1.0, + "content": "We next compare the rate bound (3) with existing analyses for the three cases", + "type": "text" + }, + { + "bbox": [ + 417, + 562, + 501, + 576 + ], + "score": 0.79, + "content": "\\begin{array} { r } { - \\frac { 1 } { 2 L } < \\rho < 0 , \\rho = 0 } \\end{array}", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 572, + 153, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 123, + 586 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 574, + 148, + 585 + ], + "score": 0.9, + "content": "\\rho > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 572, + 153, + 586 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 558, + 501, + 586 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 598, + 399, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 400, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 195, + 613 + ], + "score": 1.0, + "content": "4.1 Comparison to", + "type": "text" + }, + { + "bbox": [ + 195, + 599, + 217, + 609 + ], + "score": 0.79, + "content": "\\mathbf { E G + }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 597, + 367, + 613 + ], + "score": 1.0, + "content": "under the negative comonotonicity", + "type": "text" + }, + { + "bbox": [ + 367, + 599, + 396, + 610 + ], + "score": 0.81, + "content": "( \\rho < 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 597, + 400, + 613 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 504, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 266, + 638 + ], + "score": 1.0, + "content": "Under the negative comonotonicity with", + "type": "text" + }, + { + "bbox": [ + 266, + 618, + 324, + 632 + ], + "score": 0.92, + "content": "\\begin{array} { r } { - \\frac { 1 } { 8 L } < \\rho < 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 613, + 342, + 638 + ], + "score": 1.0, + "content": ", the", + "type": "text" + }, + { + "bbox": [ + 343, + 619, + 368, + 630 + ], + "score": 0.85, + "content": "( \\mathrm { E G + } )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 613, + 421, + 638 + ], + "score": 1.0, + "content": "method with", + "type": "text" + }, + { + "bbox": [ + 421, + 618, + 459, + 632 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\alpha _ { k } = \\frac { 1 } { 2 L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 613, + 477, + 638 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 618, + 504, + 632 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\beta = \\frac { 1 } { 2 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 166, + 630, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 166, + 630, + 506, + 642 + ], + "score": 1.0, + "content": "rate on the squared gradient norm. 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Note that while the ρ-cocoercivity implies the 1ρ -Lipschitz continuity,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 196, + 464 + ], + "score": 1.0, + "content": "there is case where the", + "type": "text" + }, + { + "bbox": [ + 197, + 454, + 203, + 464 + ], + "score": 0.83, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "-cocoercive (and thus Lipschitz continuous) operator has a Lipschitz constant", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 459, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 115, + 476 + ], + "score": 0.75, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 461, + 171, + 482 + ], + "score": 1.0, + "content": "smaller than", + "type": "text" + }, + { + "bbox": [ + 171, + 465, + 178, + 480 + ], + "score": 0.85, + "content": "\\frac { 1 } { \\rho }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 461, + 210, + 482 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 211, + 465, + 241, + 480 + ], + "score": 0.92, + "content": "\\begin{array} { r } { L \\le \\frac { 1 } { \\rho } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 461, + 328, + 482 + ], + "score": 1.0, + "content": ", the FEG has a rate", + "type": "text" + }, + { + "bbox": [ + 329, + 462, + 485, + 481 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\| \\boldsymbol { F } \\boldsymbol { z } _ { k } \\| ^ { 2 } \\le \\frac { 4 \\| \\boldsymbol { z } _ { 0 } - \\boldsymbol { z } _ { * } \\| ^ { 2 } } { ( 1 / L + 2 \\rho ) ^ { 2 } k ^ { 2 } } = \\frac { 4 \\| \\boldsymbol { z } _ { 0 } - \\boldsymbol { z } _ { * } \\| ^ { 2 } } { 9 \\rho ^ { 2 } k ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 459, + 507, + 480 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "is faster than that of Halpern iteration. 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If we narrow down to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 102, + 509, + 507, + 534 + ], + "spans": [ + { + "bbox": [ + 102, + 509, + 142, + 534 + ], + "score": 1.0, + "content": "the case", + "type": "text" + }, + { + "bbox": [ + 142, + 514, + 176, + 529 + ], + "score": 0.93, + "content": "\\begin{array} { r } { L < \\frac { 1 } { 2 \\rho } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 509, + 289, + 534 + ], + "score": 1.0, + "content": ", the FEG has a faster rate,", + "type": "text" + }, + { + "bbox": [ + 289, + 511, + 439, + 530 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\| \\ b { F } \\ b { z } _ { k } \\| ^ { 2 } \\leq \\frac { 4 \\| \\ b { z } _ { 0 } - \\ b { z } _ { * } \\| ^ { 2 } } { ( 1 / L + 2 \\rho ) ^ { 2 } k ^ { 2 } } < \\frac { \\| \\ b { z } _ { 0 } - \\ b { z } _ { * } \\| ^ { 2 } } { 4 \\rho ^ { 2 } k ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 511, + 507, + 529 + ], + "score": 1.0, + "content": ". 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This example", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "presents that the existing guarantees on convergence and acceleration of the aforementioned methods", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 647, + 465, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 465, + 660 + ], + "score": 1.0, + "content": "under the convex-concave setting do not generalize to the nonconvex-nonconcave setting.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 106, + 675, + 306, + 689 + ], + "lines": [ + { + "bbox": [ + 104, + 672, + 306, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 672, + 306, + 691 + ], + "score": 1.0, + "content": "5 FEG with backtracking line-search", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 357, + 712 + ], + "score": 1.0, + "content": "The FEG requires the knowledge of the two global parameters", + "type": "text" + }, + { + "bbox": [ + 358, + 701, + 366, + 710 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 699, + 383, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 384, + 702, + 390, + 712 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "for Lipschitz continuity and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "comonotonicity, respectively. 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Note that while the ρ-cocoercivity implies the 1ρ -Lipschitz continuity,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 196, + 464 + ], + "score": 1.0, + "content": "there is case where the", + "type": "text" + }, + { + "bbox": [ + 197, + 454, + 203, + 464 + ], + "score": 0.83, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "-cocoercive (and thus Lipschitz continuous) operator has a Lipschitz constant", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 459, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 115, + 476 + ], + "score": 0.75, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 461, + 171, + 482 + ], + "score": 1.0, + "content": "smaller than", + "type": "text" + }, + { + "bbox": [ + 171, + 465, + 178, + 480 + ], + "score": 0.85, + "content": "\\frac { 1 } { \\rho }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 461, + 210, + 482 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 211, + 465, + 241, + 480 + ], + "score": 0.92, + "content": "\\begin{array} { r } { L \\le \\frac { 1 } { \\rho } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 461, + 328, + 482 + ], + "score": 1.0, + "content": ", the FEG has a rate", + "type": "text" + }, + { + "bbox": [ + 329, + 462, + 485, + 481 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\| \\boldsymbol { F } \\boldsymbol { z } _ { k } \\| ^ { 2 } \\le \\frac { 4 \\| \\boldsymbol { z } _ { 0 } - \\boldsymbol { z } _ { * } \\| ^ { 2 } } { ( 1 / L + 2 \\rho ) ^ { 2 } k ^ { 2 } } = \\frac { 4 \\| \\boldsymbol { z } _ { 0 } - \\boldsymbol { z } _ { * } \\| ^ { 2 } } { 9 \\rho ^ { 2 } k ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 459, + 507, + 480 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "is faster than that of Halpern iteration. 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Let", + "type": "text" + }, + { + "bbox": [ + 184, + 326, + 284, + 341 + ], + "score": 0.92, + "content": "\\tilde { F } z _ { k / 2 } = F z _ { k / 2 } + \\xi _ { k / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 326, + 316, + 342 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 316, + 328, + 360, + 341 + ], + "score": 0.94, + "content": "\\{ \\xi _ { k / 2 } \\} _ { k \\geq 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 326, + 506, + 342 + ], + "score": 1.0, + "content": "are independent random variables", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 339, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 148, + 356 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 148, + 341, + 198, + 354 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\xi _ { k / 2 } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 339, + 217, + 356 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 217, + 341, + 294, + 355 + ], + "score": 0.88, + "content": "\\mathbb { E } [ \\| \\xi _ { k / 2 } \\| ^ { 2 } ] = \\sigma _ { k / 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 339, + 323, + 356 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 324, + 342, + 349, + 352 + ], + "score": 0.89, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 339, + 408, + 356 + ], + "score": 1.0, + "content": ". Then, for the", + "type": "text" + }, + { + "bbox": [ + 409, + 341, + 416, + 351 + ], + "score": 0.56, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 339, + 506, + 356 + ], + "score": 1.0, + "content": "-Lipschitz continuous", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 353, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 205, + 368 + ], + "score": 1.0, + "content": "and monotone operator", + "type": "text" + }, + { + "bbox": [ + 206, + 355, + 215, + 364 + ], + "score": 0.81, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 353, + 267, + 368 + ], + "score": 1.0, + "content": "and for any", + "type": "text" + }, + { + "bbox": [ + 267, + 354, + 320, + 366 + ], + "score": 0.94, + "content": "z _ { * } \\in Z _ { * } ( F )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 353, + 380, + 368 + ], + "score": 1.0, + "content": ", the sequence", + "type": "text" + }, + { + "bbox": [ + 380, + 354, + 417, + 367 + ], + "score": 0.92, + "content": "\\{ z _ { k } \\} _ { k \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 353, + 505, + 368 + ], + "score": 1.0, + "content": "generated by S-FEG", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 365, + 142, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 142, + 378 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 378, + 455, + 413 + ], + "lines": [ + { + "bbox": [ + 155, + 378, + 455, + 413 + ], + "spans": [ + { + "bbox": [ + 155, + 378, + 455, + 413 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\| F z _ { k } \\| ^ { 2 } ] \\le \\frac { 4 L ^ { 2 } \\| z _ { 0 } - z _ { * } \\| ^ { 2 } } { k ^ { 2 } } + \\frac { 6 } { k ^ { 2 } } \\left[ \\sigma _ { 0 } ^ { 2 } + \\sum _ { l = 1 } ^ { k - 1 } ( l ^ { 2 } \\sigma _ { l } ^ { 2 } + ( l + 1 ) ^ { 2 } \\sigma _ { l + 1 / 2 } ^ { 2 } ) \\right]", + "type": "interline_equation", + "image_path": "7f9606efa3d23c5eceeaa2b97aa883b0f48834a7c9f80110e6314d9c3d452b63.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 155, + 378, + 455, + 389.6666666666667 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 155, + 389.6666666666667, + 455, + 401.33333333333337 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 155, + 401.33333333333337, + 455, + 413.00000000000006 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 415, + 506, + 439 + ], + "lines": [ + { + "bbox": [ + 102, + 412, + 507, + 432 + ], + "spans": [ + { + "bbox": [ + 102, + 412, + 133, + 432 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 134, + 416, + 158, + 427 + ], + "score": 0.89, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 412, + 226, + 432 + ], + "score": 1.0, + "content": ". Furthermore, if", + "type": "text" + }, + { + "bbox": [ + 227, + 415, + 257, + 429 + ], + "score": 0.8, + "content": "\\sigma _ { 0 } ^ { 2 } \\le \\frac { \\epsilon } { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 412, + 261, + 432 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 262, + 415, + 297, + 429 + ], + "score": 0.85, + "content": "\\sigma _ { k } ^ { 2 } \\le \\frac { \\epsilon } { 6 k }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 412, + 316, + 432 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 415, + 386, + 430 + ], + "score": 0.94, + "content": "\\textstyle \\sigma _ { k + 1 / 2 } ^ { 2 } \\leq \\frac { \\epsilon } { 6 ( k + 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 412, + 415, + 432 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 415, + 416, + 439, + 427 + ], + "score": 0.89, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 412, + 507, + 432 + ], + "score": 1.0, + "content": ", then the bound", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 428, + 164, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 428, + 164, + 441 + ], + "score": 1.0, + "content": "(4) reduces to", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 441, + 378, + 467 + ], + "lines": [ + { + "bbox": [ + 233, + 441, + 378, + 467 + ], + "spans": [ + { + "bbox": [ + 233, + 441, + 378, + 467 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\| F z _ { k } \\| ^ { 2 } ] \\le \\frac { 4 L ^ { 2 } \\| z _ { 0 } - z _ { * } \\| ^ { 2 } } { k ^ { 2 } } + \\epsilon", + "type": "interline_equation", + "image_path": "673a740db6b460eeffefa1d512531991080b20378946835845d2546c335a2530.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 233, + 441, + 378, + 467 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 468, + 161, + 480 + ], + "lines": [ + { + "bbox": [ + 104, + 467, + 158, + 481 + ], + "spans": [ + { + "bbox": [ + 104, + 467, + 133, + 481 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 134, + 469, + 158, + 479 + ], + "score": 0.89, + "content": "k \\geq 1", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 506, + 570 + ], + "lines": [ + { + "bbox": [ + 103, + 484, + 508, + 504 + ], + "spans": [ + { + "bbox": [ + 103, + 484, + 245, + 504 + ], + "score": 1.0, + "content": "Here, we needed the noise variance", + "type": "text" + }, + { + "bbox": [ + 245, + 487, + 265, + 501 + ], + "score": 0.93, + "content": "\\sigma _ { k / 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 484, + 368, + 504 + ], + "score": 1.0, + "content": "to decrease in the order of", + "type": "text" + }, + { + "bbox": [ + 369, + 487, + 400, + 500 + ], + "score": 0.95, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 484, + 508, + 504 + ], + "score": 1.0, + "content": "so that the stochastic error", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 102, + 496, + 510, + 521 + ], + "spans": [ + { + "bbox": [ + 102, + 496, + 305, + 521 + ], + "score": 1.0, + "content": "of the S-FEG does not accumulate. Otherwise, if", + "type": "text" + }, + { + "bbox": [ + 306, + 501, + 325, + 516 + ], + "score": 0.93, + "content": "\\sigma _ { k / 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 496, + 405, + 521 + ], + "score": 1.0, + "content": "is a constant for all", + "type": "text" + }, + { + "bbox": [ + 406, + 502, + 412, + 511 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 496, + 510, + 521 + ], + "score": 1.0, + "content": ", the error accumulates", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 144, + 527 + ], + "score": 1.0, + "content": "with rate", + "type": "text" + }, + { + "bbox": [ + 144, + 514, + 166, + 527 + ], + "score": 0.91, + "content": "{ \\mathcal { O } } ( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 515, + 505, + 527 + ], + "score": 1.0, + "content": ". In short, the S-FEG will suffer from error accumulation, unless the stochastic error", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 185, + 538 + ], + "score": 1.0, + "content": "decreases with rate", + "type": "text" + }, + { + "bbox": [ + 186, + 526, + 218, + 538 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 526, + 505, + 538 + ], + "score": 1.0, + "content": ". Such error accumulation behavior also appears in a stochastic version", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "of Nesterov’s fast gradient method [35, 36] for smooth convex minimization [5, 8]. Similar to [5],", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "we believe that adjusting the step coefficients of the S-FEG can make the S-FEG become relatively", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 559, + 368, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 368, + 569 + ], + "score": 1.0, + "content": "stable even with a constant noise, which we leave as future work.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 106, + 584, + 421, + 599 + ], + "lines": [ + { + "bbox": [ + 104, + 583, + 423, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 583, + 423, + 601 + ], + "score": 1.0, + "content": "7 Convergence analysis with nonincreasing potential lemma", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 608, + 506, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 478, + 621 + ], + "score": 1.0, + "content": "We analyze FEG and FEG-A by finding a nonincreasing potential function in a form", + "type": "text" + }, + { + "bbox": [ + 478, + 609, + 505, + 620 + ], + "score": 0.89, + "content": "V _ { k } ~ =", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 619, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 107, + 619, + 241, + 632 + ], + "score": 0.89, + "content": "a _ { k } \\| F z _ { k } \\| ^ { 2 } - b _ { k } \\left. F z _ { k } , z _ { 0 } - z _ { k } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 619, + 506, + 632 + ], + "score": 1.0, + "content": "in the lemma below. We provide a similar potential lemma for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 630, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 643 + ], + "score": 1.0, + "content": "S-FEG in Appendix D.2. The convergence analyses of EAG and Halpern iteration are also based on", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 642, + 232, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 232, + 654 + ], + "score": 1.0, + "content": "such potential function [6, 43].", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 104, + 652, + 503, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 652, + 181, + 669 + ], + "score": 1.0, + "content": "Lemma 7.1. Let", + "type": "text" + }, + { + "bbox": [ + 181, + 654, + 217, + 667 + ], + "score": 0.92, + "content": "\\{ z _ { k } \\} _ { k \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 652, + 423, + 669 + ], + "score": 1.0, + "content": "be the sequence generated by (Class FEG) with", + "type": "text" + }, + { + "bbox": [ + 423, + 654, + 503, + 667 + ], + "score": 0.54, + "content": "\\{ \\alpha _ { k } \\} _ { k \\geq 0 } , \\{ \\beta _ { k } \\} _ { k \\geq 0 } ,", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 665, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 107, + 667, + 185, + 680 + ], + "score": 0.91, + "content": "\\{ L _ { k } \\} _ { k \\ge 0 } \\subset ( 0 , \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 665, + 204, + 682 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 205, + 667, + 263, + 680 + ], + "score": 0.91, + "content": "\\{ \\rho _ { k } \\} _ { k \\ge 0 } \\subset \\mathbb { R } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 665, + 308, + 682 + ], + "score": 1.0, + "content": ", satisfying", + "type": "text" + }, + { + "bbox": [ + 308, + 667, + 359, + 679 + ], + "score": 0.89, + "content": "\\alpha _ { 0 } \\in ( 0 , \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 665, + 363, + 682 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 363, + 666, + 418, + 681 + ], + "score": 0.91, + "content": "\\textstyle \\alpha _ { k } \\in { \\bigl ( } 0 , { \\frac { 1 } { L _ { k } } } { \\bigr ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 665, + 423, + 682 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 423, + 667, + 452, + 680 + ], + "score": 0.67, + "content": "\\beta _ { 0 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 665, + 456, + 682 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 457, + 667, + 505, + 680 + ], + "score": 0.83, + "content": "\\{ \\beta _ { k } \\} _ { k \\ge 1 } \\subseteq", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 679, + 205, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 128, + 692 + ], + "score": 0.9, + "content": "( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 679, + 157, + 694 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 157, + 681, + 182, + 691 + ], + "score": 0.88, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 679, + 205, + 694 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 694, + 429, + 721 + ], + "lines": [ + { + "bbox": [ + 181, + 694, + 429, + 721 + ], + "spans": [ + { + "bbox": [ + 181, + 694, + 429, + 721 + ], + "score": 0.93, + "content": "\\frac { ( 1 - \\beta _ { k + 1 } ) } { 2 \\beta _ { k + 1 } } ( \\alpha _ { k + 1 } + 2 \\rho _ { k + 1 } ) - \\rho _ { k + 1 } \\leq \\frac { 1 } { 2 \\beta _ { k } } ( \\alpha _ { k } + 2 \\rho _ { k } ) - \\rho _ { k }", + "type": "interline_equation", + "image_path": "e295973a7169ca644968b92ff5d1a1fa708621d1fbf16782684725f666275069.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 181, + 694, + 429, + 721 + ], + "spans": [], + "index": 43 + } + ] + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 740, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 740, + 309, + 752 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 70, + 274, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 275, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 275, + 88 + ], + "score": 1.0, + "content": "6 FEG under stochastic setting", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 94, + 506, + 178 + ], + "lines": [ + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 212, + 108 + ], + "score": 1.0, + "content": "When exactly computing", + "type": "text" + }, + { + "bbox": [ + 212, + 96, + 228, + 105 + ], + "score": 0.85, + "content": "\\pmb { F } z", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 94, + 505, + 108 + ], + "score": 1.0, + "content": "is expensive in practice, one usually instead consider its stochastic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "estimate for computational efficiency (see, e.g., [13, 16, 26, 33, 40, 42, 44]). This section also", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "considers using a stochastic oracle in FEG for smooth convex-concave problems. In specific, this", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 507, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 405, + 144 + ], + "score": 1.0, + "content": "section assumes that we only have access to a noisy saddle gradient oracle,", + "type": "text" + }, + { + "bbox": [ + 405, + 128, + 502, + 142 + ], + "score": 0.92, + "content": "\\tilde { F } z _ { k / 2 } = F \\bar { z } _ { k / 2 } + \\xi _ { k / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 127, + 507, + 144 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 103, + 136, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 103, + 136, + 133, + 160 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 141, + 177, + 155 + ], + "score": 0.93, + "content": "\\{ \\xi _ { k / 2 } \\} _ { k \\geq 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 136, + 359, + 160 + ], + "score": 1.0, + "content": "are independent random variables satisfying", + "type": "text" + }, + { + "bbox": [ + 360, + 141, + 408, + 155 + ], + "score": 0.94, + "content": "\\mathbb { E } [ \\xi _ { k / 2 } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 136, + 426, + 160 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 427, + 142, + 504, + 156 + ], + "score": 0.9, + "content": "\\mathbb { E } [ \\| \\xi _ { k / 2 } \\| ^ { 2 } ] = \\sigma _ { k / 2 } ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 134, + 167 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 134, + 155, + 161, + 166 + ], + "score": 0.9, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 154, + 506, + 167 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 184, + 326, + 284, + 341 + ], + "score": 0.92, + "content": "\\tilde { F } z _ { k / 2 } = F z _ { k / 2 } + \\xi _ { k / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 326, + 316, + 342 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 316, + 328, + 360, + 341 + ], + "score": 0.94, + "content": "\\{ \\xi _ { k / 2 } \\} _ { k \\geq 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 326, + 506, + 342 + ], + "score": 1.0, + "content": "are independent random variables", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 339, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 148, + 356 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 148, + 341, + 198, + 354 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\xi _ { k / 2 } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 339, + 217, + 356 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 217, + 341, + 294, + 355 + ], + "score": 0.88, + "content": "\\mathbb { E } [ \\| \\xi _ { k / 2 } \\| ^ { 2 } ] = \\sigma _ { k / 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 339, + 323, + 356 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 324, + 342, + 349, + 352 + ], + "score": 0.89, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 339, + 408, + 356 + ], + "score": 1.0, + "content": ". 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Otherwise, if", + "type": "text" + }, + { + "bbox": [ + 306, + 501, + 325, + 516 + ], + "score": 0.93, + "content": "\\sigma _ { k / 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 496, + 405, + 521 + ], + "score": 1.0, + "content": "is a constant for all", + "type": "text" + }, + { + "bbox": [ + 406, + 502, + 412, + 511 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 496, + 510, + 521 + ], + "score": 1.0, + "content": ", the error accumulates", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 144, + 527 + ], + "score": 1.0, + "content": "with rate", + "type": "text" + }, + { + "bbox": [ + 144, + 514, + 166, + 527 + ], + "score": 0.91, + "content": "{ \\mathcal { O } } ( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 515, + 505, + 527 + ], + "score": 1.0, + "content": ". In short, the S-FEG will suffer from error accumulation, unless the stochastic error", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 185, + 538 + ], + "score": 1.0, + "content": "decreases with rate", + "type": "text" + }, + { + "bbox": [ + 186, + 526, + 218, + 538 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 526, + 505, + 538 + ], + "score": 1.0, + "content": ". Such error accumulation behavior also appears in a stochastic version", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "of Nesterov’s fast gradient method [35, 36] for smooth convex minimization [5, 8]. Similar to [5],", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "we believe that adjusting the step coefficients of the S-FEG can make the S-FEG become relatively", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 559, + 368, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 368, + 569 + ], + "score": 1.0, + "content": "stable even with a constant noise, which we leave as future work.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 102, + 484, + 510, + 569 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 584, + 421, + 599 + ], + "lines": [ + { + "bbox": [ + 104, + 583, + 423, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 583, + 423, + 601 + ], + "score": 1.0, + "content": "7 Convergence analysis with nonincreasing potential lemma", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 608, + 506, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 478, + 621 + ], + "score": 1.0, + "content": "We analyze FEG and FEG-A by finding a nonincreasing potential function in a form", + "type": "text" + }, + { + "bbox": [ + 478, + 609, + 505, + 620 + ], + "score": 0.89, + "content": "V _ { k } ~ =", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 619, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 107, + 619, + 241, + 632 + ], + "score": 0.89, + "content": "a _ { k } \\| F z _ { k } \\| ^ { 2 } - b _ { k } \\left. F z _ { k } , z _ { 0 } - z _ { k } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 619, + 506, + 632 + ], + "score": 1.0, + "content": "in the lemma below. We provide a similar potential lemma for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 630, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 643 + ], + "score": 1.0, + "content": "S-FEG in Appendix D.2. 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F z _ { k } , z _ { 0 } - z _ { k } \\right.", + "type": "interline_equation", + "image_path": "1d43dcacaaf4f5dd94eb7f8ec44f9db705f678335179477f50b44f6a1ddb97a2.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 226, + 146, + 385, + 160 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 163, + 264, + 180 + ], + "lines": [ + { + "bbox": [ + 101, + 159, + 261, + 184 + ], + "spans": [ + { + "bbox": [ + 101, + 159, + 126, + 184 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 163, + 196, + 180 + ], + "score": 0.92, + "content": "\\begin{array} { r } { a _ { 0 } = \\frac { \\alpha _ { 0 } ( L _ { 0 } ^ { 2 } \\alpha _ { 0 } ^ { 2 } - 1 ) } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 159, + 200, + 184 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 200, + 166, + 261, + 179 + ], + "score": 0.44, + "content": "b _ { 0 } = 0 , b _ { 1 } = 1 ,", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 182, + 430, + 209 + ], + "lines": [ + { + "bbox": [ + 181, + 182, + 430, + 209 + ], + "spans": [ + { + "bbox": [ + 181, + 182, + 430, + 209 + ], + "score": 0.93, + "content": "a _ { k } = { \\frac { b _ { k } ( 1 - \\beta _ { k } ) } { 2 \\beta _ { k } } } ( \\alpha _ { k } + 2 \\rho _ { k } ) - b _ { k } \\rho _ { k } a n d b _ { k + 1 } = { \\frac { b _ { k } } { 1 - \\beta _ { k } } }", + "type": "interline_equation", + "image_path": "4be93e42dfda0c79f0664cc896405edf4cc225f1e62124c32060169ffdeba45f.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 181, + 182, + 430, + 209 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 295, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 293, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 133, + 225 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 134, + 211, + 159, + 222 + ], + "score": 0.89, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 210, + 194, + 225 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 194, + 211, + 240, + 223 + ], + "score": 0.91, + "content": "V _ { k } \\le V _ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 210, + 268, + 225 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 268, + 211, + 293, + 222 + ], + "score": 0.89, + "content": "k \\geq 1", + "type": "inline_equation" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 230, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "Based on the above potential lemma, we next provide a convergence analysis of FEG. The analyses", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "for the convergence rate of FEG-A and S-FEG, i.e., the proofs of Theorem 5.1 and Theorem 6.1, are", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 252, + 413, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 413, + 264 + ], + "score": 1.0, + "content": "similar to that of FEG and are provided in Appendix C.3 and Appendix D.3.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 275, + 259, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 261, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 261, + 290 + ], + "score": 1.0, + "content": "7.1 Convergence analysis for FEG", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 108, + 295, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 103, + 292, + 508, + 313 + ], + "spans": [ + { + "bbox": [ + 103, + 292, + 403, + 313 + ], + "score": 1.0, + "content": "Proof of Theorem 4.1. 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F z _ { k } , z _ { 0 } - z _ { k } \\right. .", + "type": "interline_equation", + "image_path": "98caa60be890fc6b4faf7b664b065c56e97985f31c6546a7f24b54a8a26e579e.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 162, + 410, + 448, + 438 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 150, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 151, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 151, + 453 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 453, + 462, + 523 + ], + "lines": [ + { + "bbox": [ + 149, + 453, + 462, + 523 + ], + "spans": [ + { + "bbox": [ + 149, + 453, + 462, + 523 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\displaystyle \\frac { k ^ { 2 } } { 2 } \\Big ( \\frac { 1 } { L } + 2 \\rho \\Big ) \\| F z _ { k } \\| ^ { 2 } \\leq k \\langle F z _ { k } , z _ { 0 } - z _ { k } \\rangle + k \\rho \\| F z _ { k } \\| ^ { 2 } } \\\\ & { \\quad \\quad \\quad \\quad = k \\langle F z _ { k } , z _ { 0 } - z _ { * } \\rangle + k \\langle F z _ { k } , z _ { * } - z _ { k } \\rangle + k \\rho \\| F z _ { k } \\| ^ { 2 } } \\\\ & { \\quad \\quad \\quad \\leq k \\langle F z _ { k } , z _ { 0 } - z _ { * } \\rangle \\quad \\quad \\quad \\quad ( \\because \\rho \\mathrm { - c o m o n o t o n i c i t y ~ o f ~ } F ) } \\\\ & { \\quad \\quad \\quad \\quad \\leq k \\| F z _ { k } \\| \\| z _ { 0 } - z _ { * } \\| . } \\end{array}", + "type": "interline_equation", + "image_path": "1dfdd9ea91944e95ca9d03700e1fc5e2d6630d84ba1c74aedd4c0e3604a09ecd.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 149, + 453, + 462, + 476.3333333333333 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 149, + 476.3333333333333, + 462, + 499.66666666666663 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 149, + 499.66666666666663, + 462, + 523.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 524, + 431, + 541 + ], + "lines": [ + { + "bbox": [ + 104, + 522, + 432, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 348, + 542 + ], + "score": 1.0, + "content": "The desired result follows directly by dividing both sides by", + "type": "text" + }, + { + "bbox": [ + 349, + 524, + 428, + 541 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { k ^ { 2 } } { 2 } \\left( \\frac { 1 } { L } + 2 \\rho \\right) \\| F z _ { k } \\| } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 522, + 432, + 542 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 104, + 553, + 466, + 567 + ], + "lines": [ + { + "bbox": [ + 104, + 552, + 468, + 570 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 468, + 570 + ], + "score": 1.0, + "content": "8 Discussion: first-order methods for Lipschitz continuous operators", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 104, + 577, + 504, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "score": 1.0, + "content": "Throughout this paper, we studied and constructed efficient methods in a class of first-order methods:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 592, + 374, + 605 + ], + "lines": [ + { + "bbox": [ + 237, + 592, + 374, + 605 + ], + "spans": [ + { + "bbox": [ + 237, + 592, + 374, + 605 + ], + "score": 0.91, + "content": "z _ { k } \\in z _ { 0 } + \\operatorname { s p a n } \\{ F z _ { 0 } , \\cdot \\cdot \\cdot , F z _ { k } \\}", + "type": "interline_equation", + "image_path": "03be5922e9969d2018e6ac91d4ae72346b3eff2a43120b876c721ba8237810a6.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 237, + 592, + 374, + 605 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 607, + 506, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 153, + 621 + ], + "score": 1.0, + "content": "denoted by", + "type": "text" + }, + { + "bbox": [ + 153, + 608, + 162, + 618 + ], + "score": 0.71, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 606, + 506, + 621 + ], + "score": 1.0, + "content": ", for smooth structured nonconvex-nonconcave problems. We observed that all existing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "first-order methods, including the FEG, required an additional condition, such as the negative", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 630, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 272, + 641 + ], + "score": 1.0, + "content": "comonoticity, on a Lipschitz continuous", + "type": "text" + }, + { + "bbox": [ + 272, + 630, + 282, + 639 + ], + "score": 0.82, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 630, + 506, + 641 + ], + "score": 1.0, + "content": "to guarantee convergence. One would then be curious", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 331, + 654 + ], + "score": 1.0, + "content": "whether or not there exists an (efficient) method in class", + "type": "text" + }, + { + "bbox": [ + 331, + 641, + 341, + 651 + ], + "score": 0.79, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "that guarantees convergence without any", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 299, + 664 + ], + "score": 1.0, + "content": "additional condition on a Lipschitz continuous", + "type": "text" + }, + { + "bbox": [ + 299, + 652, + 308, + 661 + ], + "score": 0.81, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 651, + 506, + 664 + ], + "score": 1.0, + "content": ". Unfortunately, the following lemma states that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 661, + 507, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 385, + 676 + ], + "score": 1.0, + "content": "there exists a worst-case7 smooth example that none of the methods in", + "type": "text" + }, + { + "bbox": [ + 385, + 663, + 394, + 672 + ], + "score": 0.8, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 661, + 507, + 676 + ], + "score": 1.0, + "content": "can find its stationary point.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 673, + 352, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 352, + 686 + ], + "score": 1.0, + "content": "The corresponding smooth function is illustrated in Figure 3.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 691, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 120, + 689, + 507, + 704 + ], + "spans": [ + { + "bbox": [ + 120, + 689, + 507, + 704 + ], + "score": 1.0, + "content": "7[15, 20] also introduce worst-case minimax examples that existing methods cannot find a stationary point.", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 701, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 506, + 713 + ], + "score": 1.0, + "content": "A key difference from our example is that their saddle-gradient operators are not Lipschitz continuous. In", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 712, + 460, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 334, + 722 + ], + "score": 1.0, + "content": "addition, the considered classes of methods in [15, 20] exclude", + "type": "text" + }, + { + "bbox": [ + 334, + 712, + 353, + 721 + ], + "score": 0.45, + "content": "\\mathrm { E G + }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 712, + 447, + 722 + ], + "score": 1.0, + "content": "and FEG, unlike the class", + "type": "text" + }, + { + "bbox": [ + 448, + 712, + 456, + 720 + ], + "score": 0.74, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 712, + 460, + 722 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 527, + 505, + 538 + ], + "lines": [ + { + "bbox": [ + 496, + 529, + 504, + 537 + ], + "spans": [ + { + "bbox": [ + 496, + 529, + 504, + 537 + ], + "score": 0.994, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 72, + 365, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 365, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 133, + 86 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 134, + 73, + 159, + 84 + ], + "score": 0.89, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 72, + 365, + 86 + ], + "score": 1.0, + "content": ". 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F z _ { k } , z _ { 0 } - z _ { k } \\right.", + "type": "interline_equation", + "image_path": "1d43dcacaaf4f5dd94eb7f8ec44f9db705f678335179477f50b44f6a1ddb97a2.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 226, + 146, + 385, + 160 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 163, + 264, + 180 + ], + "lines": [ + { + "bbox": [ + 101, + 159, + 261, + 184 + ], + "spans": [ + { + "bbox": [ + 101, + 159, + 126, + 184 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 163, + 196, + 180 + ], + "score": 0.92, + "content": "\\begin{array} { r } { a _ { 0 } = \\frac { \\alpha _ { 0 } ( L _ { 0 } ^ { 2 } \\alpha _ { 0 } ^ { 2 } - 1 ) } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 159, + 200, + 184 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 200, + 166, + 261, + 179 + ], + "score": 0.44, + "content": "b _ { 0 } = 0 , b _ { 1 } = 1 ,", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 101, + 159, + 261, + 184 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 182, + 430, + 209 + ], + "lines": [ + { + "bbox": [ + 181, + 182, + 430, + 209 + ], + "spans": [ + { + "bbox": [ + 181, + 182, + 430, + 209 + ], + "score": 0.93, + "content": "a _ { k } = { \\frac { b _ { k } ( 1 - \\beta _ { k } ) } { 2 \\beta _ { k } } } ( \\alpha _ { k } + 2 \\rho _ { k } ) - b _ { k } \\rho _ { k } a n d b _ { k + 1 } = { \\frac { b _ { k } } { 1 - \\beta _ { k } } }", + "type": "interline_equation", + "image_path": "4be93e42dfda0c79f0664cc896405edf4cc225f1e62124c32060169ffdeba45f.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 181, + 182, + 430, + 209 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 295, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 293, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 133, + 225 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 134, + 211, + 159, + 222 + ], + "score": 0.89, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 210, + 194, + 225 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 194, + 211, + 240, + 223 + ], + "score": 0.91, + "content": "V _ { k } \\le V _ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 210, + 268, + 225 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 268, + 211, + 293, + 222 + ], + "score": 0.89, + "content": "k \\geq 1", + "type": "inline_equation" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 210, + 293, + 225 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 230, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "Based on the above potential lemma, we next provide a convergence analysis of FEG. The analyses", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "for the convergence rate of FEG-A and S-FEG, i.e., the proofs of Theorem 5.1 and Theorem 6.1, are", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 252, + 413, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 413, + 264 + ], + "score": 1.0, + "content": "similar to that of FEG and are provided in Appendix C.3 and Appendix D.3.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 230, + 505, + 264 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 275, + 259, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 261, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 261, + 290 + ], + "score": 1.0, + "content": "7.1 Convergence analysis for FEG", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 108, + 295, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 103, + 292, + 508, + 313 + ], + "spans": [ + { + "bbox": [ + 103, + 292, + 403, + 313 + ], + "score": 1.0, + "content": "Proof of Theorem 4.1. Recall that FEG is equivalent to (Class FEG) with", + "type": "text" + }, + { + "bbox": [ + 404, + 295, + 484, + 309 + ], + "score": 0.78, + "content": "\\begin{array} { r } { \\alpha _ { k } = \\frac { 1 } { L } , \\beta _ { k } = \\frac { 1 } { k + 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 292, + 508, + 313 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 107, + 310, + 136, + 320 + ], + "score": 0.88, + "content": "\\rho _ { k } = \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 308, + 314, + 322 + ], + "score": 1.0, + "content": ". It is straightforward to verify that the given", + "type": "text" + }, + { + "bbox": [ + 315, + 308, + 352, + 321 + ], + "score": 0.93, + "content": "\\{ \\alpha _ { k } \\} _ { k \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 308, + 370, + 322 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 370, + 308, + 406, + 321 + ], + "score": 0.93, + "content": "\\{ \\beta _ { k } \\} _ { k \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "satisfy the conditions in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 318, + 290, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 174, + 333 + ], + "score": 1.0, + "content": "Lemma 7.1 with", + "type": "text" + }, + { + "bbox": [ + 175, + 320, + 208, + 331 + ], + "score": 0.93, + "content": "L _ { k } = L", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 318, + 235, + 333 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 235, + 320, + 261, + 330 + ], + "score": 0.91, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 318, + 290, + 333 + ], + "score": 1.0, + "content": ". 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F z _ { k } , z _ { 0 } - z _ { k } \\right. .", + "type": "interline_equation", + "image_path": "98caa60be890fc6b4faf7b664b065c56e97985f31c6546a7f24b54a8a26e579e.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 162, + 410, + 448, + 438 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 150, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 151, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 151, + 453 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 438, + 151, + 453 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 453, + 462, + 523 + ], + "lines": [ + { + "bbox": [ + 149, + 453, + 462, + 523 + ], + "spans": [ + { + "bbox": [ + 149, + 453, + 462, + 523 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\displaystyle \\frac { k ^ { 2 } } { 2 } \\Big ( \\frac { 1 } { L } + 2 \\rho \\Big ) \\| F z _ { k } \\| ^ { 2 } \\leq k \\langle F z _ { k } , z _ { 0 } - z _ { k } \\rangle + k \\rho \\| F z _ { k } \\| ^ { 2 } } \\\\ & { \\quad \\quad \\quad \\quad = k \\langle F z _ { k } , z _ { 0 } - z _ { * } \\rangle + k \\langle F z _ { k } , z _ { * } - z _ { k } \\rangle + k \\rho \\| F z _ { k } \\| ^ { 2 } } \\\\ & { \\quad \\quad \\quad \\leq k \\langle F z _ { k } , z _ { 0 } - z _ { * } \\rangle \\quad \\quad \\quad \\quad ( \\because \\rho \\mathrm { - c o m o n o t o n i c i t y ~ o f ~ } F ) } \\\\ & { \\quad \\quad \\quad \\quad \\leq k \\| F z _ { k } \\| \\| z _ { 0 } - z _ { * } \\| . } \\end{array}", + "type": "interline_equation", + "image_path": "1dfdd9ea91944e95ca9d03700e1fc5e2d6630d84ba1c74aedd4c0e3604a09ecd.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 149, + 453, + 462, + 476.3333333333333 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 149, + 476.3333333333333, + 462, + 499.66666666666663 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 149, + 499.66666666666663, + 462, + 523.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 524, + 431, + 541 + ], + "lines": [ + { + "bbox": [ + 104, + 522, + 432, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 348, + 542 + ], + "score": 1.0, + "content": "The desired result follows directly by dividing both sides by", + "type": "text" + }, + { + "bbox": [ + 349, + 524, + 428, + 541 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { k ^ { 2 } } { 2 } \\left( \\frac { 1 } { L } + 2 \\rho \\right) \\| F z _ { k } \\| } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 522, + 432, + 542 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 522, + 432, + 542 + ] + }, + { + "type": "title", + "bbox": [ + 104, + 553, + 466, + 567 + ], + "lines": [ + { + "bbox": [ + 104, + 552, + 468, + 570 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 468, + 570 + ], + "score": 1.0, + "content": "8 Discussion: first-order methods for Lipschitz continuous operators", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 104, + 577, + 504, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "score": 1.0, + "content": "Throughout this paper, we studied and constructed efficient methods in a class of first-order methods:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 576, + 506, + 590 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 592, + 374, + 605 + ], + "lines": [ + { + "bbox": [ + 237, + 592, + 374, + 605 + ], + "spans": [ + { + "bbox": [ + 237, + 592, + 374, + 605 + ], + "score": 0.91, + "content": "z _ { k } \\in z _ { 0 } + \\operatorname { s p a n } \\{ F z _ { 0 } , \\cdot \\cdot \\cdot , F z _ { k } \\}", + "type": "interline_equation", + "image_path": "03be5922e9969d2018e6ac91d4ae72346b3eff2a43120b876c721ba8237810a6.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 237, + 592, + 374, + 605 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 607, + 506, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 153, + 621 + ], + "score": 1.0, + "content": "denoted by", + "type": "text" + }, + { + "bbox": [ + 153, + 608, + 162, + 618 + ], + "score": 0.71, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 606, + 506, + 621 + ], + "score": 1.0, + "content": ", for smooth structured nonconvex-nonconcave problems. We observed that all existing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "first-order methods, including the FEG, required an additional condition, such as the negative", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 630, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 272, + 641 + ], + "score": 1.0, + "content": "comonoticity, on a Lipschitz continuous", + "type": "text" + }, + { + "bbox": [ + 272, + 630, + 282, + 639 + ], + "score": 0.82, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 630, + 506, + 641 + ], + "score": 1.0, + "content": "to guarantee convergence. One would then be curious", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 331, + 654 + ], + "score": 1.0, + "content": "whether or not there exists an (efficient) method in class", + "type": "text" + }, + { + "bbox": [ + 331, + 641, + 341, + 651 + ], + "score": 0.79, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "that guarantees convergence without any", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 299, + 664 + ], + "score": 1.0, + "content": "additional condition on a Lipschitz continuous", + "type": "text" + }, + { + "bbox": [ + 299, + 652, + 308, + 661 + ], + "score": 0.81, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 651, + 506, + 664 + ], + "score": 1.0, + "content": ". 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Let us consider the following function", + "type": "text" + }, + { + "bbox": [ + 315, + 282, + 367, + 294 + ], + "score": 0.89, + "content": "f : \\mathbb { R } ^ { 2 } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 281, + 405, + 296 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 406, + 283, + 444, + 294 + ], + "score": 0.89, + "content": "L , R > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 281, + 448, + 296 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 299, + 441, + 376 + ], + "lines": [ + { + "bbox": [ + 170, + 299, + 441, + 376 + ], + "spans": [ + { + "bbox": [ + 170, + 299, + 441, + 376 + ], + "score": 0.96, + "content": "\\begin{array} { r } { f ( x , y ) = \\left\\{ \\begin{array} { l l } { \\frac { R } { 2 } } & { f o r x < y - \\sqrt { \\frac { R } { L } } } \\\\ { - \\frac { L } { 2 } ( x - y ) ^ { 2 } - \\sqrt { L R } ( x - y ) } & { f o r y - \\sqrt { \\frac { R } { L } } \\leq x < y } \\\\ { \\frac { L } { 2 } ( x - y ) ^ { 2 } - \\sqrt { L R } ( x - y ) } & { f o r y \\leq x < y + \\sqrt { \\frac { R } { L } } } \\\\ { - \\frac { R } { 2 } } & { f o r y + \\sqrt { \\frac { R } { L } } < x . } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "337423b721db43d80d764145e8e0607d246394769080daa8296b7c981a4a7770.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 170, + 299, + 441, + 324.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 170, + 324.6666666666667, + 441, + 350.33333333333337 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 170, + 350.33333333333337, + 441, + 376.00000000000006 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 504, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 222, + 395 + ], + "score": 1.0, + "content": "Its saddle-gradient operator", + "type": "text" + }, + { + "bbox": [ + 222, + 383, + 232, + 392 + ], + "score": 0.8, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 381, + 242, + 395 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 243, + 383, + 250, + 392 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 381, + 505, + 395 + ], + "score": 1.0, + "content": "-Lipschitz continuous but not comonotone.8 Then, the sequence", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 392, + 504, + 406 + ], + "spans": [ + { + "bbox": [ + 107, + 393, + 143, + 405 + ], + "score": 0.91, + "content": "\\{ z _ { k } \\} _ { k \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 392, + 324, + 406 + ], + "score": 1.0, + "content": "generated by any first-order method in class", + "type": "text" + }, + { + "bbox": [ + 325, + 394, + 334, + 403 + ], + "score": 0.75, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 392, + 354, + 406 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 354, + 393, + 401, + 405 + ], + "score": 0.94, + "content": "z _ { 0 } = ( 0 , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 392, + 437, + 406 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 437, + 393, + 504, + 405 + ], + "score": 0.93, + "content": "\\| { \\pmb { F } } { \\pmb { z } } _ { k } \\| ^ { 2 } = 2 L R", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 404, + 162, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 133, + 416 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 134, + 405, + 158, + 415 + ], + "score": 0.9, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 404, + 162, + 416 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 427, + 505, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 136, + 443 + ], + "score": 1.0, + "content": "Proof.", + "type": "text" + }, + { + "bbox": [ + 136, + 429, + 146, + 439 + ], + "score": 0.79, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 426, + 180, + 443 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 180, + 428, + 299, + 441 + ], + "score": 0.91, + "content": "F ( x , y ) = ( - 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We also leave", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 326, + 509 + ], + "score": 1.0, + "content": "finding additional conditions for a Lipschitz continuous", + "type": "text" + }, + { + "bbox": [ + 327, + 496, + 336, + 506 + ], + "score": 0.82, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 495, + 506, + 509 + ], + "score": 1.0, + "content": ", weaker than the weak MVI condition and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 506, + 508, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 247, + 522 + ], + "score": 1.0, + "content": "the negative comonotonicity (with", + "type": "text" + }, + { + "bbox": [ + 247, + 506, + 289, + 520 + ], + "score": 0.94, + "content": "\\begin{array} { r } { { \\Dot { \\rho } } > - \\frac { 1 } { 2 L } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 506, + 508, + 522 + ], + "score": 1.0, + "content": ", which guarantee convergence or its accelerated rate,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 518, + 221, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 221, + 531 + ], + "score": 1.0, + "content": "respectively, as future work.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 106, + 544, + 183, + 558 + ], + "lines": [ + { + "bbox": [ + 104, + 542, + 184, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 542, + 184, + 561 + ], + "score": 1.0, + "content": "9 Conclusion", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 506, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "This paper proposed a two-time-scale and anchored extragradient method, named FEG, for smooth", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 448, + 593 + ], + "score": 1.0, + "content": "structured nonconvex-nonconcave problems. The proposed FEG has an accelerated", + "type": "text" + }, + { + "bbox": [ + 448, + 580, + 484, + 592 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "rate,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "with respect to the squared gradient norm, for the Lipschitz continuous and negative comonotone", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "operators for the first time. The FEG also has value for smooth convex-concave problems, compared", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "to existing works. We further studied its backtracking line-search version, named FEG-A, for the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 624, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 506, + 636 + ], + "score": 1.0, + "content": "smooth structured nonconvex-nonconcave problems and studied its stochastic version, named S-FEG,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "score": 1.0, + "content": "for smooth convex-concave problems. 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Let us consider the following function", + "type": "text" + }, + { + "bbox": [ + 315, + 282, + 367, + 294 + ], + "score": 0.89, + "content": "f : \\mathbb { R } ^ { 2 } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 281, + 405, + 296 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 406, + 283, + 444, + 294 + ], + "score": 0.89, + "content": "L , R > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 281, + 448, + 296 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 281, + 448, + 296 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 299, + 441, + 376 + ], + "lines": [ + { + "bbox": [ + 170, + 299, + 441, + 376 + ], + "spans": [ + { + "bbox": [ + 170, + 299, + 441, + 376 + ], + "score": 0.96, + "content": "\\begin{array} { r } { f ( x , y ) = \\left\\{ \\begin{array} { l l } { \\frac { R } { 2 } } & { f o r x < y - 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We also leave", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 326, + 509 + ], + "score": 1.0, + "content": "finding additional conditions for a Lipschitz continuous", + "type": "text" + }, + { + "bbox": [ + 327, + 496, + 336, + 506 + ], + "score": 0.82, + "content": "\\pmb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 495, + 506, + 509 + ], + "score": 1.0, + "content": ", weaker than the weak MVI condition and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 506, + 508, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 247, + 522 + ], + "score": 1.0, + "content": "the negative comonotonicity (with", + "type": "text" + }, + { + "bbox": [ + 247, + 506, + 289, + 520 + ], + "score": 0.94, + "content": "\\begin{array} { r } { { \\Dot { \\rho } } > - \\frac { 1 } { 2 L } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 506, + 508, + 522 + ], + "score": 1.0, + "content": ", which guarantee convergence or its accelerated rate,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 518, + 221, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 221, + 531 + ], + "score": 1.0, + "content": "respectively, as future work.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 474, + 508, + 531 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 544, + 183, + 558 + ], + "lines": [ + { + "bbox": [ + 104, + 542, + 184, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 542, + 184, + 561 + ], + "score": 1.0, + "content": "9 Conclusion", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 506, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "This paper proposed a two-time-scale and anchored extragradient method, named FEG, for smooth", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 448, + 593 + ], + "score": 1.0, + "content": "structured nonconvex-nonconcave problems. The proposed FEG has an accelerated", + "type": "text" + }, + { + "bbox": [ + 448, + 580, + 484, + 592 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "rate,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "with respect to the squared gradient norm, for the Lipschitz continuous and negative comonotone", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "operators for the first time. The FEG also has value for smooth convex-concave problems, compared", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "to existing works. We further studied its backtracking line-search version, named FEG-A, for the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 624, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 506, + 636 + ], + "score": 1.0, + "content": "smooth structured nonconvex-nonconcave problems and studied its stochastic version, named S-FEG,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "score": 1.0, + "content": "for smooth convex-concave problems. 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MethodConvex-concaveNonconvex-nonconcave
Cocoercive MonotoneNegative comonotoneMVIWeak MVI
NormalEG [4, 42]0(1/k)0(1/k）0(1/k）
EG+[7]0(1/k)0(1/k)0(1/k)0(1/k）0(1/k）
AcceleratedHalpern [12, 6]0(1/k²)(1/k²)
EAG [43] FEG (this paper)0(1/k2) 0(1/k2)0(1/k²) 0(1/k2)0(1/k2)
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Input: z0 ∈ Rd,L ∈ (0,∞o),ρ ∈(- 2,00) for k = 0,1,... do
2+1/=+1(2-（1-1）（+2）F 1
1 2 2k+1= 2k+ k+1
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Our results indicate that convolutional neural networks can operate without any loss of accuracy at less than $0 . 5 \%$ classification layer connection density, or less than $5 \%$ overall network connection density. We also investigate the effects of pre-defining the sparsity of networks with only fully connected layers. Based on our sparsifying technique, we introduce the ‘scatter’ metric to characterize the quality of a particular connection pattern. As proof of concept, we show results on CIFAR, MNIST and a new dataset on classifying Morse code symbols, which highlights some interesting trends and limits of sparse connection patterns. + +# 1 INTRODUCTION + +Neural networks (NNs) in machine learning systems are critical drivers of new technologies such as image processing and speech recognition. Modern NNs are gigantic in size with millions of parameters, such as the ones described in Alexnet (Krizhevsky et al., 2012), Overfeat (Sermanet et al., 2013) and ResNet (He et al., 2016). They therefore require an enormous amount of memory and silicon processing during usage. Optimizing a network to improve performance typically involves making it deeper and adding more parameters (Simonyan & Zisserman, 2015; Szegedy et al., 2015; Huang et al., 2016), which further exacerbates the problem of large storage complexity. While the convolutional (conv) layers in these networks do feature extraction, there are usually fully connected layers at the end performing classification. We shall henceforth refer to these layers as connected layers $( C L s )$ , of which fully connected layers $( F C L s )$ are a special case. Owing to their high density of connections, the majority of network parameters are concentrated in FCLs. For example, the FCLs in Alexnet account for $9 5 . 7 \%$ of the network parameters (Zhang et al., 2016). + +We shall refer to the spaces between CLs as CL junctions (or simply junctions), which are occupied by connections, or weights. Given the trend in modern NNs, we raise the question – “How necessary is it to have FCLs?” or, in other words, “What if most of the junction connections never existed? Would the resulting sparsely connected layers (SCLs), when trained and tested, still give competitive performance?” As an example, consider a network with 2 CLs of 100 neurons each and the junction between them has 1000 weights instead of the expected 10,000. Then this is a sparse network with connection density of $10 \%$ . Given such a sparse architecture, a natural question to ask is “How can the existing 1000 weights be best distributed so that network performance is maximized?” + +In this regard, the present work makes the following contributions. In Section 2, we formalize the concept of sparsity, or its opposite measure density, and explore its effects on different network types. We show that CL parameters are largely redundant and a network pre-defined to be sparse before starting training does not result in any performance degradation. For certain network architectures, this leads to CL parameter reduction by a factor of more than 450, or an overall parameter reduction by a factor of more than 20. In Section 2.4, we discuss techniques to distribute connections across junctions when given an overall network density. Finally, in Section 3, we formalize pre-defined sparse connectivity patterns using adjacency matrices and introduce the scatter metric. Our results show that scatter is a quick and useful indicator of how good a sparse network is. + +# 2 PRE-DEFINED SPARSITY + +As an example of the footprint of modern NNs, AlexNet has a weight size of $2 3 4 \mathrm { M B }$ and requires 635 million arithmetic operations only for feedforward processing (Zhang et al., 2016). It has been shown that NNs, particularly their FCLs, have an excess of parameters and tend to overfit to the training data (Denil et al., 2013), resulting in inferior performance on test data. The following paragraph describes several previous works that have attempted to reduce parameters in NNs. + +Dropout (deletion) of random neurons (Srivastava et al., 2014) trains multiple differently configured networks, which are finally combined to regain the original full size network. Chen et al. (2015) randomly forced the same value on collections of weights, but acknowledged that “a significant number of nodes [get] disconnected from neighboring layers.” Other sparsifying techniques such as pruning and quantization (Han et al., 2016; 2015; Zhou et al., 2016; Gong et al., 2014) first train the complete network, and then perform further computations to delete parameters. Sindhwani et al. (2015) used low rank matrices to impose structure on network parameters. Srinivas et al. (2016) proposed a regularizer to reduce parameters in the network, but acknowledged that this increased training complexity. In general, all these architectures deal with FCLs at some point of time during their usage cycle and therefore, do not permanently solve the parameter explosion problem of NNs. + +# 2.1 OUR METHODOLOGY + +Our attempt to simplify NNs is to pre-define the level of sparsity, or connection density, in a network prior to the start of training. This means that our network always has fewer connections than its FCL counterpart; the weights which are absent never make an appearance during training or inference. In our notation, a NN will have $J$ junctions, i.e. $J + 1$ layers, with $\{ N _ { 1 } , \stackrel { \textstyle - } { N } _ { 2 } , \cdot \cdot \cdot \stackrel { \textstyle - } { , } N _ { J + 1 } \}$ being the number of neurons in each layer. $N _ { i }$ and $N _ { i + 1 }$ are respectively the number of neurons in the earlier (left) and later (right) layers of junction $i$ . Every left neuron has a fixed number of edges going from it to the right, and every right neuron has a fixed number of edges coming into it from the left. These numbers are defined as fan-out $( f o _ { i } )$ and fan-in $( f i _ { i } )$ , respectively. For conventional FCLs, $f o _ { i } = N _ { i + 1 }$ and $f i _ { i } = N _ { i }$ . We propose SCLs where $f o _ { i } < N _ { i + 1 }$ and $f i _ { i } < N _ { i }$ , such that $N _ { i } \times f o _ { i } = N _ { i + 1 } \times f i _ { i } = W _ { i }$ , the number of weights in junction $i$ . Having a fixed $f o _ { i }$ and $f i _ { i }$ ensures that all neurons in a junction contribute equally and none of them get disconnected, since that would lead to a loss of information. The connection density in junction $i$ is given as $W _ { i } / ( N _ { i } N _ { i + 1 } )$ and the overall CL connection density is defined as $\textstyle \left( \sum _ { i = 1 } ^ { J } W _ { i } \right) / \left( \sum _ { i = 1 } ^ { J } N _ { i } N _ { i + 1 } \right)$ . + +Note that earlier works such as Dey et al. (2017b;a) have proposed hardware architectures that leverage pre-defined sparsity to speed up training. However, a complete analysis of methods to pre-define connections, its possible gains on different kinds of modern deep NNs and a test of its limits via a metric quantifying its goodness has been lacking. Bourely et al. (2017) introduced a metric based on eigenvalues, but ran limited tests on MNIST. The following subsections analyze our method of pre-defined sparsity in more detail. We experimented with networks operating on CIFAR, MNIST and Morse code symbol classification – a new dataset described in Dey $( 2 0 1 7 ) ^ { \bar { 1 } }$ . + +# 2.2 NETWORK EXPERIMENTS + +# 2.2.1 CIFAR + +We used the original CIFAR10 and CIFAR100 datasets without data augmentation. Our network has 6 conv layers with number of filters equal to [64, 64, 128, 128, 256, 256]. Each has window size $3 { \tt X } 3$ . The outputs are batch-normalized before applying ReLU non-linearity. A max-pooling layer of pool size 2x2 succeeds every pair of conv layers. This structure finally results in a layer of 4096 neurons, which is followed by the CLs. We used the Adam optimizer, ReLU-activated hidden layers and softmax output layer – choices which we maintained for all networks unless otherwise specified. + +Our results in Section 2.4 indicate that later CL junctions (i.e. closer to the outputs) should be denser than earlier ones (i.e. closer to the inputs). Moreover, since most CL networks have a tapering structure where $N _ { i }$ monotonically decreases as $i$ increases, more parameter savings can be achieved by making earlier layers less dense. Accordingly we did a grid search and picked CL junction densities as given in Table 1. The phrase ‘conv $+ 2 \mathrm { C L s } ^ { \prime }$ ’ denotes $2 \mathrm { C L }$ junctions corresponding to a CL neuron configuration of (4096, 512, 16) for CIFAR10, (4096, 512, 128) for $\mathbf { C I F A R 1 0 0 } ^ { \tilde { 2 } }$ , and (3136, 784, 10) for MNIST (see Section 2.2.2). For ‘con $\mathbf { v } + 3 { \mathbf { C L s } } ^ { \prime }$ , an additional 256-neuron layer precedes the output. ‘MNIST CL’ and ‘Morse CL’ refer to the CL only networks described subsequently, for which we have only shown some of the more important configurations in Table 1. + +As an example, consider the first network in ‘CIFAR10 conv $+ 2 \mathrm { C L s }$ ’ which has $f o _ { 1 } = f o _ { 2 } = 1$ . This means that the individual junction densities are $( 4 0 9 6 \times 1 ) / ( 4 0 9 6 \times 5 1 2 ) = 0 . 2 \%$ and $( 5 1 2 \times$ 1 $) / ( 5 1 2 \times 1 6 ) = 6 . 3 \%$ , to give an overall CL density of $( 4 0 9 6 \times 1 + 5 1 2 \times 1 ) / ( 4 0 9 6 \times 5 1 2 + 5 1 2 \times$ $1 6 ) = 0 . 2 2 \%$ . In other words, while FCLs would have been $100 \%$ dense with 2, 097, $1 5 2 + 8 1 9 2 =$ 2, 105, 344 weights, the SCLs use $4 0 9 6 + 5 1 2 = 4 6 0 8$ weights, which is 457 times less. Note that weights in the sparse junction are distributed as fixed, but randomly generated patterns, with the constraints of fixed fan-in and fan-out. + +Table 1: Densities for some of our sparse networks + +
NetJunctionfan-outsCL JunctionDensities (%)Overall CLDensity (%)NetJunctionfan-outsCL JunctionDensities (%)Overall CLDensity (%)
CIFAR10conv+2CLs1,11,80.2,6.30.2,500.220.39CIFAR10conv+3CLs1,1,11,1,81,2,160.2,0.4,6.30.2,0.4,500.2,0.8,1000.220.30.41
CIFAR100conv+2CLs1,11.81,320.2,0.80.2,6.30.2,250.210.380.95CIFAR100conv+3CLs1,1,11,1,161,2,320.2,0.4,0.80.2,0.4,130.2,0.8,250.220.390.59
MNISTconv+2CLs1,54,1016,100.1,500.5,1002,1000.290.832.35MNIST CL(x=224)4,10112,101.79,10050,1003.0250.63
Morse CL512,3250,5050
+ +![](images/830349f25e0f21f487804d81609b236598d69551392521a091ca06621539bd3f.jpg) +Figure 1: Performance results of pre-defined sparsity for (a)–(c) CIFAR10, and (d)–(f) CIFAR100, trained for 30 epochs using different network densities and varying number of CLs. (a),(b),(d),(e) Validation accuracy across epochs. (c),(f) Best validation accuracies after 1, 5 and 30 epochs. + +Figure 1 shows the results for CIFAR. Subfigures (a), (b), (d) and (e) show classification performance on validation data as the network is trained for 30 epochs (note that the final accuracies stayed almost constant after 20 epochs). The different lines correspond to different overall CL densities. Subfigures (c) and (f) show the best validation accuracies after 1, 5 and 30 epochs for the different CL densities. We see that the final accuracies (the numbers at the top of each column) show negligible performance degradation for these extremely low levels of density, not to mention some cases where SCLs outperform FCLs. These results point to the promise of sparsity. Also notice from subfigures (c) and (f) that SCLs generally start training quicker than FCLs, as evidenced by their higher accuracies after 1 epoch of training. See Appendix Section 5.3 for more discussion. + +# 2.2.2 MNIST + +We used 2 different kinds of networks when experimenting on MNIST (no data augmentation). The first was ‘conv+2CLs’ – 2 conv layers having 32 and 64 filters of size 5x5 each, alternating with $2 \mathbf { x } 2$ max pooling layers. This results in a layer of 3136 neurons, which is followed by 2 CLs having 784 and 10 neurons, i.e. 2 junctions overall. Fig. 2(a) and (b) show the results. Due to the simplicity of the overall network, performance starts degrading at higher densities compared to CIFAR. However, a network with CL density $2 . 3 5 \%$ still matches FCLs in performance. Note that the total number of weights $( \mathrm { c o n v + S C L s } )$ ) is 0.11M for this network, which is only $4 . 3 7 \%$ of the original $( \mathrm { c o n v + F C L s } )$ . + +The second was a family of networks with only CLs, either having a single junction with a neuron configuration of (1024, 16), or 2 junctions configured as $( 7 8 4 , x , 1 0 )$ , where $x$ varies. The results are shown in Fig. 2(c), which offers two insights. Firstly, performance drops off at higher densities for CL only MNIST networks as compared to the one with conv layers. However, half the parameters can still be dropped without appreciable performance degradation. This aspect is further discussed in Section 2.3. Secondly, large SCLs perform better than small FCLs with similar number of parameters. Considering the black-circled points as an example, performance drops when switching from 224 hidden neurons at $12 . 5 \%$ density to 112 at $2 5 \%$ to 56 at $50 \%$ to 28 at $100 \%$ , even though all these networks have similar number of parameters. So increasing the number of hidden neurons is desirable, albeit with diminishing returns. + +![](images/3fc47770f88838dbbe0788e071df56ec0e82e712b051bc101380511a9ed923a2.jpg) +Figure 2: (a)–(b) Performance results of pre-defined sparsity on an MNIST conv network with different densities, each trained for 30 epochs. (c) Performance vs. connection density for different MNIST CL only networks, each trained for 100 epochs. + +# 2.2.3 MORSE + +The Morse code dataset presents a harder challenge for sparsity. It only has 64-valued inputs (as compared to 784 for MNIST and 3072 for CIFAR), so each input neuron encodes a significant amount of information. The outputs are Morse codewords and there are 64 classes. Distinctions between inputs belonging to different classes is small. For example, the input pattern for the Morse codeword ‘. can be easily confused with the codeword ‘. . -’. As a result, performance degrades quickly as connections are removed. Our network had 64 input and output neurons and 1024 hidden layer neurons, i.e. 3 CLs and 2 junctions, trained using stochastic gradient descent. The results are shown in Fig. 3(a). As with MNIST CL only, $50 \%$ density can be achieved with negligible degradation in accuracy. + +# 2.3 ANALYZING THE RESULTS OF PRE-DEFINED SPARSITY + +Our results indicate that for deep networks having several conv layers, there is severe redundancy in the CLs. As a result, they can be made extremely sparse without hampering network performance, which leads to significant memory savings. If the network only has CLs, the amount of density reduction achievable without performance degradation is smaller. This can be explained using the argument of relative importance. For a network which extensively extracts features and processes its raw input data via conv filters, the input to the CLs can already substantially discriminate between inputs belonging to different classes. As a result, the importance of the CLs’ functioning is less as compared to a network where they process the raw inputs. + +![](images/4d5f08f20aa7a2dc26e7f9dac1269bb84fcadca8338b6534c3bf088c476bd486.jpg) +Figure 3: (a) Performance vs. connection density for a Morse CL only 2 junction network. (b) and (c) Performance results by varying individual junction densities while overall density is fixed at (b) $2 5 \%$ (c) $50 \%$ . All cases trained for 30 epochs. + +The computational savings by sparsifying CLs, however, are not as large because the conv layers dominate the computational complexity. Other types of NNs, such as restricted Boltzmann machines, have higher prominence of CLs than CNNs and would thus benefit more from our approach. Table 2 shows the overall memory and computational gains obtained from pre-defining CLs to be sparse for our networks. The number of SCL parameters (params) are calculated by taking the minimum overall CL density at which there is no accuracy loss. Note that the number of operations (ops) for CLs is nearly the same as their number of parameters, hence are not explicitly shown. + +Table 2: Savings in some of our NN architectures due to pre-defined sparsity + +
NetCLs/TotalLayersConvParams(M)ConvOps(B)FC CLParams(M)SparseCL Par-ams (M)OverallParam %ReductionOverallOp%Reduction
Morse CL2/2000.1310.0665050
MNIST CL (x = 224)2/2000.1780.0895050
MNIST conv+2CLs2/60.050.12.470.0695.6318.29
CIFAR10 conv+2CLs2/171.150.152.110.00564.631.35
CIFAR100 conv+2CLs2/171.150.152.160.0264.761.38
CIFAR10 conv+3CLs3/181.150.152.230.00965.831.43
CIFAR100 conv+3CLs3/181.150.152.260.01365.991.45
+ +# 2.4 DISTRIBUTING INDIVIDUAL JUNCTION DENSITIES + +Note that the Morse code network has symmetric junctions since each will have $6 4 \times 1 0 2 4 = 6 5 ,$ , 536 weights to give a total of 131,072 FCL weights. Consider an example where overall density of $50 \%$ (i.e. 65,536 total SCL weights) is desired. This can be achieved in multiple ways, such as making both junctions $50 \%$ dense, i.e. 32,768 weights in each. Here we explore if individual junction densities contribute equally to network performance. + +Figures 3(b) and (c) sweep junction 1 and 2 connectivity densities on the $\mathbf { X }$ -axis such that the resulting overall density is fixed at $2 5 \%$ for (b) and $50 \%$ for (c). The black vertical line denotes where the densities are equal. Note that peak performance in both cases is achieved to the left of the black line, such as in (c) where junction 2 is $7 5 \%$ dense and junction 1 is $2 5 \%$ dense. This suggests that later junctions need more connections than earlier ones. See Appendix Section 5.1 for more details. + +# 3 CONNECTIVITY PATTERNS + +We now introduce adjacency matrices to describe junction connection patterns. Let $A _ { i } \in$ $\{ 0 , 1 \} ^ { N _ { i + 1 } \times N _ { i } }$ be the (simplified) adjacency matrix of junction $i$ , such that element $[ A _ { i } ] _ { j , k }$ indicates whether there is a connection between the $j$ th right neuron and $k$ th left neuron. $A _ { i }$ will have $f i _ { i }$ 1’s on each row and $f o _ { i }$ 1’s on each column. These adjacency matrices can be multiplied to yield the effective connectwhere element n between any 2 junctions denotes the number of pa $X$ and fro $Y$ , i.he . t $\begin{array} { r } { A _ { X : Y } = \prod _ { i = Y } ^ { X } A _ { i } \in \mathbb { Z } _ { \geq 0 } ^ { N _ { Y + 1 } \times N _ { X } } } \end{array}$ $[ A _ { X : Y } ] _ { j , k }$ $k$ $X$ $j$ ron in layer $( Y + 1 )$ . For the special case where $X = 1$ and $Y = J$ (total number of junctions), we obtain the input-output adjacency matrix $A _ { 1 : J }$ . As a simple example, consider the $( 8 , 4 , 4 )$ network shown in Fig. 4 where $f o _ { 1 } = 1$ and $f o _ { 2 } = 2$ , which implies that $f i _ { 1 } = f i _ { 2 } = 2$ . $A _ { 1 }$ and $A _ { 2 }$ are adjacency matrices of single junctions. We obtain the input-output adjacency matrix $A _ { 1 : 2 } = A _ { 2 } A _ { 1 }$ , equivalent $f o _ { 1 : 2 } = f o _ { 1 } f o _ { 2 } = 2$ , and equivalent $f i _ { 1 : 2 } = f i _ { 1 } f i _ { 2 } = 4$ . Note that this equivalent junction 1:2 is only an abstract concept that aids visualizing how neurons connect from the inputs to the outputs. It has no relation to the overall network density. + +![](images/e1c5bcca6ab72e26ab730e0d19ffa5addbdcc35314daf1f4aa3dd1234dac838e.jpg) +Figure 4: An example of adjacency matrices and equivalent junctions. + +We now attempt to characterize the quality of a sparse connection pattern, i.e. we try to find the best possible way to connect neurons to optimize performance. Since sparsity gives good performance, we hypothesize that there exists redundancy / correlated information between neurons. Intuitively, we assume that left neurons of a junction can be grouped into windows depending on the dimensionality of the left layer output. For example, the input layer in an MNIST CL only network would have 2D windows, each of which might correspond to a fraction of the image, as shown in Fig. 5(a). When outputs from a CL have an additional dimension for features, such as in CIFAR or the MNIST conv network, each window is a cuboid capturing fractions of both spatial extent and features, as shown in Fig. 5(b). Given such windows, we will try to maximize the number of left windows to which each right neuron connects, the idea being that each right neuron should get some information from all portions of the left layer in order to capture global view. To realize the importance of this, consider the MNIST output neuron representing digit 2. Let’s say the sparse connection pattern is such that when the connections to output 3 are traced back to the input layer, they all come from the top half of the image. This would be undesirable since the top half of an image of a 2 can be mistaken for a 3. A good sparse connection pattern will try to avoid such scenarios by spreading the connections to any right neuron across as many input windows as possible. The problem can also be mirrored so that every left neuron connects to as many different right windows as possible. This ensures that local information from left neurons is spread to different parts of the right layer. The grouping of right windows will depend on the dimensionality of the input to the right layer. + +The window size is chosen to be the minimum possible such that the ideal number of connections +from or to it remains integral. The example from Fig. 4 is reproduced in Fig. 6. Since $f i _ { 1 } = 2$ , +the inputs must be grouped into 2 windows so that ideally 1 connection from each reaches every +hidden neuron. If instead the inputs are grouped into 4 windows, the ideal number would be half of a connection, which is not achievable. In order to achieve the minimum window size, we let the number of left windows be $f i$ and the number of right windows be $f o$ . So in junction $i$ , the number +of neurons in each left and right window ileft- and right-window adjacency matrices $N _ { i } / f i _ { i }$ $N _ { i + 1 } / f o _ { i }$ en we constructby summing up $A _ { i } ^ { w _ { i l } } \in \mathbb { Z } _ { \geq 0 } ^ { N _ { i + 1 } \times f i _ { i } }$ $A _ { i } ^ { w _ { i r } } \in \mathbb { Z } _ { \geq 0 } ^ { f o _ { i } \times N _ { i } }$ as shown in Fig. 5(c). The window adjacency matrices describe connectivity between windows and neurons on the opposite side. Ideally, every window adjacency matrix for a single +junction should be the all 1s matrix, which signifies exactly 1 connection from every window to + +![](images/9002b38b5c3038201bc56b399cec1cdd03d989734f6eafa0d2319b0167a19f89.jpg) +Figure 5: (a) Example of 16 2D windows for an MNIST input image. (b) Example of 3D windows when the output from a layer also includes features. (c) Construction of window adjacency matrices. + +![](images/d7aef1f920d5edb6700ec8a8ba2e01c8c1718b06d3a42cd758343d2807880660.jpg) +Figure 6: Window adjacency matrices and scatter. Green neurons indicate ideal connectivity. The hidden layer is split into 2 to show separate constructions of $A _ { 1 } ^ { w _ { 1 r } }$ and $A _ { 2 } ^ { w _ { 2 l } }$ + +every neuron on the opposite side. Note that these matrices can also be constructed for multiple junctions, i.e. $A _ { X : Y } ^ { w _ { X l } }$ and $A _ { X : Y } ^ { w _ { Y } }$ , by multiplying matrices for individual junctions. See Appendix Section 5.2 for more discussion. + +# 3.1 SCATTER + +Scatter is a proxy for the performance of a NN. It is useful because it can be computed in a fraction of a second and used to predict how good or bad a sparse network is without spending time training it. To compute scatter, we count the number of entries greater than or equal to 1 in the window adjacency matrix. If a particular window gets more than its fair share of connections to a neuron on the opposite side, then it is depriving some other window from getting its fair share. This should not be encouraged, so we treat entries greater than 1 the same as 1. Scatter is the average of the count, i.e. for junction $i$ : + +$$ +S _ { i f } = \frac { 1 } { f i _ { i } N _ { i + 1 } } \sum _ { j = 1 } ^ { N _ { i + 1 } } \sum _ { k = 1 } ^ { f i _ { i } } \mathbb { I } \Big ( [ A _ { i } ^ { w _ { i } } ] _ { j , k } \geq 1 \Big ) , \qquad S _ { i b } = \frac { 1 } { N _ { i } f o _ { i } } \sum _ { j = 1 } ^ { f o _ { i } } \sum _ { k = 1 } ^ { N _ { i } } \mathbb { I } \Big ( [ A _ { i } ^ { w _ { i } } ] _ { j , k } \geq 1 \Big ) . +$$ + +Subscripts $f$ and $b$ denote forward (left windows to right neurons) and backward (right neurons to left windows), indicating the direction of data flow. As an example, we consider $A _ { 1 } ^ { w _ { 1 l } }$ in Fig. 6, which has a scatter value $S _ { 1 f } = 6 / 8 = 0 . 7 5$ . The other scatter values can be computed similarly to form the scatter vector $\bar { S } = [ S _ { 1 f } , S _ { 1 b } , S _ { 2 f } , S _ { 2 b } , S _ { f } , S _ { b } ]$ , where the final 2 values correspond to junction 1:2. Notice that $\bar { S }$ will be all 1s for FCLs, which is the ideal case. Incorporating sparsity leads to reduced $\bar { S }$ values. The final scatter metric $S \in [ 0 , 1 ]$ is the minimum value in $\bar { S }$ , i.e. 0.75 for Fig. 6. Our experiments indicate that any low value in $\bar { S }$ leads to bad performance, so we picked the critical minimum value. + +# 3.2 ANALYSIS AND RESULTS OF SCATTER + +We ran experiments to evaluate scatter using a) the Morse CL only network with $f o \ = \ 1 2 8 , 8$ , b) an MNIST CL only network with (1024, 64, 16) neuron configuration and $f o = 1 , 4$ , and c) the ‘conv $+ 2 \mathrm { C L s } ^ { \prime }$ CIFAR10 network with $f o = 1 , 2$ . We found that high scatter indicates good performance and the correlation is stronger for networks where CLs have more importance, i.e. CL only networks as opposed to conv. This is shown in the performance vs. scatter plots in Fig. 7, where (a) and (b) show the performance predicting ability of scatter better than (c). Note that the random connection patterns used so far have the highest scatter and occur as the rightmost points in each subfigure. The other points are obtained by specifically planning connections. We found that when 1 junction was planned to give corresponding high values in $\bar { S }$ , it invariably led to low values for another junction, leading to a low $S$ . This explains why random patterns generally perform well. + +![](images/2a1cf611997723e1bdc19ec13f7a4be0119a684042dbb3dd50eeffe37ea4e89c.jpg) +Figure 7: Network performance vs. scatter for CL only networks of (a) Morse (b) MNIST, and convolutional network with $2 \mathrm { C L }$ junctions of (c) CIFAR10. All minimum values that need to be considered to differentiate between connection patterns are bolded. + +$\bar { S }$ is shown alongside each point. When $S$ is equal for different connection patterns, the next minimum value in $\breve { \bar { S } }$ needs to be considered to differentiate the networks, and so on. Considering the Morse results, the leftmost 3 points all have $\begin{array} { r } { S = \frac { 1 } { 8 } } \end{array}$ , but the number of occurrences of $\frac { 1 } { 8 }$ in $\bar { S }$ is 3 for the lowest point $8 \%$ accuracy), 2 for the second lowest ( $12 \%$ accuracy) and 1 for the highest point $46 \%$ accuracy). For the MNIST results, both the leftmost points have a single minimum value of $\textstyle { \frac { 1 } { 1 6 } }$ in $\bar { S }$ , but the lower has two occurrences of $\textstyle { \frac { 1 } { 4 } }$ while the upper has one. + +We draw several insights from these results. Firstly, although we defined $S$ as a single value for convenience, there may arise cases when other (non-minimum) elements in $\bar { S }$ are important. Secondly, perhaps contrary to intuition, the concept of windows and scatter is important for all CLs, not simply the first. As shown in Fig. 7a), a network with $\begin{array} { r } { S _ { 1 b } = \frac { 1 } { 8 } } \end{array}$ performs equally poorly as a network with $\begin{array} { r } { S _ { 2 f } = ~ \frac { 1 } { 8 } } \end{array}$ . Thirdly, scatter is a sufficient metric for performance, not necessary. A network with a high $S$ value will perform well, but a network with a slightly lower $S$ than another cannot be conclusively dismissed as being worse. But if a network has multiple low values in $\bar { S }$ , it should be rejected. Finally, carefully choosing which neurons to group in a window will increase the predictive power of scatter. A priori knowledge of the dataset will lead to better window choices. + +# 4 CONCLUSION AND FUTURE WORK + +This paper discusses the merits of pre-defining sparsity in CLs of neural networks, which leads to significant reduction in parameters without performance loss. 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ACM, 2016. +Xichuan Zhou, Shengli Li, Kai Qin, Kunping Li, Fang Tang, Shengdong Hu, Shujun Liu, and Zhi Lin. Deep adaptive network: An efficient deep neural network with sparse binary connections. In arXiv:1604.06154, 2016. + +# 5 APPENDIX + +# 5.1 MORE ON DISTRIBUTING INDIVIDUAL JUNCTION DENSITIES + +Section 2.4 showed that when overall CL density is fixed, it is desirable to make junction 2 denser than junction 1. It is also interesting to note, however, that performance falls off more sharply when junction 1 density is reduced to the bare minimum as compared to treating junction 2 similarly. This is not shown in Fig. 3 due to space constraints. We found that when junction 1 had the minimum possible density and junction 2 had the maximum possible while still satisfying the fixed overall, the accuracy was about $36 \%$ for both subfigures (b) and (c). When the densities were flipped, the accuracies were $67 \%$ for subfigure (b) and $7 5 \%$ for (c) in Figure 3. + +# 5.2 DENSE CASES OF WINDOW ADJACENCY MATRICES + +As stated in Section 3.1, window output matrices for several junctions can be constructed by multiplying the individual matrices for each component junction. Consider the Morse network as described in Section 3.2. Note that $f o _ { 1 : 2 } = 1 2 8 \times 8 = 1 0 2 4$ and $f i _ { 1 : 2 } = 8 \times 1 2 8 = 1 0 2 4$ . Thus, for the equivalent junction 1:2 which has $N _ { 1 } = 6 4$ left neurons and $N _ { 3 } = 6 4$ right neurons, we have $f o _ { 1 : 2 } > N _ { 3 }$ and $f i _ { 1 : 2 } > N _ { 1 }$ . So in this case the number of neurons in each window will be rounded up to 1, and both the ideal window adjacency matrices $A _ { 1 : 2 } ^ { w _ { 1 l } }$ and $A _ { 1 : 2 } ^ { w _ { 2 r } }$ will be all 16’s matrices since the ideal number of connections from each window to a neuron on the opposite side is $1 0 2 4 / 6 4 = 1 6$ . This is a result of the network having sufficient density so that several paths exist from every input neuron to every output neuron. + +# 5.3 POSSIBLE REASONS FOR SCLS CONVERGING FASTER THAN FCLS + +Training a neural network is essentially an exercise in finding the minimum of the cost function, which is a function of all the network parameters. The graph for cost as a function of parameters may have saddle points which masquerade as minima. It could also be poorly conditioned, wherein the gradient of cost with respect to two different parameters have widely different magnitudes, making simultaneous optimization difficult. These effects are non-idealities and training the network often takes more time because of the length of the trajectory needed to overcome these and arrive at the optimum point. The probability of encountering these non-idealities increases as the number of network parameters increase, i.e. less parameters leads to a higher ratio of minima : saddle points, which can make the network converge faster. We hypothesize that SCLs train faster than FCLs due to the former having fewer parameters. \ No newline at end of file diff --git a/parse/train/BJgPCveAW/BJgPCveAW_content_list.json b/parse/train/BJgPCveAW/BJgPCveAW_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..31f3e68f5356295e463c25567365022c6f26923f --- /dev/null +++ b/parse/train/BJgPCveAW/BJgPCveAW_content_list.json @@ -0,0 +1,798 @@ +[ + { + "type": "text", + "text": "CHARACTERIZING SPARSE CONNECTIVITY PATTERNS IN NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 236, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We propose a novel way of reducing the number of parameters in the storagehungry fully connected layers of a neural network by using pre-defined sparsity, where the majority of connections are absent prior to starting training. Our results indicate that convolutional neural networks can operate without any loss of accuracy at less than $0 . 5 \\%$ classification layer connection density, or less than $5 \\%$ overall network connection density. We also investigate the effects of pre-defining the sparsity of networks with only fully connected layers. Based on our sparsifying technique, we introduce the ‘scatter’ metric to characterize the quality of a particular connection pattern. As proof of concept, we show results on CIFAR, MNIST and a new dataset on classifying Morse code symbols, which highlights some interesting trends and limits of sparse connection patterns. ", + "bbox": [ + 233, + 272, + 764, + 426 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 469, + 334, + 484 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Neural networks (NNs) in machine learning systems are critical drivers of new technologies such as image processing and speech recognition. Modern NNs are gigantic in size with millions of parameters, such as the ones described in Alexnet (Krizhevsky et al., 2012), Overfeat (Sermanet et al., 2013) and ResNet (He et al., 2016). They therefore require an enormous amount of memory and silicon processing during usage. Optimizing a network to improve performance typically involves making it deeper and adding more parameters (Simonyan & Zisserman, 2015; Szegedy et al., 2015; Huang et al., 2016), which further exacerbates the problem of large storage complexity. While the convolutional (conv) layers in these networks do feature extraction, there are usually fully connected layers at the end performing classification. We shall henceforth refer to these layers as connected layers $( C L s )$ , of which fully connected layers $( F C L s )$ are a special case. Owing to their high density of connections, the majority of network parameters are concentrated in FCLs. For example, the FCLs in Alexnet account for $9 5 . 7 \\%$ of the network parameters (Zhang et al., 2016). ", + "bbox": [ + 174, + 507, + 825, + 672 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We shall refer to the spaces between CLs as CL junctions (or simply junctions), which are occupied by connections, or weights. Given the trend in modern NNs, we raise the question – “How necessary is it to have FCLs?” or, in other words, “What if most of the junction connections never existed? Would the resulting sparsely connected layers (SCLs), when trained and tested, still give competitive performance?” As an example, consider a network with 2 CLs of 100 neurons each and the junction between them has 1000 weights instead of the expected 10,000. Then this is a sparse network with connection density of $10 \\%$ . Given such a sparse architecture, a natural question to ask is “How can the existing 1000 weights be best distributed so that network performance is maximized?” ", + "bbox": [ + 174, + 680, + 825, + 791 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this regard, the present work makes the following contributions. In Section 2, we formalize the concept of sparsity, or its opposite measure density, and explore its effects on different network types. We show that CL parameters are largely redundant and a network pre-defined to be sparse before starting training does not result in any performance degradation. For certain network architectures, this leads to CL parameter reduction by a factor of more than 450, or an overall parameter reduction by a factor of more than 20. In Section 2.4, we discuss techniques to distribute connections across junctions when given an overall network density. Finally, in Section 3, we formalize pre-defined sparse connectivity patterns using adjacency matrices and introduce the scatter metric. Our results show that scatter is a quick and useful indicator of how good a sparse network is. ", + "bbox": [ + 174, + 797, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "2 PRE-DEFINED SPARSITY ", + "text_level": 1, + "bbox": [ + 176, + 102, + 406, + 118 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "As an example of the footprint of modern NNs, AlexNet has a weight size of $2 3 4 \\mathrm { M B }$ and requires 635 million arithmetic operations only for feedforward processing (Zhang et al., 2016). It has been shown that NNs, particularly their FCLs, have an excess of parameters and tend to overfit to the training data (Denil et al., 2013), resulting in inferior performance on test data. The following paragraph describes several previous works that have attempted to reduce parameters in NNs. ", + "bbox": [ + 174, + 133, + 825, + 203 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Dropout (deletion) of random neurons (Srivastava et al., 2014) trains multiple differently configured networks, which are finally combined to regain the original full size network. Chen et al. (2015) randomly forced the same value on collections of weights, but acknowledged that “a significant number of nodes [get] disconnected from neighboring layers.” Other sparsifying techniques such as pruning and quantization (Han et al., 2016; 2015; Zhou et al., 2016; Gong et al., 2014) first train the complete network, and then perform further computations to delete parameters. Sindhwani et al. (2015) used low rank matrices to impose structure on network parameters. Srinivas et al. (2016) proposed a regularizer to reduce parameters in the network, but acknowledged that this increased training complexity. In general, all these architectures deal with FCLs at some point of time during their usage cycle and therefore, do not permanently solve the parameter explosion problem of NNs. ", + "bbox": [ + 173, + 210, + 825, + 349 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 OUR METHODOLOGY ", + "text_level": 1, + "bbox": [ + 176, + 367, + 362, + 381 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our attempt to simplify NNs is to pre-define the level of sparsity, or connection density, in a network prior to the start of training. This means that our network always has fewer connections than its FCL counterpart; the weights which are absent never make an appearance during training or inference. In our notation, a NN will have $J$ junctions, i.e. $J + 1$ layers, with $\\{ N _ { 1 } , \\stackrel { \\textstyle - } { N } _ { 2 } , \\cdot \\cdot \\cdot \\stackrel { \\textstyle - } { , } N _ { J + 1 } \\}$ being the number of neurons in each layer. $N _ { i }$ and $N _ { i + 1 }$ are respectively the number of neurons in the earlier (left) and later (right) layers of junction $i$ . Every left neuron has a fixed number of edges going from it to the right, and every right neuron has a fixed number of edges coming into it from the left. These numbers are defined as fan-out $( f o _ { i } )$ and fan-in $( f i _ { i } )$ , respectively. For conventional FCLs, $f o _ { i } = N _ { i + 1 }$ and $f i _ { i } = N _ { i }$ . We propose SCLs where $f o _ { i } < N _ { i + 1 }$ and $f i _ { i } < N _ { i }$ , such that $N _ { i } \\times f o _ { i } = N _ { i + 1 } \\times f i _ { i } = W _ { i }$ , the number of weights in junction $i$ . Having a fixed $f o _ { i }$ and $f i _ { i }$ ensures that all neurons in a junction contribute equally and none of them get disconnected, since that would lead to a loss of information. The connection density in junction $i$ is given as $W _ { i } / ( N _ { i } N _ { i + 1 } )$ and the overall CL connection density is defined as $\\textstyle \\left( \\sum _ { i = 1 } ^ { J } W _ { i } \\right) / \\left( \\sum _ { i = 1 } ^ { J } N _ { i } N _ { i + 1 } \\right)$ . ", + "bbox": [ + 173, + 392, + 825, + 583 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Note that earlier works such as Dey et al. (2017b;a) have proposed hardware architectures that leverage pre-defined sparsity to speed up training. However, a complete analysis of methods to pre-define connections, its possible gains on different kinds of modern deep NNs and a test of its limits via a metric quantifying its goodness has been lacking. Bourely et al. (2017) introduced a metric based on eigenvalues, but ran limited tests on MNIST. The following subsections analyze our method of pre-defined sparsity in more detail. We experimented with networks operating on CIFAR, MNIST and Morse code symbol classification – a new dataset described in Dey $( 2 0 1 7 ) ^ { \\bar { 1 } }$ . ", + "bbox": [ + 174, + 588, + 825, + 686 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 NETWORK EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 704, + 392, + 718 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2.1 CIFAR ", + "text_level": 1, + "bbox": [ + 176, + 729, + 279, + 744 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We used the original CIFAR10 and CIFAR100 datasets without data augmentation. Our network has 6 conv layers with number of filters equal to [64, 64, 128, 128, 256, 256]. Each has window size $3 { \\tt X } 3$ . The outputs are batch-normalized before applying ReLU non-linearity. A max-pooling layer of pool size 2x2 succeeds every pair of conv layers. This structure finally results in a layer of 4096 neurons, which is followed by the CLs. We used the Adam optimizer, ReLU-activated hidden layers and softmax output layer – choices which we maintained for all networks unless otherwise specified. ", + "bbox": [ + 174, + 753, + 825, + 838 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our results in Section 2.4 indicate that later CL junctions (i.e. closer to the outputs) should be denser than earlier ones (i.e. closer to the inputs). Moreover, since most CL networks have a tapering structure where $N _ { i }$ monotonically decreases as $i$ increases, more parameter savings can be achieved by making earlier layers less dense. Accordingly we did a grid search and picked CL junction densities as given in Table 1. The phrase ‘conv $+ 2 \\mathrm { C L s } ^ { \\prime }$ ’ denotes $2 \\mathrm { C L }$ junctions corresponding to a CL neuron configuration of (4096, 512, 16) for CIFAR10, (4096, 512, 128) for $\\mathbf { C I F A R 1 0 0 } ^ { \\tilde { 2 } }$ , and (3136, 784, 10) for MNIST (see Section 2.2.2). For ‘con $\\mathbf { v } + 3 { \\mathbf { C L s } } ^ { \\prime }$ , an additional 256-neuron layer precedes the output. ‘MNIST CL’ and ‘Morse CL’ refer to the CL only networks described subsequently, for which we have only shown some of the more important configurations in Table 1. ", + "bbox": [ + 174, + 844, + 825, + 887 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As an example, consider the first network in ‘CIFAR10 conv $+ 2 \\mathrm { C L s }$ ’ which has $f o _ { 1 } = f o _ { 2 } = 1$ . This means that the individual junction densities are $( 4 0 9 6 \\times 1 ) / ( 4 0 9 6 \\times 5 1 2 ) = 0 . 2 \\%$ and $( 5 1 2 \\times$ 1 $) / ( 5 1 2 \\times 1 6 ) = 6 . 3 \\%$ , to give an overall CL density of $( 4 0 9 6 \\times 1 + 5 1 2 \\times 1 ) / ( 4 0 9 6 \\times 5 1 2 + 5 1 2 \\times$ $1 6 ) = 0 . 2 2 \\%$ . In other words, while FCLs would have been $100 \\%$ dense with 2, 097, $1 5 2 + 8 1 9 2 =$ 2, 105, 344 weights, the SCLs use $4 0 9 6 + 5 1 2 = 4 6 0 8$ weights, which is 457 times less. Note that weights in the sparse junction are distributed as fixed, but randomly generated patterns, with the constraints of fixed fan-in and fan-out. ", + "bbox": [ + 173, + 194, + 825, + 291 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/a24591907f755b33e09ba05fc26d15c7053b4c28e3a9570086c48d3adc567c53.jpg", + "table_caption": [ + "Table 1: Densities for some of our sparse networks " + ], + "table_footnote": [], + "table_body": "
NetJunctionfan-outsCL JunctionDensities (%)Overall CLDensity (%)NetJunctionfan-outsCL JunctionDensities (%)Overall CLDensity (%)
CIFAR10conv+2CLs1,11,80.2,6.30.2,500.220.39CIFAR10conv+3CLs1,1,11,1,81,2,160.2,0.4,6.30.2,0.4,500.2,0.8,1000.220.30.41
CIFAR100conv+2CLs1,11.81,320.2,0.80.2,6.30.2,250.210.380.95CIFAR100conv+3CLs1,1,11,1,161,2,320.2,0.4,0.80.2,0.4,130.2,0.8,250.220.390.59
MNISTconv+2CLs1,54,1016,100.1,500.5,1002,1000.290.832.35MNIST CL(x=224)4,10112,101.79,10050,1003.0250.63
Morse CL512,3250,5050
", + "bbox": [ + 176, + 328, + 820, + 463 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/830349f25e0f21f487804d81609b236598d69551392521a091ca06621539bd3f.jpg", + "image_caption": [ + "Figure 1: Performance results of pre-defined sparsity for (a)–(c) CIFAR10, and (d)–(f) CIFAR100, trained for 30 epochs using different network densities and varying number of CLs. (a),(b),(d),(e) Validation accuracy across epochs. (c),(f) Best validation accuracies after 1, 5 and 30 epochs. " + ], + "image_footnote": [], + "bbox": [ + 222, + 488, + 777, + 709 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Figure 1 shows the results for CIFAR. Subfigures (a), (b), (d) and (e) show classification performance on validation data as the network is trained for 30 epochs (note that the final accuracies stayed almost constant after 20 epochs). The different lines correspond to different overall CL densities. Subfigures (c) and (f) show the best validation accuracies after 1, 5 and 30 epochs for the different CL densities. We see that the final accuracies (the numbers at the top of each column) show negligible performance degradation for these extremely low levels of density, not to mention some cases where SCLs outperform FCLs. These results point to the promise of sparsity. Also notice from subfigures (c) and (f) that SCLs generally start training quicker than FCLs, as evidenced by their higher accuracies after 1 epoch of training. See Appendix Section 5.3 for more discussion. ", + "bbox": [ + 173, + 789, + 825, + 887 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2.2 MNIST ", + "text_level": 1, + "bbox": [ + 174, + 148, + 282, + 162 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We used 2 different kinds of networks when experimenting on MNIST (no data augmentation). The first was ‘conv+2CLs’ – 2 conv layers having 32 and 64 filters of size 5x5 each, alternating with $2 \\mathbf { x } 2$ max pooling layers. This results in a layer of 3136 neurons, which is followed by 2 CLs having 784 and 10 neurons, i.e. 2 junctions overall. Fig. 2(a) and (b) show the results. Due to the simplicity of the overall network, performance starts degrading at higher densities compared to CIFAR. However, a network with CL density $2 . 3 5 \\%$ still matches FCLs in performance. Note that the total number of weights $( \\mathrm { c o n v + S C L s } )$ ) is 0.11M for this network, which is only $4 . 3 7 \\%$ of the original $( \\mathrm { c o n v + F C L s } )$ . ", + "bbox": [ + 174, + 172, + 825, + 271 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The second was a family of networks with only CLs, either having a single junction with a neuron configuration of (1024, 16), or 2 junctions configured as $( 7 8 4 , x , 1 0 )$ , where $x$ varies. The results are shown in Fig. 2(c), which offers two insights. Firstly, performance drops off at higher densities for CL only MNIST networks as compared to the one with conv layers. However, half the parameters can still be dropped without appreciable performance degradation. This aspect is further discussed in Section 2.3. Secondly, large SCLs perform better than small FCLs with similar number of parameters. Considering the black-circled points as an example, performance drops when switching from 224 hidden neurons at $12 . 5 \\%$ density to 112 at $2 5 \\%$ to 56 at $50 \\%$ to 28 at $100 \\%$ , even though all these networks have similar number of parameters. So increasing the number of hidden neurons is desirable, albeit with diminishing returns. ", + "bbox": [ + 173, + 277, + 825, + 416 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/3fc47770f88838dbbe0788e071df56ec0e82e712b051bc101380511a9ed923a2.jpg", + "image_caption": [ + "Figure 2: (a)–(b) Performance results of pre-defined sparsity on an MNIST conv network with different densities, each trained for 30 epochs. (c) Performance vs. connection density for different MNIST CL only networks, each trained for 100 epochs. " + ], + "image_footnote": [], + "bbox": [ + 222, + 433, + 776, + 558 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2.3 MORSE ", + "text_level": 1, + "bbox": [ + 174, + 645, + 281, + 660 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The Morse code dataset presents a harder challenge for sparsity. It only has 64-valued inputs (as compared to 784 for MNIST and 3072 for CIFAR), so each input neuron encodes a significant amount of information. The outputs are Morse codewords and there are 64 classes. Distinctions between inputs belonging to different classes is small. For example, the input pattern for the Morse codeword ‘. can be easily confused with the codeword ‘. . -’. As a result, performance degrades quickly as connections are removed. Our network had 64 input and output neurons and 1024 hidden layer neurons, i.e. 3 CLs and 2 junctions, trained using stochastic gradient descent. The results are shown in Fig. 3(a). As with MNIST CL only, $50 \\%$ density can be achieved with negligible degradation in accuracy. ", + "bbox": [ + 173, + 670, + 825, + 796 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 ANALYZING THE RESULTS OF PRE-DEFINED SPARSITY ", + "text_level": 1, + "bbox": [ + 173, + 813, + 591, + 828 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our results indicate that for deep networks having several conv layers, there is severe redundancy in the CLs. As a result, they can be made extremely sparse without hampering network performance, which leads to significant memory savings. If the network only has CLs, the amount of density reduction achievable without performance degradation is smaller. This can be explained using the argument of relative importance. For a network which extensively extracts features and processes its raw input data via conv filters, the input to the CLs can already substantially discriminate between inputs belonging to different classes. As a result, the importance of the CLs’ functioning is less as compared to a network where they process the raw inputs. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/4d5f08f20aa7a2dc26e7f9dac1269bb84fcadca8338b6534c3bf088c476bd486.jpg", + "image_caption": [ + "Figure 3: (a) Performance vs. connection density for a Morse CL only 2 junction network. (b) and (c) Performance results by varying individual junction densities while overall density is fixed at (b) $2 5 \\%$ (c) $50 \\%$ . All cases trained for 30 epochs. " + ], + "image_footnote": [], + "bbox": [ + 222, + 99, + 776, + 217 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 304, + 823, + 333 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The computational savings by sparsifying CLs, however, are not as large because the conv layers dominate the computational complexity. Other types of NNs, such as restricted Boltzmann machines, have higher prominence of CLs than CNNs and would thus benefit more from our approach. Table 2 shows the overall memory and computational gains obtained from pre-defining CLs to be sparse for our networks. The number of SCL parameters (params) are calculated by taking the minimum overall CL density at which there is no accuracy loss. Note that the number of operations (ops) for CLs is nearly the same as their number of parameters, hence are not explicitly shown. ", + "bbox": [ + 174, + 340, + 825, + 438 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/22185eb4d0930cf89e7b6f2f26f5146931e8355665d59dc39c4babd9df0d4cba.jpg", + "table_caption": [ + "Table 2: Savings in some of our NN architectures due to pre-defined sparsity " + ], + "table_footnote": [], + "table_body": "
NetCLs/TotalLayersConvParams(M)ConvOps(B)FC CLParams(M)SparseCL Par-ams (M)OverallParam %ReductionOverallOp%Reduction
Morse CL2/2000.1310.0665050
MNIST CL (x = 224)2/2000.1780.0895050
MNIST conv+2CLs2/60.050.12.470.0695.6318.29
CIFAR10 conv+2CLs2/171.150.152.110.00564.631.35
CIFAR100 conv+2CLs2/171.150.152.160.0264.761.38
CIFAR10 conv+3CLs3/181.150.152.230.00965.831.43
CIFAR100 conv+3CLs3/181.150.152.260.01365.991.45
", + "bbox": [ + 178, + 477, + 820, + 617 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.4 DISTRIBUTING INDIVIDUAL JUNCTION DENSITIES ", + "text_level": 1, + "bbox": [ + 173, + 642, + 563, + 656 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that the Morse code network has symmetric junctions since each will have $6 4 \\times 1 0 2 4 = 6 5 ,$ , 536 weights to give a total of 131,072 FCL weights. Consider an example where overall density of $50 \\%$ (i.e. 65,536 total SCL weights) is desired. This can be achieved in multiple ways, such as making both junctions $50 \\%$ dense, i.e. 32,768 weights in each. Here we explore if individual junction densities contribute equally to network performance. ", + "bbox": [ + 174, + 667, + 825, + 738 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Figures 3(b) and (c) sweep junction 1 and 2 connectivity densities on the $\\mathbf { X }$ -axis such that the resulting overall density is fixed at $2 5 \\%$ for (b) and $50 \\%$ for (c). The black vertical line denotes where the densities are equal. Note that peak performance in both cases is achieved to the left of the black line, such as in (c) where junction 2 is $7 5 \\%$ dense and junction 1 is $2 5 \\%$ dense. This suggests that later junctions need more connections than earlier ones. See Appendix Section 5.1 for more details. ", + "bbox": [ + 174, + 744, + 825, + 815 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 CONNECTIVITY PATTERNS ", + "text_level": 1, + "bbox": [ + 176, + 837, + 424, + 852 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now introduce adjacency matrices to describe junction connection patterns. Let $A _ { i } \\in$ $\\{ 0 , 1 \\} ^ { N _ { i + 1 } \\times N _ { i } }$ be the (simplified) adjacency matrix of junction $i$ , such that element $[ A _ { i } ] _ { j , k }$ indicates whether there is a connection between the $j$ th right neuron and $k$ th left neuron. $A _ { i }$ will have $f i _ { i }$ 1’s on each row and $f o _ { i }$ 1’s on each column. These adjacency matrices can be multiplied to yield the effective connectwhere element n between any 2 junctions denotes the number of pa $X$ and fro $Y$ , i.he . t $\\begin{array} { r } { A _ { X : Y } = \\prod _ { i = Y } ^ { X } A _ { i } \\in \\mathbb { Z } _ { \\geq 0 } ^ { N _ { Y + 1 } \\times N _ { X } } } \\end{array}$ $[ A _ { X : Y } ] _ { j , k }$ $k$ $X$ $j$ ron in layer $( Y + 1 )$ . For the special case where $X = 1$ and $Y = J$ (total number of junctions), we obtain the input-output adjacency matrix $A _ { 1 : J }$ . As a simple example, consider the $( 8 , 4 , 4 )$ network shown in Fig. 4 where $f o _ { 1 } = 1$ and $f o _ { 2 } = 2$ , which implies that $f i _ { 1 } = f i _ { 2 } = 2$ . $A _ { 1 }$ and $A _ { 2 }$ are adjacency matrices of single junctions. We obtain the input-output adjacency matrix $A _ { 1 : 2 } = A _ { 2 } A _ { 1 }$ , equivalent $f o _ { 1 : 2 } = f o _ { 1 } f o _ { 2 } = 2$ , and equivalent $f i _ { 1 : 2 } = f i _ { 1 } f i _ { 2 } = 4$ . Note that this equivalent junction 1:2 is only an abstract concept that aids visualizing how neurons connect from the inputs to the outputs. It has no relation to the overall network density. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 101, + 825, + 232 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/e1c5bcca6ab72e26ab730e0d19ffa5addbdcc35314daf1f4aa3dd1234dac838e.jpg", + "image_caption": [ + "Figure 4: An example of adjacency matrices and equivalent junctions. " + ], + "image_footnote": [], + "bbox": [ + 235, + 244, + 759, + 449 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We now attempt to characterize the quality of a sparse connection pattern, i.e. we try to find the best possible way to connect neurons to optimize performance. Since sparsity gives good performance, we hypothesize that there exists redundancy / correlated information between neurons. Intuitively, we assume that left neurons of a junction can be grouped into windows depending on the dimensionality of the left layer output. For example, the input layer in an MNIST CL only network would have 2D windows, each of which might correspond to a fraction of the image, as shown in Fig. 5(a). When outputs from a CL have an additional dimension for features, such as in CIFAR or the MNIST conv network, each window is a cuboid capturing fractions of both spatial extent and features, as shown in Fig. 5(b). Given such windows, we will try to maximize the number of left windows to which each right neuron connects, the idea being that each right neuron should get some information from all portions of the left layer in order to capture global view. To realize the importance of this, consider the MNIST output neuron representing digit 2. Let’s say the sparse connection pattern is such that when the connections to output 3 are traced back to the input layer, they all come from the top half of the image. This would be undesirable since the top half of an image of a 2 can be mistaken for a 3. A good sparse connection pattern will try to avoid such scenarios by spreading the connections to any right neuron across as many input windows as possible. The problem can also be mirrored so that every left neuron connects to as many different right windows as possible. This ensures that local information from left neurons is spread to different parts of the right layer. The grouping of right windows will depend on the dimensionality of the input to the right layer. ", + "bbox": [ + 174, + 496, + 825, + 760 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The window size is chosen to be the minimum possible such that the ideal number of connections \nfrom or to it remains integral. The example from Fig. 4 is reproduced in Fig. 6. Since $f i _ { 1 } = 2$ , \nthe inputs must be grouped into 2 windows so that ideally 1 connection from each reaches every \nhidden neuron. If instead the inputs are grouped into 4 windows, the ideal number would be half of a connection, which is not achievable. In order to achieve the minimum window size, we let the number of left windows be $f i$ and the number of right windows be $f o$ . So in junction $i$ , the number \nof neurons in each left and right window ileft- and right-window adjacency matrices $N _ { i } / f i _ { i }$ $N _ { i + 1 } / f o _ { i }$ en we constructby summing up $A _ { i } ^ { w _ { i l } } \\in \\mathbb { Z } _ { \\geq 0 } ^ { N _ { i + 1 } \\times f i _ { i } }$ $A _ { i } ^ { w _ { i r } } \\in \\mathbb { Z } _ { \\geq 0 } ^ { f o _ { i } \\times N _ { i } }$ as shown in Fig. 5(c). The window adjacency matrices describe connectivity between windows and neurons on the opposite side. Ideally, every window adjacency matrix for a single \njunction should be the all 1s matrix, which signifies exactly 1 connection from every window to ", + "bbox": [ + 173, + 766, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/9002b38b5c3038201bc56b399cec1cdd03d989734f6eafa0d2319b0167a19f89.jpg", + "image_caption": [ + "Figure 5: (a) Example of 16 2D windows for an MNIST input image. (b) Example of 3D windows when the output from a layer also includes features. (c) Construction of window adjacency matrices. " + ], + "image_footnote": [], + "bbox": [ + 228, + 103, + 774, + 213 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/d7aef1f920d5edb6700ec8a8ba2e01c8c1718b06d3a42cd758343d2807880660.jpg", + "image_caption": [ + "Figure 6: Window adjacency matrices and scatter. Green neurons indicate ideal connectivity. The hidden layer is split into 2 to show separate constructions of $A _ { 1 } ^ { w _ { 1 r } }$ and $A _ { 2 } ^ { w _ { 2 l } }$ " + ], + "image_footnote": [], + "bbox": [ + 240, + 280, + 758, + 460 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "every neuron on the opposite side. Note that these matrices can also be constructed for multiple junctions, i.e. $A _ { X : Y } ^ { w _ { X l } }$ and $A _ { X : Y } ^ { w _ { Y } }$ , by multiplying matrices for individual junctions. See Appendix Section 5.2 for more discussion. ", + "bbox": [ + 173, + 551, + 826, + 594 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.1 SCATTER ", + "text_level": 1, + "bbox": [ + 174, + 616, + 279, + 630 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Scatter is a proxy for the performance of a NN. It is useful because it can be computed in a fraction of a second and used to predict how good or bad a sparse network is without spending time training it. To compute scatter, we count the number of entries greater than or equal to 1 in the window adjacency matrix. If a particular window gets more than its fair share of connections to a neuron on the opposite side, then it is depriving some other window from getting its fair share. This should not be encouraged, so we treat entries greater than 1 the same as 1. Scatter is the average of the count, i.e. for junction $i$ : ", + "bbox": [ + 173, + 643, + 826, + 742 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/60adbc59f29c10858b3b168384689c07716b5ef3a140ecb57ea2e8dc493918db.jpg", + "text": "$$\nS _ { i f } = \\frac { 1 } { f i _ { i } N _ { i + 1 } } \\sum _ { j = 1 } ^ { N _ { i + 1 } } \\sum _ { k = 1 } ^ { f i _ { i } } \\mathbb { I } \\Big ( [ A _ { i } ^ { w _ { i } } ] _ { j , k } \\geq 1 \\Big ) , \\qquad S _ { i b } = \\frac { 1 } { N _ { i } f o _ { i } } \\sum _ { j = 1 } ^ { f o _ { i } } \\sum _ { k = 1 } ^ { N _ { i } } \\mathbb { I } \\Big ( [ A _ { i } ^ { w _ { i } } ] _ { j , k } \\geq 1 \\Big ) .\n$$", + "text_format": "latex", + "bbox": [ + 194, + 753, + 782, + 799 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Subscripts $f$ and $b$ denote forward (left windows to right neurons) and backward (right neurons to left windows), indicating the direction of data flow. As an example, we consider $A _ { 1 } ^ { w _ { 1 l } }$ in Fig. 6, which has a scatter value $S _ { 1 f } = 6 / 8 = 0 . 7 5$ . The other scatter values can be computed similarly to form the scatter vector $\\bar { S } = [ S _ { 1 f } , S _ { 1 b } , S _ { 2 f } , S _ { 2 b } , S _ { f } , S _ { b } ]$ , where the final 2 values correspond to junction 1:2. Notice that $\\bar { S }$ will be all 1s for FCLs, which is the ideal case. Incorporating sparsity leads to reduced $\\bar { S }$ values. The final scatter metric $S \\in [ 0 , 1 ]$ is the minimum value in $\\bar { S }$ , i.e. 0.75 for Fig. 6. Our experiments indicate that any low value in $\\bar { S }$ leads to bad performance, so we picked the critical minimum value. ", + "bbox": [ + 173, + 809, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.2 ANALYSIS AND RESULTS OF SCATTER ", + "text_level": 1, + "bbox": [ + 176, + 103, + 477, + 117 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We ran experiments to evaluate scatter using a) the Morse CL only network with $f o \\ = \\ 1 2 8 , 8$ , b) an MNIST CL only network with (1024, 64, 16) neuron configuration and $f o = 1 , 4$ , and c) the ‘conv $+ 2 \\mathrm { C L s } ^ { \\prime }$ CIFAR10 network with $f o = 1 , 2$ . We found that high scatter indicates good performance and the correlation is stronger for networks where CLs have more importance, i.e. CL only networks as opposed to conv. This is shown in the performance vs. scatter plots in Fig. 7, where (a) and (b) show the performance predicting ability of scatter better than (c). Note that the random connection patterns used so far have the highest scatter and occur as the rightmost points in each subfigure. The other points are obtained by specifically planning connections. We found that when 1 junction was planned to give corresponding high values in $\\bar { S }$ , it invariably led to low values for another junction, leading to a low $S$ . This explains why random patterns generally perform well. ", + "bbox": [ + 173, + 130, + 825, + 268 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/2a1cf611997723e1bdc19ec13f7a4be0119a684042dbb3dd50eeffe37ea4e89c.jpg", + "image_caption": [ + "Figure 7: Network performance vs. scatter for CL only networks of (a) Morse (b) MNIST, and convolutional network with $2 \\mathrm { C L }$ junctions of (c) CIFAR10. All minimum values that need to be considered to differentiate between connection patterns are bolded. " + ], + "image_footnote": [], + "bbox": [ + 236, + 284, + 756, + 453 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "$\\bar { S }$ is shown alongside each point. When $S$ is equal for different connection patterns, the next minimum value in $\\breve { \\bar { S } }$ needs to be considered to differentiate the networks, and so on. Considering the Morse results, the leftmost 3 points all have $\\begin{array} { r } { S = \\frac { 1 } { 8 } } \\end{array}$ , but the number of occurrences of $\\frac { 1 } { 8 }$ in $\\bar { S }$ is 3 for the lowest point $8 \\%$ accuracy), 2 for the second lowest ( $12 \\%$ accuracy) and 1 for the highest point $46 \\%$ accuracy). For the MNIST results, both the leftmost points have a single minimum value of $\\textstyle { \\frac { 1 } { 1 6 } }$ in $\\bar { S }$ , but the lower has two occurrences of $\\textstyle { \\frac { 1 } { 4 } }$ while the upper has one. ", + "bbox": [ + 174, + 529, + 825, + 613 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We draw several insights from these results. Firstly, although we defined $S$ as a single value for convenience, there may arise cases when other (non-minimum) elements in $\\bar { S }$ are important. Secondly, perhaps contrary to intuition, the concept of windows and scatter is important for all CLs, not simply the first. As shown in Fig. 7a), a network with $\\begin{array} { r } { S _ { 1 b } = \\frac { 1 } { 8 } } \\end{array}$ performs equally poorly as a network with $\\begin{array} { r } { S _ { 2 f } = ~ \\frac { 1 } { 8 } } \\end{array}$ . Thirdly, scatter is a sufficient metric for performance, not necessary. A network with a high $S$ value will perform well, but a network with a slightly lower $S$ than another cannot be conclusively dismissed as being worse. But if a network has multiple low values in $\\bar { S }$ , it should be rejected. Finally, carefully choosing which neurons to group in a window will increase the predictive power of scatter. A priori knowledge of the dataset will lead to better window choices. ", + "bbox": [ + 174, + 619, + 825, + 747 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 CONCLUSION AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 176, + 767, + 493, + 782 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "This paper discusses the merits of pre-defining sparsity in CLs of neural networks, which leads to significant reduction in parameters without performance loss. In general, the smaller the fraction of CLs in a network, the more redundancy there exists in their parameters. If we can achieve similar results (i.e., $0 . 2 \\%$ density) on Alexnet for example, we would obtain $9 5 \\%$ reduction in overall parameters. Coupled with hardware acceleration designed for pre-defined sparse networks, we believe our approach will lead to more aggressive exploration of network structure. Network connectivity can be guided by the scatter metric, which is closely related to performance, and by optimally distributing connections across junctions. Future work would involve extension to conv layers, since recent CNNs have lower values for the ratio of number of CLs to number of conv layers. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "REFERENCES \nAlfred Bourely, John Patrick Boueri, and Krzysztof Choromonski. Sparse neural network topologies. In arXiv:1706.05683, 2017. \nWenlin Chen, James T. Wilson, Stephen Tyree, Kilian Q. Weinberger, and Yixin Chen. Compressing neural networks with the hashing trick. In Proc. ICML, pp. 2285–2294. JMLR.org, 2015. \nMisha Denil, Babak Shakibi, Laurent Dinh, Marc’aurelio Ranzato, and Nando D. Freitas. Predicting parameters in deep learning. In Proc. NIPS, pp. 2148–2156, 2013. \nSourya Dey. Morse code dataset for artificial neural networks, Oct 2017. URL https://cobaltfolly.wordpress.com/2017/10/15/ morse-code-dataset-for-artificial-neural-networks/. \nSourya Dey, Peter A. Beerel, and Keith M. Chugg. Interleaver design for deep neural networks. In Proc. Asilomar Conference on Signals, Systems and Computers. IEEE, 2017a. \nSourya Dey, Yinan Shao, Keith M. Chugg, and Peter A. Beerel. Accelerating training of deep neural networks via sparse edge processing. In Proc. ICANN, pp. 273–280. Springer, 2017b. \nYunchao Gong, Liu Liu, Ming Yang, and Lubomir D. Bourdev. Compressing deep convolutional networks using vector quantization. In arXiv:1412.6115, 2014. \nSong Han, Jeff Pool, John Tran, and William Dally. Learning both weights and connections for efficient neural network. In Proc. NIPS, 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. In Proc. ICLR, 2016. \nKaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proc. CVPR, pp. 770–778, June 2016. \nGao Huang, Zhuang Liu, Kilian Q. Weinberger, and Laurens van der Maaten. Densely connected convolutional networks. In arXiv:1608.06993, 2016. \nAlex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Proc. NIPS, pp. 1097–1105, 2012. \nPierre Sermanet, David Eigen, Xiang Zhang, Michael Mathieu, Rob Fergus, and Yann LeCun. ¨ Overfeat: Integrated recognition, localization and detection using convolutional networks. In arXiv:1312.6229, 2013. \nKaren Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In Proc. ICLR, 2015. \nVikas Sindhwani, Tara Sainath, and Sanjiv Kumar. Structured transforms for small-footprint deep learning. In Proc. NIPS, pp. 3088–3096. Curran Associates, Inc., 2015. \nSuraj Srinivas, Akshayvarun Subramanya, and R. Venkatesh Babu. Training sparse neural networks. In arXiv:1611.06694, 2016. \nNitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15:1929–1958, 2014. \nC. Szegedy, Wei Liu, Yangqing Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich. Going deeper with convolutions. In Proc. CVPR, pp. 1–9, 2015. \nChen Zhang, Di Wu, Jiayu Sun, Guangyu Sun, Guojie Luo, and Jason Cong. Energy-efficient CNN implementation on a deeply pipelined FPGA cluster. In Proc. ISLPED, pp. 326–331. ACM, 2016. \nXichuan Zhou, Shengli Li, Kai Qin, Kunping Li, Fang Tang, Shengdong Hu, Shujun Liu, and Zhi Lin. Deep adaptive network: An efficient deep neural network with sparse binary connections. In arXiv:1604.06154, 2016. ", + "bbox": [ + 171, + 79, + 828, + 925 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 APPENDIX ", + "text_level": 1, + "bbox": [ + 174, + 102, + 294, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5.1 MORE ON DISTRIBUTING INDIVIDUAL JUNCTION DENSITIES ", + "text_level": 1, + "bbox": [ + 176, + 133, + 633, + 147 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Section 2.4 showed that when overall CL density is fixed, it is desirable to make junction 2 denser than junction 1. It is also interesting to note, however, that performance falls off more sharply when junction 1 density is reduced to the bare minimum as compared to treating junction 2 similarly. This is not shown in Fig. 3 due to space constraints. We found that when junction 1 had the minimum possible density and junction 2 had the maximum possible while still satisfying the fixed overall, the accuracy was about $36 \\%$ for both subfigures (b) and (c). When the densities were flipped, the accuracies were $67 \\%$ for subfigure (b) and $7 5 \\%$ for (c) in Figure 3. ", + "bbox": [ + 174, + 160, + 825, + 257 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5.2 DENSE CASES OF WINDOW ADJACENCY MATRICES ", + "text_level": 1, + "bbox": [ + 174, + 273, + 570, + 287 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "As stated in Section 3.1, window output matrices for several junctions can be constructed by multiplying the individual matrices for each component junction. Consider the Morse network as described in Section 3.2. Note that $f o _ { 1 : 2 } = 1 2 8 \\times 8 = 1 0 2 4$ and $f i _ { 1 : 2 } = 8 \\times 1 2 8 = 1 0 2 4$ . Thus, for the equivalent junction 1:2 which has $N _ { 1 } = 6 4$ left neurons and $N _ { 3 } = 6 4$ right neurons, we have $f o _ { 1 : 2 } > N _ { 3 }$ and $f i _ { 1 : 2 } > N _ { 1 }$ . So in this case the number of neurons in each window will be rounded up to 1, and both the ideal window adjacency matrices $A _ { 1 : 2 } ^ { w _ { 1 l } }$ and $A _ { 1 : 2 } ^ { w _ { 2 r } }$ will be all 16’s matrices since the ideal number of connections from each window to a neuron on the opposite side is $1 0 2 4 / 6 4 = 1 6$ . This is a result of the network having sufficient density so that several paths exist from every input neuron to every output neuron. ", + "bbox": [ + 174, + 299, + 825, + 424 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5.3 POSSIBLE REASONS FOR SCLS CONVERGING FASTER THAN FCLS ", + "text_level": 1, + "bbox": [ + 178, + 443, + 671, + 457 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Training a neural network is essentially an exercise in finding the minimum of the cost function, which is a function of all the network parameters. The graph for cost as a function of parameters may have saddle points which masquerade as minima. It could also be poorly conditioned, wherein the gradient of cost with respect to two different parameters have widely different magnitudes, making simultaneous optimization difficult. These effects are non-idealities and training the network often takes more time because of the length of the trajectory needed to overcome these and arrive at the optimum point. The probability of encountering these non-idealities increases as the number of network parameters increase, i.e. less parameters leads to a higher ratio of minima : saddle points, which can make the network converge faster. We hypothesize that SCLs train faster than FCLs due to the former having fewer parameters. 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We shall henceforth refer to these layers as connected", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 133, + 514 + ], + "score": 1.0, + "content": "layers", + "type": "text" + }, + { + "bbox": [ + 133, + 501, + 157, + 512 + ], + "score": 0.74, + "content": "( C L s )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 500, + 289, + 514 + ], + "score": 1.0, + "content": ", of which fully connected layers", + "type": "text" + }, + { + "bbox": [ + 289, + 501, + 318, + 511 + ], + "score": 0.54, + "content": "( F C L s )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "are a special case. 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For example, the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 523, + 441, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 222, + 535 + ], + "score": 1.0, + "content": "FCLs in Alexnet account for", + "type": "text" + }, + { + "bbox": [ + 223, + 523, + 250, + 533 + ], + "score": 0.85, + "content": "9 5 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 523, + 441, + 535 + ], + "score": 1.0, + "content": "of the network parameters (Zhang et al., 2016).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "We shall refer to the spaces between CLs as CL junctions (or simply junctions), which are occupied", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "by connections, or weights. 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As proof of concept, we show results on CIFAR,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 315, + 470, + 328 + ], + "spans": [ + { + "bbox": [ + 141, + 315, + 470, + 328 + ], + "score": 1.0, + "content": "MNIST and a new dataset on classifying Morse code symbols, which highlights", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 327, + 400, + 339 + ], + "spans": [ + { + "bbox": [ + 141, + 327, + 400, + 339 + ], + "score": 1.0, + "content": "some interesting trends and limits of sparse connection patterns.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10, + "bbox_fs": [ + 141, + 216, + 470, + 339 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 372, + 205, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 208, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 208, + 388 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "Neural networks (NNs) in machine learning systems are critical drivers of new technologies such", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "as image processing and speech recognition. Modern NNs are gigantic in size with millions of pa-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 424, + 504, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 504, + 436 + ], + "score": 1.0, + "content": "rameters, such as the ones described in Alexnet (Krizhevsky et al., 2012), Overfeat (Sermanet et al.,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "2013) and ResNet (He et al., 2016). They therefore require an enormous amount of memory and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "silicon processing during usage. Optimizing a network to improve performance typically involves", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "making it deeper and adding more parameters (Simonyan & Zisserman, 2015; Szegedy et al., 2015;", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "Huang et al., 2016), which further exacerbates the problem of large storage complexity. While the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "convolutional (conv) layers in these networks do feature extraction, there are usually fully connected", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "layers at the end performing classification. We shall henceforth refer to these layers as connected", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 133, + 514 + ], + "score": 1.0, + "content": "layers", + "type": "text" + }, + { + "bbox": [ + 133, + 501, + 157, + 512 + ], + "score": 0.74, + "content": "( C L s )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 500, + 289, + 514 + ], + "score": 1.0, + "content": ", of which fully connected layers", + "type": "text" + }, + { + "bbox": [ + 289, + 501, + 318, + 511 + ], + "score": 0.54, + "content": "( F C L s )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "are a special case. Owing to their high density", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "of connections, the majority of network parameters are concentrated in FCLs. For example, the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 523, + 441, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 222, + 535 + ], + "score": 1.0, + "content": "FCLs in Alexnet account for", + "type": "text" + }, + { + "bbox": [ + 223, + 523, + 250, + 533 + ], + "score": 0.85, + "content": "9 5 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 523, + 441, + 535 + ], + "score": 1.0, + "content": "of the network parameters (Zhang et al., 2016).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 402, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "We shall refer to the spaces between CLs as CL junctions (or simply junctions), which are occupied", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "by connections, or weights. Given the trend in modern NNs, we raise the question – “How necessary", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "is it to have FCLs?” or, in other words, “What if most of the junction connections never existed?", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "Would the resulting sparsely connected layers (SCLs), when trained and tested, still give competitive", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "performance?” As an example, consider a network with 2 CLs of 100 neurons each and the junction", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "between them has 1000 weights instead of the expected 10,000. Then this is a sparse network with", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 195, + 617 + ], + "score": 1.0, + "content": "connection density of", + "type": "text" + }, + { + "bbox": [ + 196, + 605, + 214, + 615 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 605, + 505, + 617 + ], + "score": 1.0, + "content": ". Given such a sparse architecture, a natural question to ask is “How can", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 615, + 468, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 468, + 628 + ], + "score": 1.0, + "content": "the existing 1000 weights be best distributed so that network performance is maximized?”", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 539, + 505, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "In this regard, the present work makes the following contributions. In Section 2, we formalize the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "concept of sparsity, or its opposite measure density, and explore its effects on different network", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "types. We show that CL parameters are largely redundant and a network pre-defined to be sparse", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "before starting training does not result in any performance degradation. For certain network archi-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "tectures, this leads to CL parameter reduction by a factor of more than 450, or an overall parameter", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "reduction by a factor of more than 20. In Section 2.4, we discuss techniques to distribute connec-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "tions across junctions when given an overall network density. Finally, in Section 3, we formalize", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "pre-defined sparse connectivity patterns using adjacency matrices and introduce the scatter metric.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 478, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 478, + 733 + ], + "score": 1.0, + "content": "Our results show that scatter is a quick and useful indicator of how good a sparse network is.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 633, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 249, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 251, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 251, + 95 + ], + "score": 1.0, + "content": "2 PRE-DEFINED SPARSITY", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 161 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 417, + 119 + ], + "score": 1.0, + "content": "As an example of the footprint of modern NNs, AlexNet has a weight size of", + "type": "text" + }, + { + "bbox": [ + 417, + 106, + 452, + 117 + ], + "score": 0.5, + "content": "2 3 4 \\mathrm { M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "and requires", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "635 million arithmetic operations only for feedforward processing (Zhang et al., 2016). It has been", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "score": 1.0, + "content": "shown that NNs, particularly their FCLs, have an excess of parameters and tend to overfit to the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 154 + ], + "score": 1.0, + "content": "training data (Denil et al., 2013), resulting in inferior performance on test data. The following", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 482, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 482, + 163 + ], + "score": 1.0, + "content": "paragraph describes several previous works that have attempted to reduce parameters in NNs.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "score": 1.0, + "content": "Dropout (deletion) of random neurons (Srivastava et al., 2014) trains multiple differently configured", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "networks, which are finally combined to regain the original full size network. Chen et al. (2015)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "randomly forced the same value on collections of weights, but acknowledged that “a significant", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 505, + 213 + ], + "score": 1.0, + "content": "number of nodes [get] disconnected from neighboring layers.” Other sparsifying techniques such", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "as pruning and quantization (Han et al., 2016; 2015; Zhou et al., 2016; Gong et al., 2014) first train", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "the complete network, and then perform further computations to delete parameters. Sindhwani et al.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 506, + 246 + ], + "score": 1.0, + "content": "(2015) used low rank matrices to impose structure on network parameters. Srinivas et al. (2016)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 244, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 256 + ], + "score": 1.0, + "content": "proposed a regularizer to reduce parameters in the network, but acknowledged that this increased", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 253, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 104, + 253, + 505, + 268 + ], + "score": 1.0, + "content": "training complexity. In general, all these architectures deal with FCLs at some point of time during", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "their usage cycle and therefore, do not permanently solve the parameter explosion problem of NNs.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 108, + 291, + 222, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 225, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 225, + 304 + ], + "score": 1.0, + "content": "2.1 OUR METHODOLOGY", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "Our attempt to simplify NNs is to pre-define the level of sparsity, or connection density, in a network", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "prior to the start of training. This means that our network always has fewer connections than its FCL", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 333, + 504, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 504, + 346 + ], + "score": 1.0, + "content": "counterpart; the weights which are absent never make an appearance during training or inference.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 240, + 357 + ], + "score": 1.0, + "content": "In our notation, a NN will have", + "type": "text" + }, + { + "bbox": [ + 240, + 345, + 248, + 354 + ], + "score": 0.76, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 344, + 311, + 357 + ], + "score": 1.0, + "content": "junctions, i.e.", + "type": "text" + }, + { + "bbox": [ + 311, + 344, + 337, + 355 + ], + "score": 0.9, + "content": "J + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 344, + 390, + 357 + ], + "score": 1.0, + "content": "layers, with", + "type": "text" + }, + { + "bbox": [ + 390, + 344, + 478, + 356 + ], + "score": 0.92, + "content": "\\{ N _ { 1 } , \\stackrel { \\textstyle - } { N } _ { 2 } , \\cdot \\cdot \\cdot \\stackrel { \\textstyle - } { , } N _ { J + 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "being", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 261, + 367 + ], + "score": 1.0, + "content": "the number of neurons in each layer.", + "type": "text" + }, + { + "bbox": [ + 261, + 355, + 273, + 366 + ], + "score": 0.88, + "content": "N _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 354, + 293, + 367 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 293, + 356, + 316, + 367 + ], + "score": 0.92, + "content": "N _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "are respectively the number of neurons in the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 300, + 379 + ], + "score": 1.0, + "content": "earlier (left) and later (right) layers of junction", + "type": "text" + }, + { + "bbox": [ + 301, + 367, + 305, + 376 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 366, + 505, + 379 + ], + "score": 1.0, + "content": ". Every left neuron has a fixed number of edges", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "score": 1.0, + "content": "going from it to the right, and every right neuron has a fixed number of edges coming into it from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 292, + 401 + ], + "score": 1.0, + "content": "the left. These numbers are defined as fan-out", + "type": "text" + }, + { + "bbox": [ + 292, + 388, + 314, + 399 + ], + "score": 0.88, + "content": "( f o _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 388, + 358, + 401 + ], + "score": 1.0, + "content": "and fan-in", + "type": "text" + }, + { + "bbox": [ + 359, + 388, + 379, + 399 + ], + "score": 0.88, + "content": "( f i _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 388, + 505, + 401 + ], + "score": 1.0, + "content": ", respectively. For conventional", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 134, + 411 + ], + "score": 1.0, + "content": "FCLs,", + "type": "text" + }, + { + "bbox": [ + 135, + 399, + 185, + 410 + ], + "score": 0.93, + "content": "f o _ { i } = N _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 398, + 204, + 411 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 205, + 399, + 244, + 410 + ], + "score": 0.92, + "content": "f i _ { i } = N _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 398, + 352, + 411 + ], + "score": 1.0, + "content": ". We propose SCLs where", + "type": "text" + }, + { + "bbox": [ + 352, + 399, + 403, + 411 + ], + "score": 0.93, + "content": "f o _ { i } < N _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 398, + 423, + 411 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 399, + 462, + 411 + ], + "score": 0.92, + "content": "f i _ { i } < N _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 398, + 505, + 411 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 409, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 237, + 422 + ], + "score": 0.89, + "content": "N _ { i } \\times f o _ { i } = N _ { i + 1 } \\times f i _ { i } = W _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 409, + 382, + 423 + ], + "score": 1.0, + "content": ", the number of weights in junction", + "type": "text" + }, + { + "bbox": [ + 382, + 411, + 387, + 420 + ], + "score": 0.47, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 409, + 455, + 423 + ], + "score": 1.0, + "content": ". Having a fixed", + "type": "text" + }, + { + "bbox": [ + 455, + 411, + 470, + 421 + ], + "score": 0.89, + "content": "f o _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 409, + 489, + 423 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 410, + 504, + 422 + ], + "score": 0.89, + "content": "f i _ { i }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "ensures that all neurons in a junction contribute equally and none of them get disconnected, since that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 430, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 393, + 446 + ], + "score": 1.0, + "content": "would lead to a loss of information. The connection density in junction", + "type": "text" + }, + { + "bbox": [ + 394, + 433, + 398, + 442 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 430, + 444, + 446 + ], + "score": 1.0, + "content": "is given as", + "type": "text" + }, + { + "bbox": [ + 445, + 432, + 504, + 444 + ], + "score": 0.91, + "content": "W _ { i } / ( N _ { i } N _ { i + 1 } )", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 450, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 313, + 464 + ], + "score": 1.0, + "content": "and the overall CL connection density is defined as", + "type": "text" + }, + { + "bbox": [ + 313, + 443, + 445, + 463 + ], + "score": 0.89, + "content": "\\textstyle \\left( \\sum _ { i = 1 } ^ { J } W _ { i } \\right) / \\left( \\sum _ { i = 1 } ^ { J } N _ { i } N _ { i + 1 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 442, + 450, + 464 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 466, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "Note that earlier works such as Dey et al. (2017b;a) have proposed hardware architectures that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "leverage pre-defined sparsity to speed up training. However, a complete analysis of methods to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "pre-define connections, its possible gains on different kinds of modern deep NNs and a test of its", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "limits via a metric quantifying its goodness has been lacking. Bourely et al. (2017) introduced a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "metric based on eigenvalues, but ran limited tests on MNIST. The following subsections analyze our", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "method of pre-defined sparsity in more detail. 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Our network", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "has 6 conv layers with number of filters equal to [64, 64, 128, 128, 256, 256]. Each has window size", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 618, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 123, + 630 + ], + "score": 0.37, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 618, + 505, + 633 + ], + "score": 1.0, + "content": ". The outputs are batch-normalized before applying ReLU non-linearity. A max-pooling layer", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "score": 1.0, + "content": "of pool size 2x2 succeeds every pair of conv layers. This structure finally results in a layer of 4096", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "neurons, which is followed by the CLs. 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For Morse, the difference with", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 458, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 445, + 732 + ], + "score": 1.0, + "content": "and without regularization is negligible, while for CIFAR, the accuracy results differ by about", + "type": "text" + }, + { + "bbox": [ + 445, + 721, + 458, + 731 + ], + "score": 0.8, + "content": "1 \\%", + "type": "inline_equation" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 249, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 251, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 251, + 95 + ], + "score": 1.0, + "content": "2 PRE-DEFINED SPARSITY", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 161 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 417, + 119 + ], + "score": 1.0, + "content": "As an example of the footprint of modern NNs, AlexNet has a weight size of", + "type": "text" + }, + { + "bbox": [ + 417, + 106, + 452, + 117 + ], + "score": 0.5, + "content": "2 3 4 \\mathrm { M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "and requires", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "635 million arithmetic operations only for feedforward processing (Zhang et al., 2016). It has been", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "score": 1.0, + "content": "shown that NNs, particularly their FCLs, have an excess of parameters and tend to overfit to the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 154 + ], + "score": 1.0, + "content": "training data (Denil et al., 2013), resulting in inferior performance on test data. The following", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 482, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 482, + 163 + ], + "score": 1.0, + "content": "paragraph describes several previous works that have attempted to reduce parameters in NNs.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 105, + 506, + 163 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "score": 1.0, + "content": "Dropout (deletion) of random neurons (Srivastava et al., 2014) trains multiple differently configured", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "networks, which are finally combined to regain the original full size network. 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(2015)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "randomly forced the same value on collections of weights, but acknowledged that “a significant", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 505, + 213 + ], + "score": 1.0, + "content": "number of nodes [get] disconnected from neighboring layers.” Other sparsifying techniques such", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "as pruning and quantization (Han et al., 2016; 2015; Zhou et al., 2016; Gong et al., 2014) first train", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "the complete network, and then perform further computations to delete parameters. Sindhwani et al.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 506, + 246 + ], + "score": 1.0, + "content": "(2015) used low rank matrices to impose structure on network parameters. Srinivas et al. (2016)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 244, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 256 + ], + "score": 1.0, + "content": "proposed a regularizer to reduce parameters in the network, but acknowledged that this increased", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 253, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 104, + 253, + 505, + 268 + ], + "score": 1.0, + "content": "training complexity. In general, all these architectures deal with FCLs at some point of time during", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "their usage cycle and therefore, do not permanently solve the parameter explosion problem of NNs.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5, + "bbox_fs": [ + 104, + 167, + 506, + 279 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 291, + 222, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 225, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 225, + 304 + ], + "score": 1.0, + "content": "2.1 OUR METHODOLOGY", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "Our attempt to simplify NNs is to pre-define the level of sparsity, or connection density, in a network", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "prior to the start of training. This means that our network always has fewer connections than its FCL", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 333, + 504, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 504, + 346 + ], + "score": 1.0, + "content": "counterpart; the weights which are absent never make an appearance during training or inference.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 240, + 357 + ], + "score": 1.0, + "content": "In our notation, a NN will have", + "type": "text" + }, + { + "bbox": [ + 240, + 345, + 248, + 354 + ], + "score": 0.76, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 344, + 311, + 357 + ], + "score": 1.0, + "content": "junctions, i.e.", + "type": "text" + }, + { + "bbox": [ + 311, + 344, + 337, + 355 + ], + "score": 0.9, + "content": "J + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 344, + 390, + 357 + ], + "score": 1.0, + "content": "layers, with", + "type": "text" + }, + { + "bbox": [ + 390, + 344, + 478, + 356 + ], + "score": 0.92, + "content": "\\{ N _ { 1 } , \\stackrel { \\textstyle - } { N } _ { 2 } , \\cdot \\cdot \\cdot \\stackrel { \\textstyle - } { , } N _ { J + 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "being", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 261, + 367 + ], + "score": 1.0, + "content": "the number of neurons in each layer.", + "type": "text" + }, + { + "bbox": [ + 261, + 355, + 273, + 366 + ], + "score": 0.88, + "content": "N _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 354, + 293, + 367 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 293, + 356, + 316, + 367 + ], + "score": 0.92, + "content": "N _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "are respectively the number of neurons in the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 300, + 379 + ], + "score": 1.0, + "content": "earlier (left) and later (right) layers of junction", + "type": "text" + }, + { + "bbox": [ + 301, + 367, + 305, + 376 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 366, + 505, + 379 + ], + "score": 1.0, + "content": ". Every left neuron has a fixed number of edges", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "score": 1.0, + "content": "going from it to the right, and every right neuron has a fixed number of edges coming into it from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 292, + 401 + ], + "score": 1.0, + "content": "the left. These numbers are defined as fan-out", + "type": "text" + }, + { + "bbox": [ + 292, + 388, + 314, + 399 + ], + "score": 0.88, + "content": "( f o _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 388, + 358, + 401 + ], + "score": 1.0, + "content": "and fan-in", + "type": "text" + }, + { + "bbox": [ + 359, + 388, + 379, + 399 + ], + "score": 0.88, + "content": "( f i _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 388, + 505, + 401 + ], + "score": 1.0, + "content": ", respectively. For conventional", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 134, + 411 + ], + "score": 1.0, + "content": "FCLs,", + "type": "text" + }, + { + "bbox": [ + 135, + 399, + 185, + 410 + ], + "score": 0.93, + "content": "f o _ { i } = N _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 398, + 204, + 411 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 205, + 399, + 244, + 410 + ], + "score": 0.92, + "content": "f i _ { i } = N _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 398, + 352, + 411 + ], + "score": 1.0, + "content": ". We propose SCLs where", + "type": "text" + }, + { + "bbox": [ + 352, + 399, + 403, + 411 + ], + "score": 0.93, + "content": "f o _ { i } < N _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 398, + 423, + 411 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 399, + 462, + 411 + ], + "score": 0.92, + "content": "f i _ { i } < N _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 398, + 505, + 411 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 409, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 237, + 422 + ], + "score": 0.89, + "content": "N _ { i } \\times f o _ { i } = N _ { i + 1 } \\times f i _ { i } = W _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 409, + 382, + 423 + ], + "score": 1.0, + "content": ", the number of weights in junction", + "type": "text" + }, + { + "bbox": [ + 382, + 411, + 387, + 420 + ], + "score": 0.47, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 409, + 455, + 423 + ], + "score": 1.0, + "content": ". Having a fixed", + "type": "text" + }, + { + "bbox": [ + 455, + 411, + 470, + 421 + ], + "score": 0.89, + "content": "f o _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 409, + 489, + 423 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 410, + 504, + 422 + ], + "score": 0.89, + "content": "f i _ { i }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "ensures that all neurons in a junction contribute equally and none of them get disconnected, since that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 430, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 393, + 446 + ], + "score": 1.0, + "content": "would lead to a loss of information. The connection density in junction", + "type": "text" + }, + { + "bbox": [ + 394, + 433, + 398, + 442 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 430, + 444, + 446 + ], + "score": 1.0, + "content": "is given as", + "type": "text" + }, + { + "bbox": [ + 445, + 432, + 504, + 444 + ], + "score": 0.91, + "content": "W _ { i } / ( N _ { i } N _ { i + 1 } )", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 450, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 313, + 464 + ], + "score": 1.0, + "content": "and the overall CL connection density is defined as", + "type": "text" + }, + { + "bbox": [ + 313, + 443, + 445, + 463 + ], + "score": 0.89, + "content": "\\textstyle \\left( \\sum _ { i = 1 } ^ { J } W _ { i } \\right) / \\left( \\sum _ { i = 1 } ^ { J } N _ { i } N _ { i + 1 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 442, + 450, + 464 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 311, + 506, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 466, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "Note that earlier works such as Dey et al. (2017b;a) have proposed hardware architectures that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "leverage pre-defined sparsity to speed up training. However, a complete analysis of methods to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "pre-define connections, its possible gains on different kinds of modern deep NNs and a test of its", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "limits via a metric quantifying its goodness has been lacking. Bourely et al. (2017) introduced a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "metric based on eigenvalues, but ran limited tests on MNIST. The following subsections analyze our", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "method of pre-defined sparsity in more detail. We experimented with networks operating on CIFAR,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 532, + 460, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 426, + 545 + ], + "score": 1.0, + "content": "MNIST and Morse code symbol classification – a new dataset described in Dey", + "type": "text" + }, + { + "bbox": [ + 426, + 532, + 457, + 544 + ], + "score": 0.27, + "content": "( 2 0 1 7 ) ^ { \\bar { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 533, + 460, + 545 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 466, + 506, + 545 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 558, + 240, + 569 + ], + "lines": [ + { + "bbox": [ + 106, + 558, + 241, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 241, + 570 + ], + "score": 1.0, + "content": "2.2 NETWORK EXPERIMENTS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 108, + 578, + 171, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 173, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 173, + 592 + ], + "score": 1.0, + "content": "2.2.1 CIFAR", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 597, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 505, + 609 + ], + "score": 1.0, + "content": "We used the original CIFAR10 and CIFAR100 datasets without data augmentation. Our network", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "has 6 conv layers with number of filters equal to [64, 64, 128, 128, 256, 256]. Each has window size", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 618, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 123, + 630 + ], + "score": 0.37, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 618, + 505, + 633 + ], + "score": 1.0, + "content": ". The outputs are batch-normalized before applying ReLU non-linearity. A max-pooling layer", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "score": 1.0, + "content": "of pool size 2x2 succeeds every pair of conv layers. This structure finally results in a layer of 4096", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "neurons, which is followed by the CLs. We used the Adam optimizer, ReLU-activated hidden layers", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "and softmax output layer – choices which we maintained for all networks unless otherwise specified.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 598, + 506, + 665 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 669, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "Our results in Section 2.4 indicate that later CL junctions (i.e. closer to the outputs) should be", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "denser than earlier ones (i.e. closer to the inputs). Moreover, since most CL networks have a ta-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 692, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 200, + 704 + ], + "score": 1.0, + "content": "pering structure where", + "type": "text" + }, + { + "bbox": [ + 200, + 692, + 213, + 703 + ], + "score": 0.89, + "content": "N _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 692, + 328, + 704 + ], + "score": 1.0, + "content": "monotonically decreases as", + "type": "text" + }, + { + "bbox": [ + 328, + 692, + 333, + 701 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 692, + 505, + 704 + ], + "score": 1.0, + "content": "increases, more parameter savings can be", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "achieved by making earlier layers less dense. Accordingly we did a grid search and picked CL junc-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 316, + 106 + ], + "score": 1.0, + "content": "tion densities as given in Table 1. The phrase ‘conv", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 316, + 94, + 345, + 104 + ], + "score": 0.56, + "content": "+ 2 \\mathrm { C L s } ^ { \\prime }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 346, + 93, + 382, + 106 + ], + "score": 1.0, + "content": "’ denotes", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 382, + 94, + 405, + 104 + ], + "score": 0.47, + "content": "2 \\mathrm { C L }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 405, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "junctions corresponding", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 453, + 117 + ], + "score": 1.0, + "content": "to a CL neuron configuration of (4096, 512, 16) for CIFAR10, (4096, 512, 128) for", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 453, + 104, + 501, + 115 + ], + "score": 0.35, + "content": "\\mathbf { C I F A R 1 0 0 } ^ { \\tilde { 2 } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 501, + 104, + 505, + 117 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 359, + 128 + ], + "score": 1.0, + "content": "and (3136, 784, 10) for MNIST (see Section 2.2.2). For ‘con", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 360, + 116, + 395, + 126 + ], + "score": 0.44, + "content": "\\mathbf { v } + 3 { \\mathbf { C L s } } ^ { \\prime }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 395, + 115, + 506, + 128 + ], + "score": 1.0, + "content": ", an additional 256-neuron", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "layer precedes the output. ‘MNIST CL’ and ‘Morse CL’ refer to the CL only networks described", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "subsequently, for which we have only shown some of the more important configurations in Table 1.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 668, + 505, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "achieved by making earlier layers less dense. Accordingly we did a grid search and picked CL junc-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 316, + 106 + ], + "score": 1.0, + "content": "tion densities as given in Table 1. The phrase ‘conv", + "type": "text" + }, + { + "bbox": [ + 316, + 94, + 345, + 104 + ], + "score": 0.56, + "content": "+ 2 \\mathrm { C L s } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 93, + 382, + 106 + ], + "score": 1.0, + "content": "’ denotes", + "type": "text" + }, + { + "bbox": [ + 382, + 94, + 405, + 104 + ], + "score": 0.47, + "content": "2 \\mathrm { C L }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "junctions corresponding", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 453, + 117 + ], + "score": 1.0, + "content": "to a CL neuron configuration of (4096, 512, 16) for CIFAR10, (4096, 512, 128) for", + "type": "text" + }, + { + "bbox": [ + 453, + 104, + 501, + 115 + ], + "score": 0.35, + "content": "\\mathbf { C I F A R 1 0 0 } ^ { \\tilde { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 104, + 505, + 117 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 359, + 128 + ], + "score": 1.0, + "content": "and (3136, 784, 10) for MNIST (see Section 2.2.2). For ‘con", + "type": "text" + }, + { + "bbox": [ + 360, + 116, + 395, + 126 + ], + "score": 0.44, + "content": "\\mathbf { v } + 3 { \\mathbf { C L s } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 115, + 506, + 128 + ], + "score": 1.0, + "content": ", an additional 256-neuron", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "layer precedes the output. ‘MNIST CL’ and ‘Morse CL’ refer to the CL only networks described", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "subsequently, for which we have only shown some of the more important configurations in Table 1.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 355, + 166 + ], + "score": 1.0, + "content": "As an example, consider the first network in ‘CIFAR10 conv", + "type": "text" + }, + { + "bbox": [ + 355, + 155, + 382, + 164 + ], + "score": 0.31, + "content": "+ 2 \\mathrm { C L s }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 154, + 433, + 166 + ], + "score": 1.0, + "content": "’ which has", + "type": "text" + }, + { + "bbox": [ + 433, + 154, + 501, + 165 + ], + "score": 0.91, + "content": "f o _ { 1 } = f o _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 154, + 505, + 166 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 316, + 177 + ], + "score": 1.0, + "content": "This means that the individual junction densities are", + "type": "text" + }, + { + "bbox": [ + 317, + 165, + 456, + 176 + ], + "score": 0.87, + "content": "( 4 0 9 6 \\times 1 ) / ( 4 0 9 6 \\times 5 1 2 ) = 0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 165, + 475, + 177 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 476, + 166, + 505, + 176 + ], + "score": 0.86, + "content": "( 5 1 2 \\times", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 504, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 112, + 189 + ], + "score": 1.0, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 113, + 176, + 199, + 187 + ], + "score": 0.84, + "content": ") / ( 5 1 2 \\times 1 6 ) = 6 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 176, + 330, + 189 + ], + "score": 1.0, + "content": ", to give an overall CL density of", + "type": "text" + }, + { + "bbox": [ + 330, + 177, + 504, + 188 + ], + "score": 0.79, + "content": "( 4 0 9 6 \\times 1 + 5 1 2 \\times 1 ) / ( 4 0 9 6 \\times 5 1 2 + 5 1 2 \\times", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 161, + 199 + ], + "score": 0.88, + "content": "1 6 ) = 0 . 2 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 187, + 346, + 199 + ], + "score": 1.0, + "content": ". In other words, while FCLs would have been", + "type": "text" + }, + { + "bbox": [ + 347, + 188, + 371, + 198 + ], + "score": 0.83, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 187, + 446, + 199 + ], + "score": 1.0, + "content": "dense with 2, 097,", + "type": "text" + }, + { + "bbox": [ + 446, + 187, + 505, + 198 + ], + "score": 0.74, + "content": "1 5 2 + 8 1 9 2 =", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 245, + 210 + ], + "score": 1.0, + "content": "2, 105, 344 weights, the SCLs use", + "type": "text" + }, + { + "bbox": [ + 246, + 198, + 328, + 209 + ], + "score": 0.92, + "content": "4 0 9 6 + 5 1 2 = 4 6 0 8", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "weights, which is 457 times less. Note that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "weights in the sparse junction are distributed as fixed, but randomly generated patterns, with the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 262, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 262, + 232 + ], + "score": 1.0, + "content": "constraints of fixed fan-in and fan-out.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "table", + "bbox": [ + 108, + 260, + 502, + 367 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 202, + 249, + 408, + 259 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 202, + 246, + 408, + 262 + ], + "spans": [ + { + "bbox": [ + 202, + 246, + 408, + 262 + ], + "score": 1.0, + "content": "Table 1: Densities for some of our sparse networks", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "table_body", + "bbox": [ + 108, + 260, + 502, + 367 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 260, + 502, + 367 + ], + "spans": [ + { + "bbox": [ + 108, + 260, + 502, + 367 + ], + "score": 0.978, + "html": "
NetJunctionfan-outsCL JunctionDensities (%)Overall CLDensity (%)NetJunctionfan-outsCL JunctionDensities (%)Overall CLDensity (%)
CIFAR10conv+2CLs1,11,80.2,6.30.2,500.220.39CIFAR10conv+3CLs1,1,11,1,81,2,160.2,0.4,6.30.2,0.4,500.2,0.8,1000.220.30.41
CIFAR100conv+2CLs1,11.81,320.2,0.80.2,6.30.2,250.210.380.95CIFAR100conv+3CLs1,1,11,1,161,2,320.2,0.4,0.80.2,0.4,130.2,0.8,250.220.390.59
MNISTconv+2CLs1,54,1016,100.1,500.5,1002,1000.290.832.35MNIST CL(x=224)4,10112,101.79,10050,1003.0250.63
Morse CL512,3250,5050
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(a),(b),(d),(e)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 600, + 480, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 480, + 614 + ], + "score": 1.0, + "content": "Validation accuracy across epochs. (c),(f) Best validation accuracies after 1, 5 and 30 epochs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 625, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "score": 1.0, + "content": "Figure 1 shows the results for CIFAR. Subfigures (a), (b), (d) and (e) show classification perfor-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 637, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 648 + ], + "score": 1.0, + "content": "mance on validation data as the network is trained for 30 epochs (note that the final accuracies", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "stayed almost constant after 20 epochs). The different lines correspond to different overall CL den-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 659, + 504, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 504, + 670 + ], + "score": 1.0, + "content": "sities. Subfigures (c) and (f) show the best validation accuracies after 1, 5 and 30 epochs for the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 668, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 682 + ], + "score": 1.0, + "content": "different CL densities. We see that the final accuracies (the numbers at the top of each column)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "show negligible performance degradation for these extremely low levels of density, not to mention", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 691, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 704 + ], + "score": 1.0, + "content": "some cases where SCLs outperform FCLs. These results point to the promise of sparsity. Also", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 712, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "2Powers of 2 are used for ease of testing sparsity. The extra output neurons have a ‘false’ ground truth", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 720, + 338, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 338, + 734 + ], + "score": 1.0, + "content": "labeling and thus do not impact the final classification accuracy.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 150 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 355, + 166 + ], + "score": 1.0, + "content": "As an example, consider the first network in ‘CIFAR10 conv", + "type": "text" + }, + { + "bbox": [ + 355, + 155, + 382, + 164 + ], + "score": 0.31, + "content": "+ 2 \\mathrm { C L s }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 154, + 433, + 166 + ], + "score": 1.0, + "content": "’ which has", + "type": "text" + }, + { + "bbox": [ + 433, + 154, + 501, + 165 + ], + "score": 0.91, + "content": "f o _ { 1 } = f o _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 154, + 505, + 166 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 316, + 177 + ], + "score": 1.0, + "content": "This means that the individual junction densities are", + "type": "text" + }, + { + "bbox": [ + 317, + 165, + 456, + 176 + ], + "score": 0.87, + "content": "( 4 0 9 6 \\times 1 ) / ( 4 0 9 6 \\times 5 1 2 ) = 0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 165, + 475, + 177 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 476, + 166, + 505, + 176 + ], + "score": 0.86, + "content": "( 5 1 2 \\times", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 504, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 112, + 189 + ], + "score": 1.0, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 113, + 176, + 199, + 187 + ], + "score": 0.84, + "content": ") / ( 5 1 2 \\times 1 6 ) = 6 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 176, + 330, + 189 + ], + "score": 1.0, + "content": ", to give an overall CL density of", + "type": "text" + }, + { + "bbox": [ + 330, + 177, + 504, + 188 + ], + "score": 0.79, + "content": "( 4 0 9 6 \\times 1 + 5 1 2 \\times 1 ) / ( 4 0 9 6 \\times 5 1 2 + 5 1 2 \\times", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 161, + 199 + ], + "score": 0.88, + "content": "1 6 ) = 0 . 2 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 187, + 346, + 199 + ], + "score": 1.0, + "content": ". In other words, while FCLs would have been", + "type": "text" + }, + { + "bbox": [ + 347, + 188, + 371, + 198 + ], + "score": 0.83, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 187, + 446, + 199 + ], + "score": 1.0, + "content": "dense with 2, 097,", + "type": "text" + }, + { + "bbox": [ + 446, + 187, + 505, + 198 + ], + "score": 0.74, + "content": "1 5 2 + 8 1 9 2 =", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 245, + 210 + ], + "score": 1.0, + "content": "2, 105, 344 weights, the SCLs use", + "type": "text" + }, + { + "bbox": [ + 246, + 198, + 328, + 209 + ], + "score": 0.92, + "content": "4 0 9 6 + 5 1 2 = 4 6 0 8", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "weights, which is 457 times less. Note that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "weights in the sparse junction are distributed as fixed, but randomly generated patterns, with the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 262, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 262, + 232 + ], + "score": 1.0, + "content": "constraints of fixed fan-in and fan-out.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 154, + 505, + 232 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 260, + 502, + 367 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 202, + 249, + 408, + 259 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 202, + 246, + 408, + 262 + ], + "spans": [ + { + "bbox": [ + 202, + 246, + 408, + 262 + ], + "score": 1.0, + "content": "Table 1: Densities for some of our sparse networks", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "table_body", + "bbox": [ + 108, + 260, + 502, + 367 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 260, + 502, + 367 + ], + "spans": [ + { + "bbox": [ + 108, + 260, + 502, + 367 + ], + "score": 0.978, + "html": "
NetJunctionfan-outsCL JunctionDensities (%)Overall CLDensity (%)NetJunctionfan-outsCL JunctionDensities (%)Overall CLDensity (%)
CIFAR10conv+2CLs1,11,80.2,6.30.2,500.220.39CIFAR10conv+3CLs1,1,11,1,81,2,160.2,0.4,6.30.2,0.4,500.2,0.8,1000.220.30.41
CIFAR100conv+2CLs1,11.81,320.2,0.80.2,6.30.2,250.210.380.95CIFAR100conv+3CLs1,1,11,1,161,2,320.2,0.4,0.80.2,0.4,130.2,0.8,250.220.390.59
MNISTconv+2CLs1,54,1016,100.1,500.5,1002,1000.290.832.35MNIST CL(x=224)4,10112,101.79,10050,1003.0250.63
Morse CL512,3250,5050
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(a),(b),(d),(e)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 600, + 480, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 480, + 614 + ], + "score": 1.0, + "content": "Validation accuracy across epochs. (c),(f) Best validation accuracies after 1, 5 and 30 epochs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 625, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "score": 1.0, + "content": "Figure 1 shows the results for CIFAR. Subfigures (a), (b), (d) and (e) show classification perfor-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 637, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 648 + ], + "score": 1.0, + "content": "mance on validation data as the network is trained for 30 epochs (note that the final accuracies", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "stayed almost constant after 20 epochs). The different lines correspond to different overall CL den-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 659, + 504, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 504, + 670 + ], + "score": 1.0, + "content": "sities. Subfigures (c) and (f) show the best validation accuracies after 1, 5 and 30 epochs for the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 668, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 682 + ], + "score": 1.0, + "content": "different CL densities. We see that the final accuracies (the numbers at the top of each column)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "show negligible performance degradation for these extremely low levels of density, not to mention", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 691, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 704 + ], + "score": 1.0, + "content": "some cases where SCLs outperform FCLs. These results point to the promise of sparsity. Also", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "notice from subfigures (c) and (f) that SCLs generally start training quicker than FCLs, as evidenced", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 503, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 503, + 106 + ], + "score": 1.0, + "content": "by their higher accuracies after 1 epoch of training. See Appendix Section 5.3 for more discussion.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 626, + 505, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "notice from subfigures (c) and (f) that SCLs generally start training quicker than FCLs, as evidenced", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 503, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 503, + 106 + ], + "score": 1.0, + "content": "by their higher accuracies after 1 epoch of training. See Appendix Section 5.3 for more discussion.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 107, + 118, + 173, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 115, + 176, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 176, + 132 + ], + "score": 1.0, + "content": "2.2.2 MNIST", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 137, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "We used 2 different kinds of networks when experimenting on MNIST (no data augmentation). The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 149, + 504, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 487, + 161 + ], + "score": 1.0, + "content": "first was ‘conv+2CLs’ – 2 conv layers having 32 and 64 filters of size 5x5 each, alternating with", + "type": "text" + }, + { + "bbox": [ + 487, + 149, + 504, + 159 + ], + "score": 0.67, + "content": "2 \\mathbf { x } 2", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "max pooling layers. This results in a layer of 3136 neurons, which is followed by 2 CLs having 784", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "and 10 neurons, i.e. 2 junctions overall. Fig. 2(a) and (b) show the results. Due to the simplicity of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 194 + ], + "score": 1.0, + "content": "the overall network, performance starts degrading at higher densities compared to CIFAR. However,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 193, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 215, + 204 + ], + "score": 1.0, + "content": "a network with CL density", + "type": "text" + }, + { + "bbox": [ + 215, + 193, + 243, + 203 + ], + "score": 0.87, + "content": "2 . 3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 193, + 506, + 204 + ], + "score": 1.0, + "content": "still matches FCLs in performance. Note that the total number of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 504, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 142, + 216 + ], + "score": 1.0, + "content": "weights", + "type": "text" + }, + { + "bbox": [ + 142, + 204, + 193, + 214 + ], + "score": 0.67, + "content": "( \\mathrm { c o n v + S C L s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 204, + 358, + 216 + ], + "score": 1.0, + "content": ") is 0.11M for this network, which is only", + "type": "text" + }, + { + "bbox": [ + 358, + 204, + 385, + 214 + ], + "score": 0.85, + "content": "4 . 3 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 204, + 447, + 216 + ], + "score": 1.0, + "content": "of the original", + "type": "text" + }, + { + "bbox": [ + 448, + 204, + 499, + 215 + ], + "score": 0.29, + "content": "( \\mathrm { c o n v + F C L s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 204, + 504, + 216 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 220, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "The second was a family of networks with only CLs, either having a single junction with a neuron", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 330, + 244 + ], + "score": 1.0, + "content": "configuration of (1024, 16), or 2 junctions configured as", + "type": "text" + }, + { + "bbox": [ + 331, + 232, + 378, + 243 + ], + "score": 0.81, + "content": "( 7 8 4 , x , 1 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 231, + 408, + 244 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 408, + 234, + 415, + 241 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "varies. The results are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "shown in Fig. 2(c), which offers two insights. Firstly, performance drops off at higher densities for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "CL only MNIST networks as compared to the one with conv layers. However, half the parameters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "can still be dropped without appreciable performance degradation. This aspect is further discussed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "in Section 2.3. Secondly, large SCLs perform better than small FCLs with similar number of param-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "eters. Considering the black-circled points as an example, performance drops when switching from", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 200, + 309 + ], + "score": 1.0, + "content": "224 hidden neurons at", + "type": "text" + }, + { + "bbox": [ + 200, + 297, + 227, + 308 + ], + "score": 0.87, + "content": "12 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 296, + 299, + 309 + ], + "score": 1.0, + "content": "density to 112 at", + "type": "text" + }, + { + "bbox": [ + 300, + 297, + 320, + 308 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 296, + 355, + 309 + ], + "score": 1.0, + "content": "to 56 at", + "type": "text" + }, + { + "bbox": [ + 355, + 297, + 375, + 308 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 296, + 411, + 309 + ], + "score": 1.0, + "content": "to 28 at", + "type": "text" + }, + { + "bbox": [ + 412, + 297, + 435, + 308 + ], + "score": 0.88, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 296, + 505, + 309 + ], + "score": 1.0, + "content": ", even though all", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "these networks have similar number of parameters. So increasing the number of hidden neurons is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 318, + 275, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 275, + 332 + ], + "score": 1.0, + "content": "desirable, albeit with diminishing returns.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + }, + { + "type": "image", + "bbox": [ + 136, + 343, + 475, + 442 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 136, + 343, + 475, + 442 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 136, + 343, + 475, + 442 + ], + "spans": [ + { + "bbox": [ + 136, + 343, + 475, + 442 + ], + "score": 0.959, + "type": "image", + "image_path": "3fc47770f88838dbbe0788e071df56ec0e82e712b051bc101380511a9ed923a2.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 136, + 343, + 475, + 376.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 136, + 376.0, + 475, + 409.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 136, + 409.0, + 475, + 442.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 457, + 504, + 492 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "Figure 2: (a)–(b) Performance results of pre-defined sparsity on an MNIST conv network with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "score": 1.0, + "content": "different densities, each trained for 30 epochs. (c) Performance vs. connection density for different", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 480, + 331, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 331, + 492 + ], + "score": 1.0, + "content": "MNIST CL only networks, each trained for 100 epochs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 511, + 172, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 173, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 173, + 524 + ], + "score": 1.0, + "content": "2.2.3 MORSE", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "The Morse code dataset presents a harder challenge for sparsity. It only has 64-valued inputs (as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "compared to 784 for MNIST and 3072 for CIFAR), so each input neuron encodes a significant", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "amount of information. The outputs are Morse codewords and there are 64 classes. Distinctions", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "between inputs belonging to different classes is small. For example, the input pattern for the Morse", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 158, + 586 + ], + "score": 1.0, + "content": "codeword ‘.", + "type": "text" + }, + { + "bbox": [ + 186, + 576, + 367, + 587 + ], + "score": 1.0, + "content": "can be easily confused with the codeword ‘.", + "type": "text" + }, + { + "bbox": [ + 382, + 574, + 505, + 588 + ], + "score": 1.0, + "content": ". -’. As a result, performance", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "degrades quickly as connections are removed. Our network had 64 input and output neurons and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "1024 hidden layer neurons, i.e. 3 CLs and 2 junctions, trained using stochastic gradient descent.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 363, + 620 + ], + "score": 1.0, + "content": "The results are shown in Fig. 3(a). As with MNIST CL only,", + "type": "text" + }, + { + "bbox": [ + 363, + 608, + 383, + 618 + ], + "score": 0.84, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "density can be achieved with", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 618, + 248, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 248, + 632 + ], + "score": 1.0, + "content": "negligible degradation in accuracy.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 106, + 644, + 362, + 656 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 363, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 363, + 657 + ], + "score": 1.0, + "content": "2.3 ANALYZING THE RESULTS OF PRE-DEFINED SPARSITY", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Our results indicate that for deep networks having several conv layers, there is severe redundancy in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "the CLs. As a result, they can be made extremely sparse without hampering network performance,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "which leads to significant memory savings. If the network only has CLs, the amount of density", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "reduction achievable without performance degradation is smaller. This can be explained using the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "argument of relative importance. For a network which extensively extracts features and processes its", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "raw input data via conv filters, the input to the CLs can already substantially discriminate between", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 106 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 118, + 173, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 115, + 176, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 176, + 132 + ], + "score": 1.0, + "content": "2.2.2 MNIST", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 137, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "We used 2 different kinds of networks when experimenting on MNIST (no data augmentation). The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 149, + 504, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 487, + 161 + ], + "score": 1.0, + "content": "first was ‘conv+2CLs’ – 2 conv layers having 32 and 64 filters of size 5x5 each, alternating with", + "type": "text" + }, + { + "bbox": [ + 487, + 149, + 504, + 159 + ], + "score": 0.67, + "content": "2 \\mathbf { x } 2", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "max pooling layers. This results in a layer of 3136 neurons, which is followed by 2 CLs having 784", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "and 10 neurons, i.e. 2 junctions overall. Fig. 2(a) and (b) show the results. Due to the simplicity of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 194 + ], + "score": 1.0, + "content": "the overall network, performance starts degrading at higher densities compared to CIFAR. However,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 193, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 215, + 204 + ], + "score": 1.0, + "content": "a network with CL density", + "type": "text" + }, + { + "bbox": [ + 215, + 193, + 243, + 203 + ], + "score": 0.87, + "content": "2 . 3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 193, + 506, + 204 + ], + "score": 1.0, + "content": "still matches FCLs in performance. Note that the total number of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 504, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 142, + 216 + ], + "score": 1.0, + "content": "weights", + "type": "text" + }, + { + "bbox": [ + 142, + 204, + 193, + 214 + ], + "score": 0.67, + "content": "( \\mathrm { c o n v + S C L s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 204, + 358, + 216 + ], + "score": 1.0, + "content": ") is 0.11M for this network, which is only", + "type": "text" + }, + { + "bbox": [ + 358, + 204, + 385, + 214 + ], + "score": 0.85, + "content": "4 . 3 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 204, + 447, + 216 + ], + "score": 1.0, + "content": "of the original", + "type": "text" + }, + { + "bbox": [ + 448, + 204, + 499, + 215 + ], + "score": 0.29, + "content": "( \\mathrm { c o n v + F C L s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 204, + 504, + 216 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 137, + 506, + 216 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 220, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "The second was a family of networks with only CLs, either having a single junction with a neuron", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 330, + 244 + ], + "score": 1.0, + "content": "configuration of (1024, 16), or 2 junctions configured as", + "type": "text" + }, + { + "bbox": [ + 331, + 232, + 378, + 243 + ], + "score": 0.81, + "content": "( 7 8 4 , x , 1 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 231, + 408, + 244 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 408, + 234, + 415, + 241 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "varies. The results are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "shown in Fig. 2(c), which offers two insights. Firstly, performance drops off at higher densities for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "CL only MNIST networks as compared to the one with conv layers. However, half the parameters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "can still be dropped without appreciable performance degradation. This aspect is further discussed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "in Section 2.3. Secondly, large SCLs perform better than small FCLs with similar number of param-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "eters. Considering the black-circled points as an example, performance drops when switching from", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 200, + 309 + ], + "score": 1.0, + "content": "224 hidden neurons at", + "type": "text" + }, + { + "bbox": [ + 200, + 297, + 227, + 308 + ], + "score": 0.87, + "content": "12 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 296, + 299, + 309 + ], + "score": 1.0, + "content": "density to 112 at", + "type": "text" + }, + { + "bbox": [ + 300, + 297, + 320, + 308 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 296, + 355, + 309 + ], + "score": 1.0, + "content": "to 56 at", + "type": "text" + }, + { + "bbox": [ + 355, + 297, + 375, + 308 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 296, + 411, + 309 + ], + "score": 1.0, + "content": "to 28 at", + "type": "text" + }, + { + "bbox": [ + 412, + 297, + 435, + 308 + ], + "score": 0.88, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 296, + 505, + 309 + ], + "score": 1.0, + "content": ", even though all", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "these networks have similar number of parameters. 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(c) Performance vs. connection density for different", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 480, + 331, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 331, + 492 + ], + "score": 1.0, + "content": "MNIST CL only networks, each trained for 100 epochs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 511, + 172, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 173, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 173, + 524 + ], + "score": 1.0, + "content": "2.2.3 MORSE", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "The Morse code dataset presents a harder challenge for sparsity. It only has 64-valued inputs (as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "compared to 784 for MNIST and 3072 for CIFAR), so each input neuron encodes a significant", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "amount of information. The outputs are Morse codewords and there are 64 classes. Distinctions", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "between inputs belonging to different classes is small. 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CIFAR10 conv+3CLs3/181.150.152.230.00965.831.43
CIFAR100 conv+3CLs3/181.150.152.260.01365.991.45
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NetCLs/TotalLayersConvParams(M)ConvOps(B)FC CLParams(M)SparseCL Par-ams (M)OverallParam %ReductionOverallOp%Reduction
Morse CL2/2000.1310.0665050
MNIST CL (x = 224)2/2000.1780.0895050
MNIST conv+2CLs2/60.050.12.470.0695.6318.29
CIFAR10 conv+2CLs2/171.150.152.110.00564.631.35
CIFAR100 conv+2CLs2/171.150.152.160.0264.761.38
CIFAR10 conv+3CLs3/181.150.152.230.00965.831.43
CIFAR100 conv+3CLs3/181.150.152.260.01365.991.45
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For the special case where", + "type": "text" + }, + { + "bbox": [ + 302, + 107, + 330, + 117 + ], + "score": 0.91, + "content": "X = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 106, + 348, + 118 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 348, + 106, + 378, + 117 + ], + "score": 0.9, + "content": "Y = J", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "(total number of junctions), we", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 271, + 130 + ], + "score": 1.0, + "content": "obtain the input-output adjacency matrix", + "type": "text" + }, + { + "bbox": [ + 272, + 118, + 292, + 128 + ], + "score": 0.9, + "content": "A _ { 1 : J }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 116, + 436, + 130 + ], + "score": 1.0, + "content": ". As a simple example, consider the", + "type": "text" + }, + { + "bbox": [ + 437, + 117, + 468, + 129 + ], + "score": 0.89, + "content": "( 8 , 4 , 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 116, + 506, + 130 + ], + "score": 1.0, + "content": "network", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 128, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 201, + 140 + ], + "score": 1.0, + "content": "shown in Fig. 4 where", + "type": "text" + }, + { + "bbox": [ + 201, + 128, + 237, + 140 + ], + "score": 0.92, + "content": "f o _ { 1 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 128, + 256, + 140 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 257, + 129, + 293, + 140 + ], + "score": 0.93, + "content": "f o _ { 2 } = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 128, + 374, + 140 + ], + "score": 1.0, + "content": ", which implies that", + "type": "text" + }, + { + "bbox": [ + 375, + 128, + 437, + 140 + ], + "score": 0.91, + "content": "f i _ { 1 } = f i _ { 2 } = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 128, + 443, + 140 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 443, + 129, + 456, + 139 + ], + "score": 0.75, + "content": "A _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 128, + 475, + 140 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 476, + 129, + 489, + 139 + ], + "score": 0.88, + "content": "A _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 128, + 505, + 140 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 444, + 152 + ], + "score": 1.0, + "content": "adjacency matrices of single junctions. We obtain the input-output adjacency matrix", + "type": "text" + }, + { + "bbox": [ + 445, + 140, + 501, + 150 + ], + "score": 0.93, + "content": "A _ { 1 : 2 } = A _ { 2 } A _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 138, + 505, + 152 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 151, + 163 + ], + "score": 1.0, + "content": "equivalent", + "type": "text" + }, + { + "bbox": [ + 151, + 150, + 242, + 162 + ], + "score": 0.93, + "content": "f o _ { 1 : 2 } = f o _ { 1 } f o _ { 2 } = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 150, + 309, + 163 + ], + "score": 1.0, + "content": ", and equivalent", + "type": "text" + }, + { + "bbox": [ + 309, + 150, + 396, + 162 + ], + "score": 0.93, + "content": "f i _ { 1 : 2 } = f i _ { 1 } f i _ { 2 } = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 150, + 506, + 163 + ], + "score": 1.0, + "content": ". Note that this equivalent", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 161, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 506, + 173 + ], + "score": 1.0, + "content": "junction 1:2 is only an abstract concept that aids visualizing how neurons connect from the inputs to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 349, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 349, + 185 + ], + "score": 1.0, + "content": "the outputs. It has no relation to the overall network density.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "image", + "bbox": [ + 144, + 194, + 465, + 356 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 144, + 194, + 465, + 356 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 144, + 194, + 463, + 356 + ], + "spans": [ + { + "bbox": [ + 144, + 194, + 463, + 356 + ], + "score": 0.969, + "type": "image", + "image_path": "e1c5bcca6ab72e26ab730e0d19ffa5addbdcc35314daf1f4aa3dd1234dac838e.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 144, + 194, + 465, + 248.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 144, + 248.0, + 465, + 302.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 144, + 302.0, + 465, + 356.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 164, + 371, + 445, + 384 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 164, + 370, + 447, + 386 + ], + "spans": [ + { + "bbox": [ + 164, + 370, + 447, + 386 + ], + "score": 1.0, + "content": "Figure 4: An example of adjacency matrices and equivalent junctions.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + } + ], + "index": 11.0 + }, + { + "type": "text", + "bbox": [ + 107, + 393, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "We now attempt to characterize the quality of a sparse connection pattern, i.e. we try to find the best", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 403, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 418 + ], + "score": 1.0, + "content": "possible way to connect neurons to optimize performance. Since sparsity gives good performance,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 414, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 429 + ], + "score": 1.0, + "content": "we hypothesize that there exists redundancy / correlated information between neurons. Intuitively,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 427, + 504, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 504, + 438 + ], + "score": 1.0, + "content": "we assume that left neurons of a junction can be grouped into windows depending on the dimen-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "sionality of the left layer output. For example, the input layer in an MNIST CL only network would", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "have 2D windows, each of which might correspond to a fraction of the image, as shown in Fig. 5(a).", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "score": 1.0, + "content": "When outputs from a CL have an additional dimension for features, such as in CIFAR or the MNIST", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 470, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 482 + ], + "score": 1.0, + "content": "conv network, each window is a cuboid capturing fractions of both spatial extent and features, as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "shown in Fig. 5(b). Given such windows, we will try to maximize the number of left windows to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "which each right neuron connects, the idea being that each right neuron should get some information", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "from all portions of the left layer in order to capture global view. To realize the importance of this,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "consider the MNIST output neuron representing digit 2. Let’s say the sparse connection pattern is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 525, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 536 + ], + "score": 1.0, + "content": "such that when the connections to output 3 are traced back to the input layer, they all come from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "the top half of the image. This would be undesirable since the top half of an image of a 2 can be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "mistaken for a 3. A good sparse connection pattern will try to avoid such scenarios by spreading the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "connections to any right neuron across as many input windows as possible. The problem can also", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "be mirrored so that every left neuron connects to as many different right windows as possible. This", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "score": 1.0, + "content": "ensures that local information from left neurons is spread to different parts of the right layer. The", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 590, + 473, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 473, + 604 + ], + "score": 1.0, + "content": "grouping of right windows will depend on the dimensionality of the input to the right layer.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 607, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "The window size is chosen to be the minimum possible such that the ideal number of connections", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 466, + 631 + ], + "score": 1.0, + "content": "from or to it remains integral. The example from Fig. 4 is reproduced in Fig. 6. Since", + "type": "text" + }, + { + "bbox": [ + 466, + 618, + 501, + 630 + ], + "score": 0.91, + "content": "f i _ { 1 } = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 617, + 506, + 631 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 628, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 643 + ], + "score": 1.0, + "content": "the inputs must be grouped into 2 windows so that ideally 1 connection from each reaches every", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "hidden neuron. If instead the inputs are grouped into 4 windows, the ideal number would be half", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "of a connection, which is not achievable. In order to achieve the minimum window size, we let the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 662, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 216, + 675 + ], + "score": 1.0, + "content": "number of left windows be", + "type": "text" + }, + { + "bbox": [ + 216, + 662, + 227, + 673 + ], + "score": 0.87, + "content": "f i", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 662, + 374, + 675 + ], + "score": 1.0, + "content": "and the number of right windows be", + "type": "text" + }, + { + "bbox": [ + 375, + 662, + 387, + 673 + ], + "score": 0.8, + "content": "f o", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 662, + 449, + 675 + ], + "score": 1.0, + "content": ". So in junction", + "type": "text" + }, + { + "bbox": [ + 449, + 663, + 453, + 672 + ], + "score": 0.65, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 662, + 506, + 675 + ], + "score": 1.0, + "content": ", the number", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 100, + 671, + 509, + 708 + ], + "spans": [ + { + "bbox": [ + 100, + 671, + 278, + 708 + ], + "score": 1.0, + "content": "of neurons in each left and right window ileft- and right-window adjacency matrices", + "type": "text" + }, + { + "bbox": [ + 281, + 673, + 311, + 685 + ], + "score": 0.91, + "content": "N _ { i } / f i _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 673, + 371, + 685 + ], + "score": 0.91, + "content": "N _ { i + 1 } / f o _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 671, + 509, + 708 + ], + "score": 1.0, + "content": "en we constructby summing up", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 278, + 684, + 438, + 700 + ], + "spans": [ + { + "bbox": [ + 278, + 685, + 351, + 700 + ], + "score": 0.93, + "content": "A _ { i } ^ { w _ { i l } } \\in \\mathbb { Z } _ { \\geq 0 } ^ { N _ { i + 1 } \\times f i _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 684, + 438, + 700 + ], + "score": 0.95, + "content": "A _ { i } ^ { w _ { i r } } \\in \\mathbb { Z } _ { \\geq 0 } ^ { f o _ { i } \\times N _ { i } }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 159, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 159, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "as shown in Fig. 5(c). The window adjacency matrices describe connectivity between", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "windows and neurons on the opposite side. 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For example, the input layer in an MNIST CL only network would", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "have 2D windows, each of which might correspond to a fraction of the image, as shown in Fig. 5(a).", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "score": 1.0, + "content": "When outputs from a CL have an additional dimension for features, such as in CIFAR or the MNIST", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 470, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 482 + ], + "score": 1.0, + "content": "conv network, each window is a cuboid capturing fractions of both spatial extent and features, as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "shown in Fig. 5(b). Given such windows, we will try to maximize the number of left windows to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "which each right neuron connects, the idea being that each right neuron should get some information", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "from all portions of the left layer in order to capture global view. To realize the importance of this,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "consider the MNIST output neuron representing digit 2. 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A good sparse connection pattern will try to avoid such scenarios by spreading the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "connections to any right neuron across as many input windows as possible. The problem can also", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "be mirrored so that every left neuron connects to as many different right windows as possible. This", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "score": 1.0, + "content": "ensures that local information from left neurons is spread to different parts of the right layer. 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Since", + "type": "text" + }, + { + "bbox": [ + 466, + 618, + 501, + 630 + ], + "score": 0.91, + "content": "f i _ { 1 } = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 617, + 506, + 631 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 628, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 643 + ], + "score": 1.0, + "content": "the inputs must be grouped into 2 windows so that ideally 1 connection from each reaches every", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "hidden neuron. 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Green neurons indicate ideal connectivity. The", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 103, + 388, + 413, + 408 + ], + "spans": [ + { + "bbox": [ + 103, + 388, + 348, + 408 + ], + "score": 1.0, + "content": "hidden layer is split into 2 to show separate constructions of", + "type": "text" + }, + { + "bbox": [ + 348, + 393, + 370, + 405 + ], + "score": 0.93, + "content": "A _ { 1 } ^ { w _ { 1 r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 388, + 388, + 408 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 389, + 392, + 409, + 405 + ], + "score": 0.91, + "content": "A _ { 2 } ^ { w _ { 2 l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 388, + 413, + 408 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 7.25 + }, + { + "type": "text", + "bbox": [ + 106, + 437, + 506, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "every neuron on the opposite side. Note that these matrices can also be constructed for multiple", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 101, + 443, + 508, + 466 + ], + "spans": [ + { + "bbox": [ + 101, + 443, + 167, + 466 + ], + "score": 1.0, + "content": "junctions, i.e.", + "type": "text" + }, + { + "bbox": [ + 167, + 448, + 192, + 461 + ], + "score": 0.91, + "content": "A _ { X : Y } ^ { w _ { X l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 443, + 212, + 466 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 212, + 448, + 236, + 461 + ], + "score": 0.92, + "content": "A _ { X : Y } ^ { w _ { Y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 443, + 508, + 466 + ], + "score": 1.0, + "content": ", by multiplying matrices for individual junctions. 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The other scatter values can be computed similarly", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 675, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 212, + 689 + ], + "score": 1.0, + "content": "to form the scatter vector", + "type": "text" + }, + { + "bbox": [ + 212, + 676, + 343, + 688 + ], + "score": 0.91, + "content": "\\bar { S } = [ S _ { 1 f } , S _ { 1 b } , S _ { 2 f } , S _ { 2 b } , S _ { f } , S _ { b } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 675, + 506, + 689 + ], + "score": 1.0, + "content": ", where the final 2 values correspond to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 686, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 209, + 700 + ], + "score": 1.0, + "content": "junction 1:2. Notice that", + "type": "text" + }, + { + "bbox": [ + 209, + 687, + 217, + 698 + ], + "score": 0.82, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 686, + 505, + 700 + ], + "score": 1.0, + "content": "will be all 1s for FCLs, which is the ideal case. Incorporating sparsity", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 174, + 711 + ], + "score": 1.0, + "content": "leads to reduced", + "type": "text" + }, + { + "bbox": [ + 175, + 698, + 183, + 709 + ], + "score": 0.84, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 698, + 312, + 711 + ], + "score": 1.0, + "content": "values. The final scatter metric", + "type": "text" + }, + { + "bbox": [ + 313, + 699, + 353, + 711 + ], + "score": 0.9, + "content": "S \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 698, + 456, + 711 + ], + "score": 1.0, + "content": "is the minimum value in", + "type": "text" + }, + { + "bbox": [ + 457, + 698, + 464, + 709 + ], + "score": 0.82, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 698, + 505, + 711 + ], + "score": 1.0, + "content": ", i.e. 0.75", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 337, + 722 + ], + "score": 1.0, + "content": "for Fig. 6. Our experiments indicate that any low value in", + "type": "text" + }, + { + "bbox": [ + 337, + 710, + 345, + 720 + ], + "score": 0.81, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "leads to bad performance, so we picked", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 721, + 218, + 731 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 218, + 731 + ], + "score": 1.0, + "content": "the critical minimum value.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 641, + 507, + 731 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 292, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 294, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 294, + 95 + ], + "score": 1.0, + "content": "3.2 ANALYSIS AND RESULTS OF SCATTER", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 213 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 447, + 115 + ], + "score": 1.0, + "content": "We ran experiments to evaluate scatter using a) the Morse CL only network with", + "type": "text" + }, + { + "bbox": [ + 447, + 103, + 502, + 114 + ], + "score": 0.88, + "content": "f o \\ = \\ 1 2 8 , 8", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 102, + 506, + 115 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 428, + 127 + ], + "score": 1.0, + "content": "b) an MNIST CL only network with (1024, 64, 16) neuron configuration and", + "type": "text" + }, + { + "bbox": [ + 428, + 114, + 472, + 126 + ], + "score": 0.9, + "content": "f o = 1 , 4", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 114, + 506, + 127 + ], + "score": 1.0, + "content": ", and c)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 145, + 138 + ], + "score": 1.0, + "content": "the ‘conv", + "type": "text" + }, + { + "bbox": [ + 145, + 126, + 176, + 136 + ], + "score": 0.3, + "content": "+ 2 \\mathrm { C L s } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 124, + 279, + 138 + ], + "score": 1.0, + "content": "CIFAR10 network with", + "type": "text" + }, + { + "bbox": [ + 279, + 126, + 322, + 136 + ], + "score": 0.9, + "content": "f o = 1 , 2", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 124, + 506, + 138 + ], + "score": 1.0, + "content": ". We found that high scatter indicates good", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "score": 1.0, + "content": "performance and the correlation is stronger for networks where CLs have more importance, i.e. CL", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 160 + ], + "score": 1.0, + "content": "only networks as opposed to conv. This is shown in the performance vs. scatter plots in Fig. 7,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "where (a) and (b) show the performance predicting ability of scatter better than (c). Note that the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 182 + ], + "score": 1.0, + "content": "random connection patterns used so far have the highest scatter and occur as the rightmost points in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "each subfigure. The other points are obtained by specifically planning connections. We found that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 190, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 372, + 204 + ], + "score": 1.0, + "content": "when 1 junction was planned to give corresponding high values in", + "type": "text" + }, + { + "bbox": [ + 373, + 190, + 381, + 201 + ], + "score": 0.8, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 191, + 505, + 204 + ], + "score": 1.0, + "content": ", it invariably led to low values", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 202, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 255, + 214 + ], + "score": 1.0, + "content": "for another junction, leading to a low", + "type": "text" + }, + { + "bbox": [ + 256, + 202, + 263, + 212 + ], + "score": 0.6, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 202, + 505, + 214 + ], + "score": 1.0, + "content": ". This explains why random patterns generally perform well.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5 + }, + { + "type": "image", + "bbox": [ + 145, + 225, + 463, + 359 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 225, + 463, + 359 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 225, + 463, + 359 + ], + "spans": [ + { + "bbox": [ + 145, + 225, + 463, + 359 + ], + "score": 0.97, + "type": "image", + "image_path": "2a1cf611997723e1bdc19ec13f7a4be0119a684042dbb3dd50eeffe37ea4e89c.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 145, + 225, + 463, + 269.6666666666667 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 145, + 269.6666666666667, + 463, + 314.33333333333337 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 145, + 314.33333333333337, + 463, + 359.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 374, + 504, + 408 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "Figure 7: Network performance vs. scatter for CL only networks of (a) Morse (b) MNIST, and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 386, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 220, + 397 + ], + "score": 1.0, + "content": "convolutional network with", + "type": "text" + }, + { + "bbox": [ + 220, + 386, + 243, + 396 + ], + "score": 0.37, + "content": "2 \\mathrm { C L }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 386, + 505, + 397 + ], + "score": 1.0, + "content": "junctions of (c) CIFAR10. All minimum values that need to be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 397, + 375, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 375, + 409 + ], + "score": 1.0, + "content": "considered to differentiate between connection patterns are bolded.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 114, + 429 + ], + "score": 0.82, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 418, + 270, + 432 + ], + "score": 1.0, + "content": "is shown alongside each point. When", + "type": "text" + }, + { + "bbox": [ + 271, + 420, + 278, + 429 + ], + "score": 0.8, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 418, + 505, + 432 + ], + "score": 1.0, + "content": "is equal for different connection patterns, the next min-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 168, + 442 + ], + "score": 1.0, + "content": "imum value in", + "type": "text" + }, + { + "bbox": [ + 168, + 429, + 176, + 440 + ], + "score": 0.82, + "content": "\\breve { \\bar { S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "needs to be considered to differentiate the networks, and so on. Considering the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 438, + 508, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 282, + 457 + ], + "score": 1.0, + "content": "Morse results, the leftmost 3 points all have", + "type": "text" + }, + { + "bbox": [ + 282, + 441, + 309, + 454 + ], + "score": 0.92, + "content": "\\begin{array} { r } { S = \\frac { 1 } { 8 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 438, + 447, + 457 + ], + "score": 1.0, + "content": ", but the number of occurrences of", + "type": "text" + }, + { + "bbox": [ + 447, + 441, + 454, + 454 + ], + "score": 0.86, + "content": "\\frac { 1 } { 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 438, + 465, + 457 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 466, + 441, + 474, + 451 + ], + "score": 0.83, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 438, + 508, + 457 + ], + "score": 1.0, + "content": "is 3 for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 176, + 465 + ], + "score": 1.0, + "content": "the lowest point", + "type": "text" + }, + { + "bbox": [ + 177, + 452, + 191, + 463 + ], + "score": 0.83, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 451, + 335, + 465 + ], + "score": 1.0, + "content": "accuracy), 2 for the second lowest (", + "type": "text" + }, + { + "bbox": [ + 335, + 452, + 354, + 463 + ], + "score": 0.84, + "content": "12 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "accuracy) and 1 for the highest point", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 110, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 110, + 463, + 129, + 473 + ], + "score": 0.84, + "content": "46 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "accuracy). For the MNIST results, both the leftmost points have a single minimum value of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 468, + 402, + 493 + ], + "spans": [ + { + "bbox": [ + 107, + 474, + 118, + 487 + ], + "score": 0.89, + "content": "\\textstyle { \\frac { 1 } { 1 6 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 468, + 129, + 493 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 129, + 473, + 137, + 484 + ], + "score": 0.81, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 468, + 290, + 493 + ], + "score": 1.0, + "content": ", but the lower has two occurrences of", + "type": "text" + }, + { + "bbox": [ + 290, + 474, + 297, + 487 + ], + "score": 0.86, + "content": "\\textstyle { \\frac { 1 } { 4 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 468, + 402, + 493 + ], + "score": 1.0, + "content": "while the upper has one.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 409, + 503 + ], + "score": 1.0, + "content": "We draw several insights from these results. Firstly, although we defined", + "type": "text" + }, + { + "bbox": [ + 410, + 492, + 418, + 501 + ], + "score": 0.8, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "as a single value for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 414, + 515 + ], + "score": 1.0, + "content": "convenience, there may arise cases when other (non-minimum) elements in", + "type": "text" + }, + { + "bbox": [ + 414, + 501, + 422, + 512 + ], + "score": 0.83, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "are important. Sec-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "score": 1.0, + "content": "ondly, perhaps contrary to intuition, the concept of windows and scatter is important for all CLs,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 522, + 507, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 347, + 538 + ], + "score": 1.0, + "content": "not simply the first. As shown in Fig. 7a), a network with", + "type": "text" + }, + { + "bbox": [ + 347, + 523, + 384, + 537 + ], + "score": 0.94, + "content": "\\begin{array} { r } { S _ { 1 b } = \\frac { 1 } { 8 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 522, + 507, + 538 + ], + "score": 1.0, + "content": "performs equally poorly as a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 535, + 507, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 535, + 163, + 552 + ], + "score": 1.0, + "content": "network with", + "type": "text" + }, + { + "bbox": [ + 164, + 536, + 203, + 549 + ], + "score": 0.92, + "content": "\\begin{array} { r } { S _ { 2 f } = ~ \\frac { 1 } { 8 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 535, + 507, + 552 + ], + "score": 1.0, + "content": ". Thirdly, scatter is a sufficient metric for performance, not necessary. A", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 548, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 190, + 560 + ], + "score": 1.0, + "content": "network with a high", + "type": "text" + }, + { + "bbox": [ + 190, + 549, + 198, + 558 + ], + "score": 0.76, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 548, + 443, + 560 + ], + "score": 1.0, + "content": "value will perform well, but a network with a slightly lower", + "type": "text" + }, + { + "bbox": [ + 443, + 548, + 451, + 558 + ], + "score": 0.78, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 548, + 505, + 560 + ], + "score": 1.0, + "content": "than another", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 557, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 485, + 571 + ], + "score": 1.0, + "content": "cannot be conclusively dismissed as being worse. But if a network has multiple low values in", + "type": "text" + }, + { + "bbox": [ + 485, + 558, + 493, + 568 + ], + "score": 0.83, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 557, + 505, + 571 + ], + "score": 1.0, + "content": ", it", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "should be rejected. Finally, carefully choosing which neurons to group in a window will increase the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 580, + 496, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 496, + 593 + ], + "score": 1.0, + "content": "predictive power of scatter. A priori knowledge of the dataset will lead to better window choices.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 608, + 302, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 304, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 304, + 622 + ], + "score": 1.0, + "content": "4 CONCLUSION AND FUTURE WORK", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "This paper discusses the merits of pre-defining sparsity in CLs of neural networks, which leads to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "significant reduction in parameters without performance loss. In general, the smaller the fraction of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "CLs in a network, the more redundancy there exists in their parameters. If we can achieve similar", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 156, + 679 + ], + "score": 1.0, + "content": "results (i.e.,", + "type": "text" + }, + { + "bbox": [ + 156, + 666, + 179, + 676 + ], + "score": 0.85, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 664, + 386, + 679 + ], + "score": 1.0, + "content": "density) on Alexnet for example, we would obtain", + "type": "text" + }, + { + "bbox": [ + 387, + 666, + 407, + 676 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 664, + 505, + 679 + ], + "score": 1.0, + "content": "reduction in overall pa-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "rameters. Coupled with hardware acceleration designed for pre-defined sparse networks, we believe", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "score": 1.0, + "content": "our approach will lead to more aggressive exploration of network structure. Network connectivity", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "can be guided by the scatter metric, which is closely related to performance, and by optimally dis-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "tributing connections across junctions. Future work would involve extension to conv layers, since", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 461, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 461, + 733 + ], + "score": 1.0, + "content": "recent CNNs have lower values for the ratio of number of CLs to number of conv layers.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 292, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 294, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 294, + 95 + ], + "score": 1.0, + "content": "3.2 ANALYSIS AND RESULTS OF SCATTER", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 213 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 447, + 115 + ], + "score": 1.0, + "content": "We ran experiments to evaluate scatter using a) the Morse CL only network with", + "type": "text" + }, + { + "bbox": [ + 447, + 103, + 502, + 114 + ], + "score": 0.88, + "content": "f o \\ = \\ 1 2 8 , 8", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 102, + 506, + 115 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 428, + 127 + ], + "score": 1.0, + "content": "b) an MNIST CL only network with (1024, 64, 16) neuron configuration and", + "type": "text" + }, + { + "bbox": [ + 428, + 114, + 472, + 126 + ], + "score": 0.9, + "content": "f o = 1 , 4", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 114, + 506, + 127 + ], + "score": 1.0, + "content": ", and c)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 145, + 138 + ], + "score": 1.0, + "content": "the ‘conv", + "type": "text" + }, + { + "bbox": [ + 145, + 126, + 176, + 136 + ], + "score": 0.3, + "content": "+ 2 \\mathrm { C L s } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 124, + 279, + 138 + ], + "score": 1.0, + "content": "CIFAR10 network with", + "type": "text" + }, + { + "bbox": [ + 279, + 126, + 322, + 136 + ], + "score": 0.9, + "content": "f o = 1 , 2", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 124, + 506, + 138 + ], + "score": 1.0, + "content": ". We found that high scatter indicates good", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "score": 1.0, + "content": "performance and the correlation is stronger for networks where CLs have more importance, i.e. CL", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 160 + ], + "score": 1.0, + "content": "only networks as opposed to conv. This is shown in the performance vs. scatter plots in Fig. 7,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "where (a) and (b) show the performance predicting ability of scatter better than (c). Note that the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 182 + ], + "score": 1.0, + "content": "random connection patterns used so far have the highest scatter and occur as the rightmost points in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "each subfigure. The other points are obtained by specifically planning connections. We found that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 190, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 372, + 204 + ], + "score": 1.0, + "content": "when 1 junction was planned to give corresponding high values in", + "type": "text" + }, + { + "bbox": [ + 373, + 190, + 381, + 201 + ], + "score": 0.8, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 191, + 505, + 204 + ], + "score": 1.0, + "content": ", it invariably led to low values", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 202, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 255, + 214 + ], + "score": 1.0, + "content": "for another junction, leading to a low", + "type": "text" + }, + { + "bbox": [ + 256, + 202, + 263, + 212 + ], + "score": 0.6, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 202, + 505, + 214 + ], + "score": 1.0, + "content": ". 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All minimum values that need to be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 397, + 375, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 375, + 409 + ], + "score": 1.0, + "content": "considered to differentiate between connection patterns are bolded.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 114, + 429 + ], + "score": 0.82, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 418, + 270, + 432 + ], + "score": 1.0, + "content": "is shown alongside each point. When", + "type": "text" + }, + { + "bbox": [ + 271, + 420, + 278, + 429 + ], + "score": 0.8, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 418, + 505, + 432 + ], + "score": 1.0, + "content": "is equal for different connection patterns, the next min-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 168, + 442 + ], + "score": 1.0, + "content": "imum value in", + "type": "text" + }, + { + "bbox": [ + 168, + 429, + 176, + 440 + ], + "score": 0.82, + "content": "\\breve { \\bar { S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "needs to be considered to differentiate the networks, and so on. Considering the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 438, + 508, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 282, + 457 + ], + "score": 1.0, + "content": "Morse results, the leftmost 3 points all have", + "type": "text" + }, + { + "bbox": [ + 282, + 441, + 309, + 454 + ], + "score": 0.92, + "content": "\\begin{array} { r } { S = \\frac { 1 } { 8 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 438, + 447, + 457 + ], + "score": 1.0, + "content": ", but the number of occurrences of", + "type": "text" + }, + { + "bbox": [ + 447, + 441, + 454, + 454 + ], + "score": 0.86, + "content": "\\frac { 1 } { 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 438, + 465, + 457 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 466, + 441, + 474, + 451 + ], + "score": 0.83, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 438, + 508, + 457 + ], + "score": 1.0, + "content": "is 3 for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 176, + 465 + ], + "score": 1.0, + "content": "the lowest point", + "type": "text" + }, + { + "bbox": [ + 177, + 452, + 191, + 463 + ], + "score": 0.83, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 451, + 335, + 465 + ], + "score": 1.0, + "content": "accuracy), 2 for the second lowest (", + "type": "text" + }, + { + "bbox": [ + 335, + 452, + 354, + 463 + ], + "score": 0.84, + "content": "12 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "accuracy) and 1 for the highest point", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 110, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 110, + 463, + 129, + 473 + ], + "score": 0.84, + "content": "46 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "accuracy). For the MNIST results, both the leftmost points have a single minimum value of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 468, + 402, + 493 + ], + "spans": [ + { + "bbox": [ + 107, + 474, + 118, + 487 + ], + "score": 0.89, + "content": "\\textstyle { \\frac { 1 } { 1 6 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 468, + 129, + 493 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 129, + 473, + 137, + 484 + ], + "score": 0.81, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 468, + 290, + 493 + ], + "score": 1.0, + "content": ", but the lower has two occurrences of", + "type": "text" + }, + { + "bbox": [ + 290, + 474, + 297, + 487 + ], + "score": 0.86, + "content": "\\textstyle { \\frac { 1 } { 4 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 468, + 402, + 493 + ], + "score": 1.0, + "content": "while the upper has one.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 418, + 508, + 493 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 409, + 503 + ], + "score": 1.0, + "content": "We draw several insights from these results. Firstly, although we defined", + "type": "text" + }, + { + "bbox": [ + 410, + 492, + 418, + 501 + ], + "score": 0.8, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "as a single value for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 414, + 515 + ], + "score": 1.0, + "content": "convenience, there may arise cases when other (non-minimum) elements in", + "type": "text" + }, + { + "bbox": [ + 414, + 501, + 422, + 512 + ], + "score": 0.83, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "are important. 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As shown in Fig. 7a), a network with", + "type": "text" + }, + { + "bbox": [ + 347, + 523, + 384, + 537 + ], + "score": 0.94, + "content": "\\begin{array} { r } { S _ { 1 b } = \\frac { 1 } { 8 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 522, + 507, + 538 + ], + "score": 1.0, + "content": "performs equally poorly as a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 535, + 507, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 535, + 163, + 552 + ], + "score": 1.0, + "content": "network with", + "type": "text" + }, + { + "bbox": [ + 164, + 536, + 203, + 549 + ], + "score": 0.92, + "content": "\\begin{array} { r } { S _ { 2 f } = ~ \\frac { 1 } { 8 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 535, + 507, + 552 + ], + "score": 1.0, + "content": ". Thirdly, scatter is a sufficient metric for performance, not necessary. A", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 548, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 190, + 560 + ], + "score": 1.0, + "content": "network with a high", + "type": "text" + }, + { + "bbox": [ + 190, + 549, + 198, + 558 + ], + "score": 0.76, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 548, + 443, + 560 + ], + "score": 1.0, + "content": "value will perform well, but a network with a slightly lower", + "type": "text" + }, + { + "bbox": [ + 443, + 548, + 451, + 558 + ], + "score": 0.78, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 548, + 505, + 560 + ], + "score": 1.0, + "content": "than another", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 557, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 485, + 571 + ], + "score": 1.0, + "content": "cannot be conclusively dismissed as being worse. But if a network has multiple low values in", + "type": "text" + }, + { + "bbox": [ + 485, + 558, + 493, + 568 + ], + "score": 0.83, + "content": "\\bar { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 557, + 505, + 571 + ], + "score": 1.0, + "content": ", it", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "should be rejected. Finally, carefully choosing which neurons to group in a window will increase the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 580, + 496, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 496, + 593 + ], + "score": 1.0, + "content": "predictive power of scatter. A priori knowledge of the dataset will lead to better window choices.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 491, + 507, + 593 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 608, + 302, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 304, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 304, + 622 + ], + "score": 1.0, + "content": "4 CONCLUSION AND FUTURE WORK", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "This paper discusses the merits of pre-defining sparsity in CLs of neural networks, which leads to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "significant reduction in parameters without performance loss. In general, the smaller the fraction of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "CLs in a network, the more redundancy there exists in their parameters. If we can achieve similar", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 156, + 679 + ], + "score": 1.0, + "content": "results (i.e.,", + "type": "text" + }, + { + "bbox": [ + 156, + 666, + 179, + 676 + ], + "score": 0.85, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 664, + 386, + 679 + ], + "score": 1.0, + "content": "density) on Alexnet for example, we would obtain", + "type": "text" + }, + { + "bbox": [ + 387, + 666, + 407, + 676 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 664, + 505, + 679 + ], + "score": 1.0, + "content": "reduction in overall pa-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "rameters. Coupled with hardware acceleration designed for pre-defined sparse networks, we believe", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "score": 1.0, + "content": "our approach will lead to more aggressive exploration of network structure. Network connectivity", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "can be guided by the scatter metric, which is closely related to performance, and by optimally dis-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "tributing connections across junctions. 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This is a result of the network having sufficient density so that several paths exist", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 325, + 300, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 300, + 338 + ], + "score": 1.0, + "content": "from every input neuron to every output neuron.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 109, + 351, + 411, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 412, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 412, + 363 + ], + "score": 1.0, + "content": "5.3 POSSIBLE REASONS FOR SCLS CONVERGING FASTER THAN FCLS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "Training a neural network is essentially an exercise in finding the minimum of the cost function,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 396 + ], + "score": 1.0, + "content": "which is a function of all the network parameters. The graph for cost as a function of parameters may", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "have saddle points which masquerade as minima. It could also be poorly conditioned, wherein the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 403, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 419 + ], + "score": 1.0, + "content": "gradient of cost with respect to two different parameters have widely different magnitudes, making", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "score": 1.0, + "content": "simultaneous optimization difficult. These effects are non-idealities and training the network often", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 427, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 505, + 438 + ], + "score": 1.0, + "content": "takes more time because of the length of the trajectory needed to overcome these and arrive at the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 437, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 449 + ], + "score": 1.0, + "content": "optimum point. 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It is also interesting to note, however, that performance falls off more sharply when", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "junction 1 density is reduced to the bare minimum as compared to treating junction 2 similarly. This", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "is not shown in Fig. 3 due to space constraints. We found that when junction 1 had the minimum", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 504, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 504, + 183 + ], + "score": 1.0, + "content": "possible density and junction 2 had the maximum possible while still satisfying the fixed overall,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 203, + 194 + ], + "score": 1.0, + "content": "the accuracy was about", + "type": "text" + }, + { + "bbox": [ + 204, + 182, + 223, + 192 + ], + "score": 0.86, + "content": "36 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "for both subfigures (b) and (c). 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Thus,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 277, + 283 + ], + "score": 1.0, + "content": "for the equivalent junction 1:2 which has", + "type": "text" + }, + { + "bbox": [ + 278, + 272, + 318, + 282 + ], + "score": 0.89, + "content": "N _ { 1 } = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 270, + 389, + 283 + ], + "score": 1.0, + "content": "left neurons and", + "type": "text" + }, + { + "bbox": [ + 389, + 272, + 429, + 282 + ], + "score": 0.89, + "content": "N _ { 3 } = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "right neurons, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 128, + 294 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 129, + 282, + 181, + 293 + ], + "score": 0.92, + "content": "f o _ { 1 : 2 } > N _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 281, + 200, + 294 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 282, + 251, + 294 + ], + "score": 0.91, + "content": "f i _ { 1 : 2 } > N _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 281, + 505, + 294 + ], + "score": 1.0, + "content": ". 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This is a result of the network having sufficient density so that several paths exist", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 325, + 300, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 300, + 338 + ], + "score": 1.0, + "content": "from every input neuron to every output neuron.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14, + "bbox_fs": [ + 103, + 238, + 509, + 338 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 351, + 411, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 412, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 412, + 363 + ], + "score": 1.0, + "content": "5.3 POSSIBLE REASONS FOR SCLS CONVERGING FASTER THAN FCLS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "Training a neural network is essentially an exercise in finding the minimum of the cost function,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 396 + ], + "score": 1.0, + "content": "which is a function of all the network parameters. 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These effects are non-idealities and training the network often", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 427, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 505, + 438 + ], + "score": 1.0, + "content": "takes more time because of the length of the trajectory needed to overcome these and arrive at the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 437, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 449 + ], + "score": 1.0, + "content": "optimum point. The probability of encountering these non-idealities increases as the number of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "network parameters increase, i.e. less parameters leads to a higher ratio of minima : saddle points,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 459, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "which can make the network converge faster. We hypothesize that SCLs train faster than FCLs due", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 470, + 263, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 263, + 482 + ], + "score": 1.0, + "content": "to the former having fewer parameters.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 371, + 506, + 482 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/BJgPCveAW/BJgPCveAW_model.json b/parse/train/BJgPCveAW/BJgPCveAW_model.json new file mode 100644 index 0000000000000000000000000000000000000000..abd3b0e4a065b7030eb8bc0313b7e7c80f7b000c --- /dev/null +++ b/parse/train/BJgPCveAW/BJgPCveAW_model.json @@ -0,0 +1,18181 @@ +[ + { + "layout_dets": [ + { + "category_id": 1, + "poly": [ + 298, + 1117, + 1404, + 1117, + 1404, + 1483, + 298, + 1483 + ], + "score": 0.984 + }, + { + "category_id": 1, + "poly": [ + 298, + 1499, + 1403, + 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NetJunctionfan-outsCL JunctionDensities (%)Overall CLDensity (%)NetJunctionfan-outsCL JunctionDensities (%)Overall CLDensity (%)
CIFAR10conv+2CLs1,11,80.2,6.30.2,500.220.39CIFAR10conv+3CLs1,1,11,1,81,2,160.2,0.4,6.30.2,0.4,500.2,0.8,1000.220.30.41
CIFAR100conv+2CLs1,11.81,320.2,0.80.2,6.30.2,250.210.380.95CIFAR100conv+3CLs1,1,11,1,161,2,320.2,0.4,0.80.2,0.4,130.2,0.8,250.220.390.59
MNISTconv+2CLs1,54,1016,100.1,500.5,1002,1000.290.832.35MNIST CL(x=224)4,10112,101.79,10050,1003.0250.63
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NetCLs/TotalLayersConvParams(M)ConvOps(B)FC CLParams(M)SparseCL Par-ams (M)OverallParam %ReductionOverallOp%Reduction
Morse CL2/2000.1310.0665050
MNIST CL (x = 224)2/2000.1780.0895050
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CIFAR10 conv+2CLs2/171.150.152.110.00564.631.35
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CIFAR10 conv+3CLs3/181.150.152.230.00965.831.43
CIFAR100 conv+3CLs3/181.150.152.260.01365.991.45
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sha256:add1b11db608ac7c8c7e116c382d8346ac6eabc83ac79b5c739a1ab8025bbb7f +size 7069 diff --git a/parse/train/H1gDNyrKDS/H1gDNyrKDS.md b/parse/train/H1gDNyrKDS/H1gDNyrKDS.md new file mode 100644 index 0000000000000000000000000000000000000000..18610628b995bc0e4940b39cc9a9df5450629486 --- /dev/null +++ b/parse/train/H1gDNyrKDS/H1gDNyrKDS.md @@ -0,0 +1,496 @@ +# UNDERSTANDING AND ROBUSTIFYING DIFFERENTIABLE ARCHITECTURE SEARCH + +Arber $\mathbf { Z e l a } ^ { 1 }$ , Thomas Elsken2,1, Tonmoy Saikia1, Yassine Marrakchi1, +Thomas Brox1 & Frank Hutter1,2 +1Department of Computer Science, University of Freiburg +{zelaa, saikiat, marrakch, brox, fh}@cs.uni-freiburg.de +2Bosch Center for Artificial Intelligence +Thomas.Elsken@de.bosch.com + +# ABSTRACT + +Differentiable Architecture Search (DARTS) has attracted a lot of attention due to its simplicity and small search costs achieved by a continuous relaxation and an approximation of the resulting bi-level optimization problem. However, DARTS does not work robustly for new problems: we identify a wide range of search spaces for which DARTS yields degenerate architectures with very poor test performance. We study this failure mode and show that, while DARTS successfully minimizes validation loss, the found solutions generalize poorly when they coincide with high validation loss curvature in the architecture space. We show that by adding one of various types of regularization we can robustify DARTS to find solutions with less curvature and better generalization properties. Based on these observations, we propose several simple variations of DARTS that perform substantially more robustly in practice. Our observations are robust across five search spaces on three image classification tasks and also hold for the very different domains of disparity estimation (a dense regression task) and language modelling. + +# 1 INTRODUCTION + +Neural Architecture Search (NAS), the process of automatically designing neural network architectures, has recently attracted attention by achieving state-of-the-art performance on a variety of tasks (Zoph & Le, 2017; Real et al., 2019). Differentiable architecture search (DARTS) (Liu et al., 2019) significantly improved the efficiency of NAS over prior work, reducing its costs to the same order of magnitude as training a single neural network. This expanded the scope of NAS substantially, allowing it to also be applied on more expensive problems, such as semantic segmentation (Chenxi et al., 2019) or disparity estimation (Saikia et al., 2019). + +However, several researchers have also reported DARTS to not work well, in some cases even no better than random search (Li & Talwalkar, 2019; Sciuto et al., 2019). Why is this? How can these seemingly contradicting results be explained? The overall goal of this paper is to understand and overcome such failure modes of DARTS. To this end, we make the following contributions: + +1. We identify 12 NAS benchmarks based on four search spaces in which standard DARTS yields degenerate architectures with poor test performance across several datasets (Section 3). +2. By computing the eigenspectrum of the Hessian of the validation loss with respect to the architectural parameters, we show that there is a strong correlation between its dominant eigenvalue and the architecture’s generalization error. Based on this finding, we propose a simple variation of DARTS with early stopping that performs substantially more robustly (Section 4). +3. We show that, related to previous work on sharp/flat local minima, regularizing the inner objective of DARTS more strongly allows it to find solutions with smaller Hessian spectrum and better generalization properties. Based on these insights, we propose two practical robustifications of DARTS that overcome its failure modes in all our 12 NAS benchmarks (Section 5). + +Our findings are robust across a wide range of NAS benchmarks based on image recognition and also hold for the very different domains of language modelling (PTB) and disparity estimation. They consolidate the findings of the various results in the literature and lead to a substantially more robust version of DARTS. We provide our implementation and scripts to facilitate reproducibility1. + +# 2 BACKGROUND AND RELATED WORK + +# 2.1 RELATION BETWEEN FLAT/SHARP MINIMA AND GENERALIZATION PERFORMANCE + +Already Hochreiter & Schmidhuber (1997) observed that flat minima of the training loss yield better generalization performance than sharp minima. Recent work (Keskar et al., 2016; Yao et al., 2018) focuses more on the settings of large/small batch size training, where observations show that small batch training tends to get attracted to flatter minima and generalizes better. Similarly, Nguyen et al. (2018) observed that this phenomenon manifests also in the hyperparameter space. They showed that whenever the hyperparameters overfit the validation data, the minima lie in a sharper region of the space. This motivated us to conduct a similar analysis in the context of differentiable architecture search later in Section 4.1, where we see the same effect in the space of neural network architectures. + +# 2.2 BI-LEVEL OPTIMIZATION + +We start by a short introduction of the bi-level optimization problem (Colson et al., 2007). These are problems which contain two optimization tasks, nested within each other. + +Definition 2.1. Given the outer objective function $F : \mathbb { R } ^ { P } \times \mathbb { R } ^ { N } \to \mathbb { R }$ and the inner objective function $f : \mathbb { R } ^ { P } \times \mathbb { R } ^ { N } \to \mathbb { R }$ , the bi-level optimization problem is given by + +$$ +\begin{array} { r l } & { \underset { y \in \mathbb { R } ^ { P } } { \operatorname* { m i n } } F ( y , \theta ^ { * } ( y ) ) } \\ & { s . t . \quad \theta ^ { * } ( y ) \in \underset { \theta \in \mathbb { R } ^ { N } } { \arg \operatorname* { m i n } } f ( y , \theta ) , } \end{array} +$$ + +where $y \in \mathbb { R } ^ { P }$ and $\boldsymbol { \theta } \in \mathbb { R } ^ { N }$ are the outer and inner variables, respectively. One may also see the bi-level problem as a constrained optimization problem, with the inner problem as a constraint. + +In general, even in the case when the inner objective (2) is strongly convex and has an unique minimizer $\theta ^ { * } ( y ) = \arg \operatorname* { m i n } _ { \theta \in \mathbb { R } ^ { N } } f ( y , \theta )$ , it is not possible to directly optimize the outer objective (1). A possible method around this issue is to use the implicit function theorem to retrieve the derivative of the solution map (or response map) $\theta ^ { * } ( y ) \in \mathbb { F } \subseteq \mathbb { R } ^ { N }$ w.r.t. $y$ (Bengio, 2000; Pedregosa, 2016; Beirami et al., 2017). Another strategy is to approximate the inner problem with a dynamical system (Domke, 2012; Maclaurin et al., 2015; Franceschi et al., 2017; 2018), where the optimization dynamics could, e.g., describe gradient descent. In the case that the minimizer of the inner problem is unique, under some conditions the set of minimizers of this approximate problem will indeed converge to the minimizers of the bilevel problem (1) (see Franceschi et al. (2018)). + +# 2.3 NEURAL ARCHITECTURE SEARCH + +Neural Architecture Search (NAS) denotes the process of automatically designing neural network architectures in order to overcome the cumbersome trial-and-error process when designing architectures manually. We briefly review NAS here and refer to the recent survey by Elsken et al. (2019b) for a more thorough overview. Prior work mostly employs either reinforcement learning techniques (Baker et al., 2017a; Zoph & Le, 2017; Zhong et al., 2018; Zoph et al., 2018) or evolutionary algorithms (Stanley & Miikkulainen, 2002; Liu et al., 2018b; Miikkulainen et al., 2017; Real et al., 2017; 2019) to optimize the discrete architecture space. As these methods are often very expensive, various works focus on reducing the search costs by, e.g., employing network morphisms (Cai et al., 2018a;b; Elsken et al., 2017; 2019a), weight sharing within search models (Saxena & Verbeek, 2016; Bender et al., 2018; Pham et al., 2018) or multi-fidelity optimization (Baker et al., 2017b; Falkner et al., 2018; Li et al., 2017; Zela et al., 2018), but their applicability still often remains restricted to rather simple tasks and small datasets. + +# 2.4 DIFFERENTIABLE ARCHITECTURE SEARCH (DARTS) + +A recent line of work focuses on relaxing the discrete neural architecture search problem to a continuous one that can be solved by gradient descent (Liu et al., 2019; Xie et al., 2019; Casale et al., 2019; Cai et al., 2019). In DARTS (Liu et al., 2019), this is achieved by simply using a weighted sum of possible candidate operations for each layer, whereas the real-valued weights then effectively parametrize the network’s architecture. We will now review DARTS in more detail, as our work builds directly upon it. + +Continuous relaxation of the search space. In agreement with prior work (Zoph et al., 2018; Real et al., 2019), DARTS optimizes only substructures called cells that are stacked to define the full network architecture. Each cell contains $N$ nodes organized in a directed acyclic graph. The graph contains two inputs nodes (given by the outputs of the previous two cells), a set of intermediate nodes, and one output node (given by concatenating all intermediate nodes). Each intermediate node $x ^ { ( j ) }$ represents a feature map. See Figure 1 for an illustration of such a cell. Instead of applying a single operation to a specific node during architecture search, Liu et al. (2019) relax the decision which operation to choose by computing the intermediate node as a mixture of candidate operations, applied to predecessor nodes x(i), i < j, x(j) = PiBenchmarkDARTSDARTS-ESC10S14.66 ± 0.713.05± 0.07S24.42 ± 0.403.41 ± 0.14S34.12 ± 0.853.71 ± 1.14S46.95±0.184.17 ± 0.21C100S129.93 ± 0.4128.90±0.81S228.75±0.9224.68±1.43S329.01 ± 0.2426.99 ± 1.79S424.77 ± 1.5123.90±2.01SVHNS19.88±5.502.80±0.09S23.69 ±0.122.68± 0.18S34.00 ± 1.012.78± 0.29S42.90±0.022.55±0.15 + +# 5 REGULARIZATION OF INNER OBJECTIVE IMPROVES GENERALIZATION OF ARCHITECTURES + +As we saw in Section 4.1, sharper minima (by means of large eigenvalues) of the validation loss lead to poor generalization performance. In our bi-level optimization setting, the outer variables’ trajectory depends on the inner optimization procedure. Therefore, we hypothesized that modifying the landscape of the inner objective $\mathcal { L } _ { t r a i n }$ could redirect the outer variables $\alpha$ to flatter areas of the architectural space. We study two ways of regularization (data augmentation in Section 5.1 and $L _ { 2 }$ regularization in Section 5.2) and find that both, along with the early stopping criterion from Section 4.3, make DARTS more robust in practice. We emphasize that we do not alter the regularization of the final training and evaluation phase, but solely that of the search phase. The setting we use for all experiments in this paper to obtain the final test performance is described in Appendix C. + +# 5.1 REGULARIZATION VIA DATA AUGMENTATION + +We first investigate the effect of regularizing via data augmentation, namely masking out parts of the input and intermediate feature maps via Cutout (CO, DeVries & Taylor (2017)) and ScheduledDropPath (DP, Zoph et al. (2018)) (ScheduledDropPath is a regularization technique, but we list it here since we apply it together with Cutout), respectively, during architecture search. We ran DARTS with CO and DP (with and without our early stopping criterion, DARTS-ES) with different maximum DP probabilities on all three image classification datasets and search spaces S1-S4. + +Figure 7 summarizes the results: regularization improves the test performance of DARTS and DARTS-ES in all cases, sometimes very substantially, and at the same time kept the dominant eigenvalue relatively low (Figure 13). This also directly results in smaller drops in accuracy after pruning, as discussed in Section 4.2; indeed, the search runs plotted in Figure 5b are the same as in this section. Figure 17 in the appendix explicitly shows how regularization relates to the accuracy drops. We also refer to further results in the appendix: Figure 11 (showing test vs. validation error) and Table 5 (showing that overfitting of the architectural parameters is reduced). + +![](images/1062267b762cf2a8dad20f3863189043a16c12493b2f4529678c55858bfe3901.jpg) +Figure 8: Effect of $L _ { 2 }$ regularization of the inner objective during architecture search for DARTS (solid lines) and DARTS-ES (dashed). + +Similar observations hold for disparity estimation on S6, where we vary the strength of standard data augmentation methods, such as shearing or brightness change, rather then masking parts of features, which is unreasonable for this task. The augmentation strength is described by an “augmentation scaling factor” (Appendix E). Table 2 summarizes the results. We report the average end point error (EPE), which is the Euclidean distance between the predicted and ground truth disparity maps. Data augmentation avoided the degenerate architectures and substantially improved results. + +# 5.2 INCREASED $L _ { 2 }$ REGULARIZATION + +As a second type of regularization, we also tested different $L _ { 2 }$ regularization factors $3 i \cdot 1 0 ^ { - 4 }$ for $i \in$ $\{ 1 , 3 , 9 , 2 7 , 8 1 \}$ . Standard DARTS in fact does already include a small amount of $L _ { 2 }$ regularization; $i = 1$ yields its default. Figure 8 shows that DARTS’ test performance (solid lines) can be significantly improved by higher $L _ { 2 }$ factors across all datasets and spaces, while keeping the dominant eigenvalue low (Figure 14). DARTS with early stopping (dashed lines) also benefits from additional regularization. Again, we observe the implicit regularization effect on the outer objective which reduces the overfitting of the architectural parameters. We again refer to + +Table 2: Effect of regularization for disparity estimation. Search was conducted on FlyingThings3D (FT) and then evaluated on both FT and Sintel. Lower is better. + +
Aug. ScaleSearchmodel valid EPEFT test EPESintel test EPEParams (M)
0.04.493.835.699.65
0.13.533.755.979.65
0.53.283.375.229.43
1.04.613.125.4712.46
1.55.232.604.1512.57
2.07.452.333.7612.25
L2 reg. factorSearchmodel validFT test EPESintel testParams
3×10-4EPE 3.953.25EPE 6.13(M) 11.00
9×10-45.972.304.1213.92
27×10-44.252.724.8310.29
81×10-44.612.343.8512.16
+ +Table 2 for disparity estimation; Appendix F shows similar results for language modelling (Penn TreeBank). + +# 5.3 PRACTICAL ROBUSTIFICATION OF DARTS BY REGULARIZING THE INNER OBJECTIVE + +Based on the insights from the aforementioned analysis and empirical results, we now propose two alternative simple modifications to make DARTS more robust in practice without having to manually tune its regularization hyperparameters. + +DARTS with adaptive regularization One option is to adapt DARTS’ regularization hyperparameters in an automated way, in order to keep the architectural weights in areas of the validation loss objective with smaller curvature. The simplest off-the-shelf procedure towards this desiderata would be to increase the regularization strength whenever the dominant eigenvalue starts increasing rapidly. Algorithm 1 (DARTS-ADA, Appendix D.1) shows such a procedure. We use the same stopping criterion as in DARTS-ES (Section 4.3), roll back DARTS to the epoch when this criterion is met, and continue the search with a larger regularization value $R$ for the remaining epochs (larger by a factor of $\eta$ ). This procedure is repeated whenever the criterion is met, unless the regularization value exceeds some maximum predefined value $R _ { m a x }$ . + +Multiple DARTS runs with different regularization strength Liu et al. (2019) already suggested to run the search phase of DARTS four times, resulting in four architectures, and to return the best of these four architectures w.r.t. validation performance when retrained from scratch for a limited number of epochs. We propose to use the same procedure, with the only difference that the four runs use different amounts of regularization. The resulting RobustDARTS (R-DARTS) method is conceptually very simple, trivial to implement and likely to work well if any of the tried regularization strengths works well. + +Table 3 evaluates the performance of our practical robustifications of DARTS, DARTS-ADA and R-DARTS (based on either L2 or ScheduledDropPath regularization), by comparing them to the original DARTS, DARTS-ES and Random Search with weight sharing (RS-ws). For each of these methods, as proposed in the DARTS paper (Liu et al., 2019), we ran the search four independent times with different random seeds and selected the architecture used for the final evaluation based on a validation run as described above. + +Table 3: Empirical evaluation of practical robustified versions of DARTS. Each entry is the test error after retraining the selected architecture as usual. The best method for each setting is boldface and underlined, the second best boldface. + +
BenchmarkRS-wsDARTSR-DARTS(DP)R-DARTS(L2)DARTS-ESDARTS-ADA
C10S13.233.843.112.783.013.10
S23.664.853.483.313.263.35
S32.953.342.932.512.742.59
S48.077.203.583.563.714.84
C100S123.3029.4625.9324.2528.3724.03
S221.2126.0522.3022.2423.2523.52
S323.7528.9022.3623.9923.7323.37
S428.1922.8522.1821.9421.2623.20
SVHNS12.594.582.554.792.722.53
S22.723.532.522.512.602.54
S32.873.412.492.482.502.50
+ +As the table shows, in accordance with Li & Talwalkar (2019), RS-ws often outperformed the original DARTS; however, with our robustifications, DARTS typically performs substantially better than RS-ws. DARTS-ADA consistently improved over standard DARTS for all benchmarks, indicating that a gradual increase of regularization during search prevents ending up in the bad regions of the architectural space. Finally, RobustDARTS yielded the best performance and since it is also easier to implement than DARTS-ES and DARTS-ADA, it is the method that we recommend to be used in practice. + +Finally, since the evaluations in this paper have so far focussed on smaller subspaces of the original DARTS search space, the reader may wonder how well RobustDARTS works on the full search spaces. As Table 4 shows, RobustDARTS performed similarly to DARTS for the two original benchmarks from the DARTS paper (PTB and CIFAR-10), on which DARTS was developed and is well tuned; however, even when only changing the dataset to CIFAR-100 or SVHN, RobustDARTS already performed significantly better than DARTS, underlining its robustness. + +Table 4: DARTS vs. RobustDARTS on the original DARTS search spaces. We show mean $\pm$ stddev for 5 repetitions (based on 4 fresh subruns each as in Table 3); for the more expensive PTB we could only afford 1 such repetition. + +
BenchmarkDARTSR-DARTS(L2)
C102.91± 0.252.95 ± 0.21
C10020.58 ± 0.4418.01 ± 0.26
SVHN2.46±0.092.17 ± 0.09
PTB58.6457.59
+ +# 6 CONCLUSIONS + +We showed that the generalization performance of architectures found by DARTS is related to the eigenvalues of the Hessian matrix of the validation loss w.r.t. the architectural parameters. Standard DARTS often results in degenerate architectures with large eigenvalues and poor generalization. Based on this observation, we proposed a simple early stopping criterion for DARTS based on tracking the largest eigenvalue. Our empirical results also show that properly regularizing the inner objective helps controlling the eigenvalue and therefore improves generalization. Our findings substantially improve our understanding of DARTS’ failure modes and lead to much more robust versions. They are consistent across many different search spaces on image recognition tasks and also for the very different domains of language modelling and disparity estimation. Our code is available for reproducibility. + +# ACKNOWLEDGMENTS + +The authors acknowledge funding by the Robert Bosch GmbH, support by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme through grant no. 716721, and by BMBF grant DeToL. + +# REFERENCES + +Bowen Baker, Otkrist Gupta, Nikhil Naik, and Ramesh Raskar. Designing neural network architectures using reinforcement learning. In International Conference on Learning Representations, 2017a. + +Bowen Baker, Otkrist Gupta, Ramesh Raskar, and Nikhil Naik. Accelerating Neural Architecture Search using Performance Prediction. In NIPS Workshop on Meta-Learning, 2017b. + +Ahmad Beirami, Meisam Razaviyayn, Shahin Shahrampour, and Vahid Tarokh. On optimal generalizability in parametric learning. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances in Neural Information Processing Systems 30, pp. 3455–3465. 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In International Conference on Learning Representations, 2017. + +Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V. Le. Learning transferable architectures for scalable image recognition. In Conference on Computer Vision and Pattern Recognition, 2018. + +# A MORE DETAIL ON DARTS + +Here we present a detailed description of DARTS architectural update steps. We firstly provide the general formalism which computes the gradient of the outer level problem in (1) by means of the implicit function theorem. Afterwards, we present how DARTS computes the gradient used to update the architectural parameters $\alpha$ . + +# A.1 DERIVATIVE WITH SMOOTHED NON-QUADRATIC LOWER LEVEL PROBLEM + +Consider the general definition of the bi-level optimization problem as given by (1) and (2). Given that $f$ is twice continuously differentiable and that all stationary points are local minimas, one can make use of the implicit function theorem to find the derivative of the solution map $\theta ^ { * } ( y )$ w.r.t. $y$ (Bengio, 2000). Under the smoothness assumption, the optimality condition of the lower level (2) is $\nabla _ { \boldsymbol { \theta } } f ( y , \boldsymbol { \theta } ) = \mathbf { 0 }$ , which defines an implicit function $\theta ^ { * } ( y )$ . With the assumption that $\mathrm { m i n } _ { \boldsymbol { \theta } } f ( \boldsymbol { y } , \boldsymbol { \theta } )$ has a solution, there exists a $( y , \theta ^ { * } )$ such that $\nabla _ { \theta } f ( y , \theta ^ { * } ) = \mathbf { 0 }$ . Under the condition that $\nabla _ { \theta } f ( y , \theta ^ { * } ) = 0$ is continuously differentiable and that $\theta ^ { * } ( y )$ is continuously differentiable at $y$ , implicitly differentiating the last equality from both sides w.r.t. y and applying the chain rule, yields: + +$$ +\frac { \partial ( \nabla _ { \theta } f ) } { \partial \theta } ( y , \theta ^ { * } ) \cdot \frac { \partial \theta ^ { * } } { \partial y } ( y ) + \frac { \partial ( \nabla _ { \theta } f ) } { \partial y } ( y , \theta ^ { * } ) = { \bf 0 } . +$$ + +Assuming that the Hessian $\nabla _ { { \theta } } ^ { 2 } f ( y , { \theta } ^ { * } )$ is invertible, we can rewrite (3) as follows: + +$$ +\frac { \partial \theta ^ { * } } { \partial y } ( y ) = - \Big ( \nabla _ { \theta } ^ { 2 } f ( y , \theta ^ { * } ) \Big ) ^ { - 1 } \cdot \frac { \partial ( \nabla _ { \theta } f ) } { \partial y } ( y , \theta ^ { * } ) . +$$ + +Applying the chain rule to (1) for computing the total derivative of $F$ with respect to $y$ yields: + +$$ +\frac { d F } { d y } = \frac { \partial F } { \partial \theta } \cdot \frac { \partial \theta ^ { * } } { \partial y } + \frac { \partial F } { \partial y } , +$$ + +where we have omitted the evaluation at $( y , \theta ^ { * } )$ . Substituting (4) into (5) and reordering yields: + +$$ +\frac { d F } { d y } = \frac { \partial F } { \partial y } - \frac { \partial F } { \partial \theta } \cdot \left( \nabla _ { \theta } ^ { 2 } f \right) ^ { - 1 } \cdot \frac { \partial ^ { 2 } f } { \partial \theta \partial y } . +$$ + +equation 6 computes the gradient of $F$ , given the function $\theta ^ { * } ( y )$ , which maps outer variables to the inner variables minimizing the inner problem. However, in most of the cases obtaining such a mapping is computationally expensive, therefore different heuristics have been proposed to approximate $d F / d y$ (Maclaurin et al., 2015; Pedregosa, 2016; Franceschi et al., 2017; 2018). + +# A.2 DARTS ARCHITECTURAL GRADIENT COMPUTATION + +DARTS optimization procedure is defined as a bi-level optimization problem where $\mathcal { L } _ { v a l i d }$ is the outer objective (1) and $\mathcal { L } _ { t r a i n }$ is the inner objective (2): + +$$ +\begin{array} { r l } & { \underset { \alpha } { \operatorname* { m i n } } \mathcal { L } _ { v a l i d } ( \alpha , w ^ { * } ( \alpha ) ) } \\ & { s . t . \quad w ^ { * } ( \alpha ) = \underset { w } { \arg \operatorname* { m i n } } \mathcal { L } _ { t r a i n } ( \alpha , w ) , } \end{array} +$$ + +where both losses are determined by both the architecture parameters $\alpha$ (outer variables) and the network weights $w$ (inner variables). Based on Appendix A.1, under some conditions, the total derivative of $\mathcal { L } _ { v a l i d }$ w.r.t. $\alpha$ evaluated on $( \alpha , w ^ { * } ( \alpha ) )$ would be: + +$$ +\frac { d \mathcal { L } _ { v a l i d } } { d \alpha } = \nabla _ { \alpha } \mathcal { L } _ { v a l i d } - \nabla _ { w } \mathcal { L } _ { v a l i d } \big ( \nabla _ { w } ^ { 2 } \mathcal { L } _ { t r a i n } \big ) ^ { - 1 } \nabla _ { \alpha , w } ^ { 2 } \mathcal { L } _ { t r a i n } , +$$ + +where ∇α = , $\begin{array} { r } { \nabla _ { w } = \frac { \partial } { \partial w } } \\ { . } \end{array}$ and $\begin{array} { r } { \nabla _ { \alpha , w } ^ { 2 } = \frac { \partial ^ { 2 } } { \partial \alpha \partial w } } \end{array}$ = ∂2∂α∂w . Computing the inverse of the Hessian is in general not possible considering the high dimensionality of the model parameters $w$ , therefore resolving to gradient-based iterative algorithms for finding $w ^ { * }$ is necessary. However, this would also require to optimize the model parameters $w$ till convergence each time $\alpha$ is updated. If our model is a deep neural network it is clear that this computation is expensive, therefore Liu et al. (2019) propose to approximate $w ^ { * } ( \alpha )$ by updating the current model parameters $w$ using a single gradient descent step: + +$$ +\begin{array} { r } { w ^ { * } ( \alpha ) \approx w - \xi \nabla _ { w } \mathcal { L } _ { t r a i n } ( \alpha , w ) , } \end{array} +$$ + +where $\xi$ is the learning rate for the virtual gradient step DARTS takes with respect to the model weights $w$ . From equation 10 the gradient of $w ^ { * } ( \alpha )$ with respect to $\alpha$ is + +$$ +\frac { \partial w ^ { * } } { \partial \alpha } ( \alpha ) = - \xi \nabla _ { \alpha , w } ^ { 2 } \mathcal { L } _ { t r a i n } ( \alpha , w ) , +$$ + +By setting the evaluation point $\boldsymbol { w ^ { * } } = \boldsymbol { w } - \xi \nabla _ { \boldsymbol { w } } \mathcal { L } _ { t r a i n } ( \alpha , \boldsymbol { w } )$ and following the same derivation as in Appendix A.1, we obtain the DARTS architectural gradient approximation: + +$$ +\frac { d \mathcal { L } _ { v a l i d } } { d \alpha } ( \alpha ) = \nabla _ { \alpha } \mathcal { L } _ { v a l i d } ( \alpha , w ^ { * } ) - \xi \nabla _ { w } \mathcal { L } _ { v a l i d } ( \alpha , w ^ { * } ) \nabla _ { \alpha , w } ^ { 2 } \mathcal { L } _ { t r a i n } ( \alpha , w ^ { * } ) , +$$ + +where the inverse Hessian however contains again an e $\nabla _ { w } ^ { 2 } \mathcal { L } _ { t r a i n } ^ { - 1 }$ in (9) is replaced by the learning rate ector-matrix product. Liu et al. (2019) re $\xi$ . This expressionuce the complexity by using the finite difference approximation around $w ^ { \pm } = w \pm \epsilon \nabla _ { w } \mathcal { L } _ { v a l i d } ( \alpha , w ^ { * } )$ for some small $\epsilon = 0 . 0 1 / \left. \nabla _ { w } \mathcal { L } _ { v a l i d } ( \alpha , w ^ { * } ) \right. _ { 2 }$ to compute the gradient of $\nabla _ { \alpha } \mathcal { L } _ { t r a i n } ( \alpha , w ^ { * } )$ with respect to $w$ as + +$$ +\begin{array} { r l r } & { } & { \nabla _ { \alpha , w } ^ { 2 } \mathcal { L } _ { t r a i n } ( \alpha , w ^ { * } ) \approx \frac { \nabla _ { \alpha } \mathcal { L } _ { t r a i n } ( \alpha , w ^ { + } ) - \nabla _ { \alpha } \mathcal { L } _ { t r a i n } ( \alpha , w ^ { - } ) } { 2 \epsilon \nabla _ { w } \mathcal { L } _ { v a l i d } ( \alpha , w ^ { * } ) } \qquad \Leftrightarrow } \\ & { } & { \nabla _ { w } \mathcal { L } _ { v a l i d } ( \alpha , w ^ { * } ) \nabla _ { \alpha , w } ^ { 2 } \mathcal { L } _ { t r a i n } ( \alpha , w ^ { * } ) \approx \frac { \nabla _ { \alpha } \mathcal { L } _ { t r a i n } ( \alpha , w ^ { + } ) - \nabla _ { \alpha } \mathcal { L } _ { t r a i n } ( \alpha , w ^ { - } ) } { 2 \epsilon } . } \end{array} +$$ + +In the end, combining equation 12 and equation 13 gives the gradient to compute the architectural updates in DARTS: + +$$ +\frac { d \mathcal { L } _ { v a l i d } } { d \alpha } ( \alpha ) = \nabla _ { \alpha } \mathcal { L } _ { v a l i d } ( \alpha , w ^ { * } ) - \frac { \xi } { 2 \epsilon } \big ( \nabla _ { \alpha } \mathcal { L } _ { t r a i n } ( \alpha , w ^ { + } ) - \nabla _ { \alpha } \mathcal { L } _ { t r a i n } ( \alpha , w ^ { - } ) \big ) +$$ + +In all our experiments we always use $\xi = \eta$ (also called second order approximation in Liu et al. +(2019)), where $\eta$ is the learning rate used in SGD for updating the parameters $w$ . + +# B CONSTRUCTION OF S1 FROM SECTION 3 + +We ran DARTS two times on the default search space to find the two most important operations per mixed operation. Initially, every mixed operation consists of 8 operations. After the first DARTS run, we drop the 4 (out of 8) least important ones. In the second DARTS run, we drop the 2 (out of the remaining 4) least important ones. S1 is then defined to contain only the two remaining most important operations per mixed op. Refer to Figure 9 for an illustration of this pre-optimized space. + +# C FINAL ARCHITECTURE EVALUATION + +Similar to the original DARTS paper (Liu et al., 2019), the architecture found during the search are scaled up by increasing the number of filters and cells and retrained from scratch to obtain the final test performance. For CIFAR-100 and SVHN we use 16 number of initial filters and 8 cells when training architectures from scratch for all the experiments we conduct. The rest of the settings is the same as in Liu et al. (2019). + +On CIFAR-10, when scaling the ScheduledDropPath drop probability, we use the same settings for training from scratch the found architectures as in the original DARTS paper, i.e. 36 initial filters and 20 stacked cells. However, for search space S2 and S4 we reduce the number of initial filters to 16 in order to avoid memory issues, since the cells found with more regularization usually are composed only with separable convolutions. When scaling the $L _ { 2 }$ factor on CIFAR-10 experiments we use 16 initial filters and 8 stacked cells, except the experiments on S1, where the settings are the same as in Liu et al. (2019), i.e. 36 initial filters and 20 stacked cells. + +![](images/ce86ca890ed3cbc8354ebd6d805b89b039a842003ed876447bfb906d9ff4525b.jpg) +(a) Normal cell space + +![](images/a5e9654b804ac972433196284b72635adb5df60dff44aee7a1e0f5c83a924f18.jpg) +Figure 9: Search space S1. + +Note that although altering the regularization factors during DARTS search, when training the final architectures from scratch we always use the same values for them as in Liu et al. (2019), i.e. ScheduledDropPath maximum drop probability linearly increases from 0 towards 0.2 throughout training, Cutout is always enabled with cutout probability 1.0, and the $L _ { 2 }$ regularization factor is set to $3 \cdot 1 \bar { 0 } ^ { - 4 }$ . + +# D ADDITIONAL EMPIRICAL RESULTS + +![](images/b7331d112ac5bc032e8b79382b010517d1b6290a6d78e201ab163ee77534f5b6.jpg) +Figure 10: Test regret and validation error of the search (one-shot) model when running DARTS on S5 and CIFAR-10 with different $L _ { 2 }$ regularization values. The architectural parameters’ overfit reduces as we increase the $L _ { 2 }$ factor and successfully finds the global minimum. However, we notice that the architectural parameters start underfitting as we increase to much the $L _ { 2 }$ factor, i.e. both validation and test error increase. + +Table 5: Validation (train) and test accuracy on CIFAR-10 of the search and final evaluation models, respectively. The values in the last column show the maximum eigenvalue $\lambda _ { m a x } ^ { \alpha }$ (computed on a random sampled mini-batch) of the Hessian, at the end of search for different maximum drop path probability). The four blocks in the table state results for the search spaces S1-S4, respectively. + +
Drop Prob.Valid acc.Test acc.ParamsXmax
seed 1seed2seed 3seed 1seed 2seed 3seed 1seed 2seed 3seed 1seed 2seed 3
S10.087.2287.0186.9896.1694.4395.432.241.932.031.0230.8350.698
0.284.2484.3284.2296.3996.6696.202.632.842.480.1480.2640.228
0.482.2882.1882.7996.4496.9496.762.632.993.170.1920.1990.149
0.679.1779.1878.8496.8996.9396.963.383.023.170.3000.2550.256
S20.088.4988.4088.3595.1595.4896.110.930.860.970.6840.4090.268
0.285.2984.8185.3695.1595.4096.141.281.441.360.2700.2170.145
0.482.0382.6683.2096.3496.5096.441.281.281.360.3040.4110.282
0.679.8680.1979.7096.5296.3596.291.211.281.360.2920.2950.281
S30.088.7889.1588.6794.7096.2796.662.212.432.850.4960.5350.446
0.285.6185.6085.5096.7896.8496.743.624.042.990.1790.1850.202
0.483.0383.2483.4397.0796.8596.484.103.743.380.1560.3700.184
0.679.8680.0379.6896.9194.5696.444.462.302.660.2390.2750.280
S40.086.3386.7286.4692.8093.2293.141.051.131.050.4000.4420.314
0.281.0182.4382.0395.8496.0896.151.441.441.440.0700.0540.079
0.479.4979.6778.9696.1196.3096.281.441.441.440.0640.0570.049
0.674.5474.7474.3796.4296.3696.641.441.441.440.0570.0600.066
+ +# D.1 ADAPTIVE DARTS DETAILS + +We evaluated DARTS-ADA (Section 5.3) with $R = 3 \cdot 1 0 ^ { - 4 }$ (DARTS default), $R _ { m a x } = 3 \cdot 1 0 ^ { - 2 }$ and $\eta = 1 0$ on all the search spaces and datasets we use for image classification. The results are shown in Table 3 (DARTS-ADA). The function train and eval conducts the normal DARTS search for one epoch and returns the architecture at the end of that epoch’s updates and the stop value if a decision was made to stop the search and rollback to stop epoch. + +# Algorithm 1: DARTS ADA + +/\* E: epochs to search; $R$ : initial regularization value; $R _ { m a x }$ : maximal regularization value; stop criter: stopping criterion; η: regularization increase factor \*/ +Input : E, $R$ , $R _ { m a x }$ , stop criter, η +$/ \star$ start search for E epochs \*/ +for epoch in $E$ do $/ \star$ run DARTS for one epoch and return stop $^ { \prime = }$ True together with the stop epoch \*/ $/ \star$ and the architecture at stop epoch if the criterion is met \*/ stop, stop epoch, arch train and eval(stop criter); if stop & $R \leq R _ { m a x }$ then /\* start DARTS from stop epoch with a larger R \*/ arch DARTS ADA(E - stop epoch, $\eta \cdot R$ , $R _ { m a x }$ , stop criter, η); break end +end + +![](images/f0a0895953ddcdd46fb30b08746b7b72292a9beed0d927ed27dd155abe25308d.jpg) +Output: arch +Figure 11: Test errors of architectures along with the validation error of the search (one-shot) model for each dataset and space when scaling the ScheduledDropPath drop probability. Note that these results (blue lines) are the same as the ones in Figure 8. + +![](images/b9aa25ab092484d336484395f65efd065feeff22d4645f14b4efdef49086e3d6.jpg) +Figure 12: Test errors of architectures along with the validation error of the search (one-shot) model for each dataset and space when scaling the $L _ { 2 }$ factor. Note that these results (blue lines) are the same as the ones in Figure 7. + +![](images/7b24bf051f9ab95335af0577fc8ce3ab713a7df590d547f94c17ed9aa84d1ef7.jpg) +Figure 13: Local average of the dominant EV $\lambda _ { m a x } ^ { \alpha }$ throughout DARTS search (for different drop path prob. values). Markers denote the early stopping point based on the criterion in Section 4.3. + +![](images/f6e2cb8af3ff642c0a7635b1e698cf0276ab7989535f7d5fa86f472276b49889.jpg) +Figure 14: Effect of $L _ { 2 }$ regularization no the EV trajectory. The figure is analogous to Figure 13. + +![](images/af9a3cc0a9a9f1fd276ef24fd923c38fefc410ade2e8c61a48b55e7b7f81feea.jpg) +Figure 15: Effect of ScheduledDropPath and Cutout on the full eigenspectrum of the Hessian at the end of architecture search for each of the search spaces. Since most of the eigenvalues after the 30-th largest one are almost zero, we plot only the largest (based on magnitude) 30 eigenvalues here. We also provide the eigenvalue distribution for these 30 eigenvalues. Notice that not only the dominant eigenvalue is larger when $d p = 0$ but in general also the others. + +![](images/2d016717a63594c416f146c80f2aaf320529f86b2416c6a34563deb2a3f22f66.jpg) +Figure 16: Effect of $L _ { 2 }$ regularization on the full eigenspectrum of the Hessian at the end of architecture search for each of the search spaces. Since most of the eigenvalues after the 30-th largest one are almost zero, we plot only the largest (based on magnitude) 30 eigenvalues here. We also provide the eigenvalue distribution for these 30 eigenvalues. Notice that not only the dominant eigenvalue is larger when $L _ { 2 } = 3 \cdot 1 0 ^ { - 4 }$ but in general also the others. + +![](images/9527133a63f756f7d48e5e25490a96e5822d103e13bd78b213a2da457e61a4f1.jpg) +Figure 17: Drop in accuracy after discretizing the search model for different spaces, datasets and drop path regularization strengths.. Example of some of the settings from Section 5. + +![](images/7b67e03e4688728a0465362fc8a588136fd41fb25d64c359054fd5106996e48b.jpg) +Figure 18: Effect of more regularization on the performance of found architectures by DARTS. + +Table 6: Performance of architectures found by DARTS (-ES / -ADA) vs. RandomNAS with weight sharing. For each of the settings we repeat the search 3 times and report the mean $\pm$ std of the 3 found architectures retrained from scratch. + +
SettingRandomNASDARTSDARTS-ESDARTS-ADA
C10S13.17 ± 0.154.66 ± 0.713.05 ± 0.073.03 ±0.08
S23.46 ± 0.154.42 ± 0.403.41 ± 0.143.59 ± 0.31
S32.92 ± 0.044.12 ± 0.853.71 ± 1.142.99 ± 0.34
S489.39 ± 0.846.95±0.184.17 ± 0.213.89 ± 0.67
C100S125.81 ± 0.3929.93 ± 0.4128.90 ± 0.8124.94 ± 0.81
S222.88 ± 0.1628.75 ± 0.9224.68 ± 1.4326.88 ± 1.11
S324.58 ± 0.6129.01 ± 0.2426.99 ± 1.7924.55± 0.63
S430.01 ± 1.5224.77 ± 1.5123.90 ± 2.0123.66 ± 0.90
SVHNS12.64±0.099.88 ± 5.502.80± 0.092.59± 0.07
S22.57 ± 0.043.69 ± 0.122.68 ± 0.182.79 ± 0.22
S32.89±0.094.00 ± 1.012.78± 0.292.58 ± 0.07
S43.42 ± 0.042.90± 0.022.55 ± 0.152.52 ± 0.06
+ +# D.2 A CLOSER LOOK AT THE EIGENVALUES + +Over the course of all experiments from the paper, we tracked the largest eigenvalue across all configuration and datasets to see how they evolve during the search. Figures 13 and 14 shows the results across all the settings for image classification. It can be clearly seen that increasing the inner objective regularization, both in terms of $L _ { 2 }$ or data augmentation, helps controlling the largest eigenvalue and keeping it to a small value, which again helps explaining why the architectures found with stronger regularization generalize better. The markers on each line highlight the epochs where DARTS is early stopped. As one can see from Figure 4, there is indeed some correlation between the average dominant eigenvalue throughout the search and the test performance of the found architectures by DARTS. + +Figures 15 and 16 (top 3 rows) show the full spectrum (sorted based on eigenvalue absolute values) at the end of search, whilst bottom 3 rows plot the distribution of eigenvalues in the eigenspectrum. As one can see, not only the dominant eigenvalue is larger compared to the cases when the regularization is stronger and the generalization of architectures is better, but also the other eigenvalues in the spectrum have larger absolute value, indicating a sharper objective landscape towards many dimensions. Furthermore, from the distribution plots note the presence of more negative eigenvalues whenever the architectures are degenerate (lower regularization value) indicating that DARTS gets stuck in a point with larger positive and negative curvature of the validation loss objective, associated with a more degenerate Hessian matrix. + +# E DISPARITY ESTIMATION + +# E.1 DATASETS + +We use the FlyingThings3D dataset (Mayer et al., 2016) for training AutoDispNet. It consists of rendered stereo image pairs and their ground truth disparity maps. The dataset provides a training and testing split consisting of 21, 818 and 4248 samples respectively with an image resolution of $9 6 0 \times 5 4 0$ . We use the Sintel dataset ( Butler et al. (2012)) for testing our networks. Sintel is another synthetic dataset from derived from an animated movie which also provides ground truth disparity maps (1064 samples) with a resolution of $1 0 2 4 \times 4 3 6$ . + +# E.2 TRAINING + +We use the AutoDispNet-C architecture as described in Saikia et al. (2019). However, we use the smaller search which consists of three operations: $M a x P o o l 3 \times 3$ , $S e p C o n v 3 \times 3$ , and SkipConnect. For training the search network, images are downsampled by a factor of two and trained for $3 0 0 k$ mini-batch iterations. During search, we use SGD and ADAM to optimize the inner and outer objectives respectively. Differently from the original AutoDispNet we do not warmstart the search model weights before starting the architectural parameter updates. The extracted network is also trained for $3 0 0 k$ mini-batch iterations but full resolution images are used. Here, ADAM is used for optimization and the learning rate is annealed to 0 from $1 e - 4$ , using a cosine decay schedule. + +# E.3 EFFECT OF REGULARIZATION ON THE INNER OBJECTIVE + +To study the effect of regularization on the inner objective for AutoDispNet-C we use experiment with two types of regularization: data augmentation and of $L 2$ regularization on network weights. + +We note that we could not test the early stopping method on AutoDispNet since AutoDispNet relies on custom operations to compute feature map correlation (Dosovitskiy et al., 2015) and resampling, for which second order derivatives are currently not available (which are required to compute the Hessian). + +Data augmentation. Inspite of fairly large number of training samples in FlyingThings3D, data augmentation is crucial for good generalization performance. Disparity estimation networks employ spatial transformations such as translation, cropping, shearing and scaling. Additionally, appearance transformations such as additive Gaussian noise, changes in brightness, contrast, gamma and color are also applied. Parameters for such transformations are sampled from a uniform or Gaussian distribution (parameterized by a mean and variance). In our experiments, we vary the data augmentation strength by multiplying the variance of these parameter distributions by a fixed factor, which we dub the augmentation scaling factor. The extracted networks are evaluated with the same augmentation parameters. The results of increasing the augmentation strength of the inner objective can be seen in Table 2. We observe that as augmentation strength increases DARTS finds networks with more number of parameters and better test performance. The best test performance is obtained for the network with maximum augmentation for the inner objective. At the same time the search model validation error increases when scaling up the augmentation factor, which again enforces the argument that the overfitting of architectural parameters is reduced by this implicit regularizer. + +L2 regularization. We study the effect of increasing regularization strength on the weights of the network. The results are shown in Table 2. Also in this case best test performance is obtained with the maximum regularization strength. + +# F RESULTS ON PENN TREEBANK + +Here we investigate the effect of more $L _ { 2 }$ regularization on the inner objective for searching recurrent cells on Penn Treebank (PTB). We again used a reduced search space with only $R e L U$ and identity mapping as possible operations. The rest of the settings is the same as in (Liu et al., 2019). + +We run DARTS search four independent times with different random seeds, each with four $L _ { 2 }$ regularization factors, namely $5 \times 1 0 ^ { - 7 }$ (DARTS default), $1 5 \times 1 0 ^ { - 7 }$ , $4 5 \times 1 0 ^ { - 7 }$ and $1 3 5 \times 1 0 ^ { - 7 }$ . Figure 19 shows the test perplexity of the architectures found by DARTS with the aforementioned $L _ { 2 }$ regularization values. As we can see, a stronger regularization factor on the inner objective makes the search procedure more robust. The median perplexity of the discovered architectures gets better as we increase the $L _ { 2 }$ factor from $5 \times 1 0 ^ { - 7 }$ to $4 5 \times 1 { \bar { 0 } } ^ { - 7 }$ , while the search model (one-shot) validation mean perplexity increases. This observation is similar to the ones on image classification shown in Figure 10, showing again that properly regularizing the inner objective helps reduce overfitting the architectural parameters. + +![](images/19e3fbc500fb56ed5e804cdea7b10e36efc91d5ea8816f800626992b249fccbd.jpg) +Figure 19: Performance of recurrent cells found with different $L _ { 2 }$ regularization factors on the inner objective on PTB. We run DARTS 4 independent times with different random seeds, train each of them from scratch with the evaluation settings for 1600 epochs and report the median test perplexity. The blue dashed line denotes the validation perplexity of the search model. + +# G DISCOVERED CELLS ON SEARCH SPACES S1-S4 FROM SECTION 3 ON OTHER DATASETS + +![](images/e5c15f36c9fa9942c509897788209cd3863b9fd6c00780c9c18d3d3dc06bf91a.jpg) +Figure 20: Reduction cells found by DARTS when ran on CIFAR-10 with its default hyperparameters on spaces S1-S4. These cells correspond with the normal ones in Figure 1. + +![](images/75b26c0638f345e827b81254c06ccdc067c56131d4dbdcff0c891a1cb3b3d83b.jpg) +Figure 21: Normal cells found by DARTS on CIFAR-100 and SVHN when ran with its default hyperparameters on spaces S1-S4. Notice the dominance of parameter-less operations such as skip connection and pooling ops. + +![](images/4f69ba60c7b0b915a16593f64b6ffced642774d088cd3858a8b0fb6b6eb88349.jpg) +Figure 22: Reduction cells found by DARTS on CIFAR-100 and SVHN when ran with its default hyperparameters on spaces S1-S4. + +![](images/d413bd1ced5e4fb8943b53b4f8050e43fd78a03e2ace84f2ce891cd15a69b2a2.jpg) +Figure 23: Normal cells found by DARTS-ES when ran with DARTS default hyperparameters on spaces S1-S4. + +![](images/42e9631125fbc3e9ea247caffd1295f6db3a68a426482be78c847ac31ab02ecc.jpg) +Figure 24: Reduction cells found by DARTS-ES when ran with DARTS default hyperparameters on spaces S1-S4. + +![](images/6577dd764de4c8c2b9d5784a3ecbd72913a296ed638b02c0dac98b6e25ecd97e.jpg) +Figure 25: Cells found by AutoDispNet when ran on S6-d. These cells correspond to the results for augmentation scale 0.0 of Table 2. + +![](images/fd14d0b51347b7b2184445cbcbb994a15a3919e1935a795fb0fc97c0ddc82238.jpg) +Figure 26: Cells found by AutoDispNet when ran on S6-d. These cells correspond to the results for augmentation scale 2.0 of Table 2. + +![](images/7b03cd9ea3a7ff99dead7df7e0f9171f3fb7adcd69e77572679afbdc83b4ff66.jpg) +Figure 27: Cells found by AutoDispNet when ran on S6-d. These cells correspond to the results for $L _ { 2 } ^ { - } = 3 \cdot 1 0 ^ { - 4 }$ of Table 2. + +![](images/da3988a460bed8c275043491a290288f8b60fe0c87afb59f601d990a1bc88c99.jpg) +Figure 28: Cells found by AutoDispNet when ran on S6-d. These cells correspond to the results for $L _ { 2 } ^ { - } = 8 1 \cdot 1 0 ^ { - 4 }$ of Table 2. + +![](images/db916f855d2c879b8897361343c9c6f5adcc6983c96909caa7de440d66fc25c0.jpg) +Figure 29: Normal (top row) and reduction (bottom) cells found by DARTS on CIFAR-10 when ran with its default hyperparameters on spaces S1-S4. Same as Figure 1 but with different random seed (seed 2). + +![](images/64a2ba721bb2305cc825e265f46d0bce30d59baf8dd30f97bfe16998f3bb0cb1.jpg) +Figure 30: Normal (top row) and reduction (bottom) cells found by DARTS on CIFAR-10 when ran with its default hyperparameters on spaces S1-S4. Same as Figure 1 but with different random seed (seed 3). \ No newline at end of file diff --git a/parse/train/H1gDNyrKDS/H1gDNyrKDS_content_list.json b/parse/train/H1gDNyrKDS/H1gDNyrKDS_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..8ea852c81de2162296bfcf7055e45361462d677e --- /dev/null +++ b/parse/train/H1gDNyrKDS/H1gDNyrKDS_content_list.json @@ -0,0 +1,2597 @@ +[ + { + "type": "text", + "text": "UNDERSTANDING AND ROBUSTIFYING DIFFERENTIABLE ARCHITECTURE SEARCH ", + "text_level": 1, + "bbox": [ + 174, + 99, + 687, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Arber $\\mathbf { Z e l a } ^ { 1 }$ , Thomas Elsken2,1, Tonmoy Saikia1, Yassine Marrakchi1, \nThomas Brox1 & Frank Hutter1,2 \n1Department of Computer Science, University of Freiburg \n{zelaa, saikiat, marrakch, brox, fh}@cs.uni-freiburg.de \n2Bosch Center for Artificial Intelligence \nThomas.Elsken@de.bosch.com ", + "bbox": [ + 184, + 167, + 722, + 255 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 291, + 544, + 306 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Differentiable Architecture Search (DARTS) has attracted a lot of attention due to its simplicity and small search costs achieved by a continuous relaxation and an approximation of the resulting bi-level optimization problem. However, DARTS does not work robustly for new problems: we identify a wide range of search spaces for which DARTS yields degenerate architectures with very poor test performance. We study this failure mode and show that, while DARTS successfully minimizes validation loss, the found solutions generalize poorly when they coincide with high validation loss curvature in the architecture space. We show that by adding one of various types of regularization we can robustify DARTS to find solutions with less curvature and better generalization properties. Based on these observations, we propose several simple variations of DARTS that perform substantially more robustly in practice. Our observations are robust across five search spaces on three image classification tasks and also hold for the very different domains of disparity estimation (a dense regression task) and language modelling. ", + "bbox": [ + 233, + 320, + 764, + 515 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 537, + 336, + 554 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Neural Architecture Search (NAS), the process of automatically designing neural network architectures, has recently attracted attention by achieving state-of-the-art performance on a variety of tasks (Zoph & Le, 2017; Real et al., 2019). Differentiable architecture search (DARTS) (Liu et al., 2019) significantly improved the efficiency of NAS over prior work, reducing its costs to the same order of magnitude as training a single neural network. This expanded the scope of NAS substantially, allowing it to also be applied on more expensive problems, such as semantic segmentation (Chenxi et al., 2019) or disparity estimation (Saikia et al., 2019). ", + "bbox": [ + 174, + 568, + 825, + 665 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, several researchers have also reported DARTS to not work well, in some cases even no better than random search (Li & Talwalkar, 2019; Sciuto et al., 2019). Why is this? How can these seemingly contradicting results be explained? The overall goal of this paper is to understand and overcome such failure modes of DARTS. To this end, we make the following contributions: ", + "bbox": [ + 174, + 672, + 825, + 728 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1. We identify 12 NAS benchmarks based on four search spaces in which standard DARTS yields degenerate architectures with poor test performance across several datasets (Section 3). \n2. By computing the eigenspectrum of the Hessian of the validation loss with respect to the architectural parameters, we show that there is a strong correlation between its dominant eigenvalue and the architecture’s generalization error. Based on this finding, we propose a simple variation of DARTS with early stopping that performs substantially more robustly (Section 4). \n3. We show that, related to previous work on sharp/flat local minima, regularizing the inner objective of DARTS more strongly allows it to find solutions with smaller Hessian spectrum and better generalization properties. Based on these insights, we propose two practical robustifications of DARTS that overcome its failure modes in all our 12 NAS benchmarks (Section 5). ", + "bbox": [ + 173, + 738, + 825, + 885 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Our findings are robust across a wide range of NAS benchmarks based on image recognition and also hold for the very different domains of language modelling (PTB) and disparity estimation. They consolidate the findings of the various results in the literature and lead to a substantially more robust version of DARTS. We provide our implementation and scripts to facilitate reproducibility1. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND AND RELATED WORK ", + "text_level": 1, + "bbox": [ + 174, + 155, + 511, + 171 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 RELATION BETWEEN FLAT/SHARP MINIMA AND GENERALIZATION PERFORMANCE ", + "text_level": 1, + "bbox": [ + 173, + 189, + 779, + 203 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Already Hochreiter & Schmidhuber (1997) observed that flat minima of the training loss yield better generalization performance than sharp minima. Recent work (Keskar et al., 2016; Yao et al., 2018) focuses more on the settings of large/small batch size training, where observations show that small batch training tends to get attracted to flatter minima and generalizes better. Similarly, Nguyen et al. (2018) observed that this phenomenon manifests also in the hyperparameter space. They showed that whenever the hyperparameters overfit the validation data, the minima lie in a sharper region of the space. This motivated us to conduct a similar analysis in the context of differentiable architecture search later in Section 4.1, where we see the same effect in the space of neural network architectures. ", + "bbox": [ + 174, + 215, + 825, + 328 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 BI-LEVEL OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 176, + 348, + 392, + 362 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We start by a short introduction of the bi-level optimization problem (Colson et al., 2007). These are problems which contain two optimization tasks, nested within each other. ", + "bbox": [ + 174, + 375, + 821, + 404 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Definition 2.1. Given the outer objective function $F : \\mathbb { R } ^ { P } \\times \\mathbb { R } ^ { N } \\to \\mathbb { R }$ and the inner objective function $f : \\mathbb { R } ^ { P } \\times \\mathbb { R } ^ { N } \\to \\mathbb { R }$ , the bi-level optimization problem is given by ", + "bbox": [ + 174, + 409, + 823, + 438 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/b761278795671c43fe1527e1d6f6a0b6c9b4613ca324f66d39f1681c3bab5662.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { y \\in \\mathbb { R } ^ { P } } { \\operatorname* { m i n } } F ( y , \\theta ^ { * } ( y ) ) } \\\\ & { s . t . \\quad \\theta ^ { * } ( y ) \\in \\underset { \\theta \\in \\mathbb { R } ^ { N } } { \\arg \\operatorname* { m i n } } f ( y , \\theta ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 395, + 445, + 601, + 501 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $y \\in \\mathbb { R } ^ { P }$ and $\\boldsymbol { \\theta } \\in \\mathbb { R } ^ { N }$ are the outer and inner variables, respectively. One may also see the bi-level problem as a constrained optimization problem, with the inner problem as a constraint. ", + "bbox": [ + 174, + 518, + 823, + 547 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In general, even in the case when the inner objective (2) is strongly convex and has an unique minimizer $\\theta ^ { * } ( y ) = \\arg \\operatorname* { m i n } _ { \\theta \\in \\mathbb { R } ^ { N } } f ( y , \\theta )$ , it is not possible to directly optimize the outer objective (1). A possible method around this issue is to use the implicit function theorem to retrieve the derivative of the solution map (or response map) $\\theta ^ { * } ( y ) \\in \\mathbb { F } \\subseteq \\mathbb { R } ^ { N }$ w.r.t. $y$ (Bengio, 2000; Pedregosa, 2016; Beirami et al., 2017). Another strategy is to approximate the inner problem with a dynamical system (Domke, 2012; Maclaurin et al., 2015; Franceschi et al., 2017; 2018), where the optimization dynamics could, e.g., describe gradient descent. In the case that the minimizer of the inner problem is unique, under some conditions the set of minimizers of this approximate problem will indeed converge to the minimizers of the bilevel problem (1) (see Franceschi et al. (2018)). ", + "bbox": [ + 173, + 554, + 825, + 680 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.3 NEURAL ARCHITECTURE SEARCH ", + "text_level": 1, + "bbox": [ + 176, + 699, + 450, + 714 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Neural Architecture Search (NAS) denotes the process of automatically designing neural network architectures in order to overcome the cumbersome trial-and-error process when designing architectures manually. We briefly review NAS here and refer to the recent survey by Elsken et al. (2019b) for a more thorough overview. Prior work mostly employs either reinforcement learning techniques (Baker et al., 2017a; Zoph & Le, 2017; Zhong et al., 2018; Zoph et al., 2018) or evolutionary algorithms (Stanley & Miikkulainen, 2002; Liu et al., 2018b; Miikkulainen et al., 2017; Real et al., 2017; 2019) to optimize the discrete architecture space. As these methods are often very expensive, various works focus on reducing the search costs by, e.g., employing network morphisms (Cai et al., 2018a;b; Elsken et al., 2017; 2019a), weight sharing within search models (Saxena & Verbeek, 2016; Bender et al., 2018; Pham et al., 2018) or multi-fidelity optimization (Baker et al., 2017b; Falkner et al., 2018; Li et al., 2017; Zela et al., 2018), but their applicability still often remains restricted to rather simple tasks and small datasets. ", + "bbox": [ + 174, + 727, + 825, + 893 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.4 DIFFERENTIABLE ARCHITECTURE SEARCH (DARTS) ", + "text_level": 1, + "bbox": [ + 173, + 103, + 586, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A recent line of work focuses on relaxing the discrete neural architecture search problem to a continuous one that can be solved by gradient descent (Liu et al., 2019; Xie et al., 2019; Casale et al., 2019; Cai et al., 2019). In DARTS (Liu et al., 2019), this is achieved by simply using a weighted sum of possible candidate operations for each layer, whereas the real-valued weights then effectively parametrize the network’s architecture. We will now review DARTS in more detail, as our work builds directly upon it. ", + "bbox": [ + 174, + 128, + 825, + 214 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Continuous relaxation of the search space. In agreement with prior work (Zoph et al., 2018; Real et al., 2019), DARTS optimizes only substructures called cells that are stacked to define the full network architecture. Each cell contains $N$ nodes organized in a directed acyclic graph. The graph contains two inputs nodes (given by the outputs of the previous two cells), a set of intermediate nodes, and one output node (given by concatenating all intermediate nodes). Each intermediate node $x ^ { ( j ) }$ represents a feature map. See Figure 1 for an illustration of such a cell. Instead of applying a single operation to a specific node during architecture search, Liu et al. (2019) relax the decision which operation to choose by computing the intermediate node as a mixture of candidate operations, applied to predecessor nodes x(i), i < j, x(j) = PiBenchmarkDARTSDARTS-ESC10S14.66 ± 0.713.05± 0.07S24.42 ± 0.403.41 ± 0.14S34.12 ± 0.853.71 ± 1.14S46.95±0.184.17 ± 0.21C100S129.93 ± 0.4128.90±0.81S228.75±0.9224.68±1.43S329.01 ± 0.2426.99 ± 1.79S424.77 ± 1.5123.90±2.01SVHNS19.88±5.502.80±0.09S23.69 ±0.122.68± 0.18S34.00 ± 1.012.78± 0.29S42.90±0.022.55±0.15", + "bbox": [ + 627, + 375, + 820, + 506 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 REGULARIZATION OF INNER OBJECTIVE IMPROVES GENERALIZATION OF ARCHITECTURES ", + "text_level": 1, + "bbox": [ + 174, + 513, + 586, + 546 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As we saw in Section 4.1, sharper minima (by means of large eigenvalues) of the validation loss lead to poor generalization performance. In our bi-level optimization setting, the outer variables’ trajectory depends on the inner optimization procedure. Therefore, we hypothesized that modifying the landscape of the inner objective $\\mathcal { L } _ { t r a i n }$ could redirect the outer variables $\\alpha$ to flatter areas of the architectural space. We study two ways of regularization (data augmentation in Section 5.1 and $L _ { 2 }$ regularization in Section 5.2) and find that both, along with the early stopping criterion from Section 4.3, make DARTS more robust in practice. We emphasize that we do not alter the regularization of the final training and evaluation phase, but solely that of the search phase. The setting we use for all experiments in this paper to obtain the final test performance is described in Appendix C. ", + "bbox": [ + 173, + 563, + 825, + 689 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 REGULARIZATION VIA DATA AUGMENTATION ", + "text_level": 1, + "bbox": [ + 174, + 709, + 527, + 723 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We first investigate the effect of regularizing via data augmentation, namely masking out parts of the input and intermediate feature maps via Cutout (CO, DeVries & Taylor (2017)) and ScheduledDropPath (DP, Zoph et al. (2018)) (ScheduledDropPath is a regularization technique, but we list it here since we apply it together with Cutout), respectively, during architecture search. We ran DARTS with CO and DP (with and without our early stopping criterion, DARTS-ES) with different maximum DP probabilities on all three image classification datasets and search spaces S1-S4. ", + "bbox": [ + 174, + 734, + 825, + 819 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 7 summarizes the results: regularization improves the test performance of DARTS and DARTS-ES in all cases, sometimes very substantially, and at the same time kept the dominant eigenvalue relatively low (Figure 13). This also directly results in smaller drops in accuracy after pruning, as discussed in Section 4.2; indeed, the search runs plotted in Figure 5b are the same as in this section. Figure 17 in the appendix explicitly shows how regularization relates to the accuracy drops. We also refer to further results in the appendix: Figure 11 (showing test vs. validation error) and Table 5 (showing that overfitting of the architectural parameters is reduced). ", + "bbox": [ + 174, + 827, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/1062267b762cf2a8dad20f3863189043a16c12493b2f4529678c55858bfe3901.jpg", + "image_caption": [ + "Figure 8: Effect of $L _ { 2 }$ regularization of the inner objective during architecture search for DARTS (solid lines) and DARTS-ES (dashed). " + ], + "image_footnote": [], + "bbox": [ + 178, + 103, + 816, + 241 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Similar observations hold for disparity estimation on S6, where we vary the strength of standard data augmentation methods, such as shearing or brightness change, rather then masking parts of features, which is unreasonable for this task. The augmentation strength is described by an “augmentation scaling factor” (Appendix E). Table 2 summarizes the results. We report the average end point error (EPE), which is the Euclidean distance between the predicted and ground truth disparity maps. Data augmentation avoided the degenerate architectures and substantially improved results. ", + "bbox": [ + 174, + 313, + 825, + 396 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 INCREASED $L _ { 2 }$ REGULARIZATION ", + "text_level": 1, + "bbox": [ + 176, + 415, + 446, + 429 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As a second type of regularization, we also tested different $L _ { 2 }$ regularization factors $3 i \\cdot 1 0 ^ { - 4 }$ for $i \\in$ $\\{ 1 , 3 , 9 , 2 7 , 8 1 \\}$ . Standard DARTS in fact does already include a small amount of $L _ { 2 }$ regularization; $i = 1$ yields its default. Figure 8 shows that DARTS’ test performance (solid lines) can be significantly improved by higher $L _ { 2 }$ factors across all datasets and spaces, while keeping the dominant eigenvalue low (Figure 14). DARTS with early stopping (dashed lines) also benefits from additional regularization. Again, we observe the implicit regularization effect on the outer objective which reduces the overfitting of the architectural parameters. We again refer to ", + "bbox": [ + 174, + 443, + 516, + 622 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/15a089628a8ebb3113fab94510b00bdb7d84adb26582cf453b508fddebba0784.jpg", + "table_caption": [ + "Table 2: Effect of regularization for disparity estimation. Search was conducted on FlyingThings3D (FT) and then evaluated on both FT and Sintel. Lower is better. " + ], + "table_footnote": [], + "table_body": "
Aug. ScaleSearchmodel valid EPEFT test EPESintel test EPEParams (M)
0.04.493.835.699.65
0.13.533.755.979.65
0.53.283.375.229.43
1.04.613.125.4712.46
1.55.232.604.1512.57
2.07.452.333.7612.25
L2 reg. factorSearchmodel validFT test EPESintel testParams
3×10-4EPE 3.953.25EPE 6.13(M) 11.00
9×10-45.972.304.1213.92
27×10-44.252.724.8310.29
81×10-44.612.343.8512.16
", + "bbox": [ + 531, + 483, + 823, + 609 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 2 for disparity estimation; Appendix F shows similar results for language modelling (Penn TreeBank). ", + "bbox": [ + 173, + 622, + 823, + 648 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 PRACTICAL ROBUSTIFICATION OF DARTS BY REGULARIZING THE INNER OBJECTIVE ", + "text_level": 1, + "bbox": [ + 174, + 667, + 808, + 683 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Based on the insights from the aforementioned analysis and empirical results, we now propose two alternative simple modifications to make DARTS more robust in practice without having to manually tune its regularization hyperparameters. ", + "bbox": [ + 174, + 694, + 823, + 736 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "DARTS with adaptive regularization One option is to adapt DARTS’ regularization hyperparameters in an automated way, in order to keep the architectural weights in areas of the validation loss objective with smaller curvature. The simplest off-the-shelf procedure towards this desiderata would be to increase the regularization strength whenever the dominant eigenvalue starts increasing rapidly. Algorithm 1 (DARTS-ADA, Appendix D.1) shows such a procedure. We use the same stopping criterion as in DARTS-ES (Section 4.3), roll back DARTS to the epoch when this criterion is met, and continue the search with a larger regularization value $R$ for the remaining epochs (larger by a factor of $\\eta$ ). This procedure is repeated whenever the criterion is met, unless the regularization value exceeds some maximum predefined value $R _ { m a x }$ . ", + "bbox": [ + 174, + 753, + 825, + 878 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Multiple DARTS runs with different regularization strength Liu et al. (2019) already suggested to run the search phase of DARTS four times, resulting in four architectures, and to return the best of these four architectures w.r.t. validation performance when retrained from scratch for a limited number of epochs. We propose to use the same procedure, with the only difference that the four runs use different amounts of regularization. The resulting RobustDARTS (R-DARTS) method is conceptually very simple, trivial to implement and likely to work well if any of the tried regularization strengths works well. ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 3 evaluates the performance of our practical robustifications of DARTS, DARTS-ADA and R-DARTS (based on either L2 or ScheduledDropPath regularization), by comparing them to the original DARTS, DARTS-ES and Random Search with weight sharing (RS-ws). For each of these methods, as proposed in the DARTS paper (Liu et al., 2019), we ran the search four independent times with different random seeds and selected the architecture used for the final evaluation based on a validation run as described above. ", + "bbox": [ + 174, + 181, + 419, + 387 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/4e1db6f573556a0bae05f2d81225c824f6e40799550aeba418d133156fc139d2.jpg", + "table_caption": [ + "Table 3: Empirical evaluation of practical robustified versions of DARTS. Each entry is the test error after retraining the selected architecture as usual. The best method for each setting is boldface and underlined, the second best boldface. " + ], + "table_footnote": [], + "table_body": "
BenchmarkRS-wsDARTSR-DARTS(DP)R-DARTS(L2)DARTS-ESDARTS-ADA
C10S13.233.843.112.783.013.10
S23.664.853.483.313.263.35
S32.953.342.932.512.742.59
S48.077.203.583.563.714.84
C100S123.3029.4625.9324.2528.3724.03
S221.2126.0522.3022.2423.2523.52
S323.7528.9022.3623.9923.7323.37
S428.1922.8522.1821.9421.2623.20
SVHNS12.594.582.554.792.722.53
S22.723.532.522.512.602.54
S32.873.412.492.482.502.50
", + "bbox": [ + 436, + 247, + 823, + 376 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "As the table shows, in accordance with Li & Talwalkar (2019), RS-ws often outperformed the original DARTS; however, with our robustifications, DARTS typically performs substantially better than RS-ws. DARTS-ADA consistently improved over standard DARTS for all benchmarks, indicating that a gradual increase of regularization during search prevents ending up in the bad regions of the architectural space. Finally, RobustDARTS yielded the best performance and since it is also easier to implement than DARTS-ES and DARTS-ADA, it is the method that we recommend to be used in practice. ", + "bbox": [ + 174, + 388, + 825, + 484 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, since the evaluations in this paper have so far focussed on smaller subspaces of the original DARTS search space, the reader may wonder how well RobustDARTS works on the full search spaces. As Table 4 shows, RobustDARTS performed similarly to DARTS for the two original benchmarks from the DARTS paper (PTB and CIFAR-10), on which DARTS was developed and is well tuned; however, even when only changing the dataset to CIFAR-100 or SVHN, RobustDARTS already performed significantly better than DARTS, underlining its robustness. ", + "bbox": [ + 174, + 492, + 547, + 645 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 4: DARTS vs. RobustDARTS on the original DARTS search spaces. We show mean $\\pm$ stddev for 5 repetitions (based on 4 fresh subruns each as in Table 3); for the more expensive PTB we could only afford 1 such repetition. ", + "bbox": [ + 562, + 493, + 823, + 575 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/be051fa43c56eb08b50f2fdec7cf3b4eec7d77c996ee89855a1d7cf7f116b6cd.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
BenchmarkDARTSR-DARTS(L2)
C102.91± 0.252.95 ± 0.21
C10020.58 ± 0.4418.01 ± 0.26
SVHN2.46±0.092.17 ± 0.09
PTB58.6457.59
", + "bbox": [ + 565, + 588, + 820, + 656 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 665, + 328, + 681 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We showed that the generalization performance of architectures found by DARTS is related to the eigenvalues of the Hessian matrix of the validation loss w.r.t. the architectural parameters. Standard DARTS often results in degenerate architectures with large eigenvalues and poor generalization. Based on this observation, we proposed a simple early stopping criterion for DARTS based on tracking the largest eigenvalue. Our empirical results also show that properly regularizing the inner objective helps controlling the eigenvalue and therefore improves generalization. Our findings substantially improve our understanding of DARTS’ failure modes and lead to much more robust versions. They are consistent across many different search spaces on image recognition tasks and also for the very different domains of language modelling and disparity estimation. Our code is available for reproducibility. ", + "bbox": [ + 174, + 696, + 825, + 837 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 853, + 326, + 866 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The authors acknowledge funding by the Robert Bosch GmbH, support by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme through grant no. 716721, and by BMBF grant DeToL. ", + "bbox": [ + 176, + 876, + 825, + 917 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Bowen Baker, Otkrist Gupta, Nikhil Naik, and Ramesh Raskar. 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IEEE Computer Society, 2018. ", + "bbox": [ + 173, + 656, + 821, + 685 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Barret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. In International Conference on Learning Representations, 2017. ", + "bbox": [ + 173, + 694, + 821, + 723 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V. Le. Learning transferable architectures for scalable image recognition. In Conference on Computer Vision and Pattern Recognition, 2018. ", + "bbox": [ + 174, + 732, + 821, + 761 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A MORE DETAIL ON DARTS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 429, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Here we present a detailed description of DARTS architectural update steps. We firstly provide the general formalism which computes the gradient of the outer level problem in (1) by means of the implicit function theorem. Afterwards, we present how DARTS computes the gradient used to update the architectural parameters $\\alpha$ . ", + "bbox": [ + 174, + 133, + 825, + 189 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 DERIVATIVE WITH SMOOTHED NON-QUADRATIC LOWER LEVEL PROBLEM ", + "text_level": 1, + "bbox": [ + 176, + 207, + 723, + 220 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Consider the general definition of the bi-level optimization problem as given by (1) and (2). Given that $f$ is twice continuously differentiable and that all stationary points are local minimas, one can make use of the implicit function theorem to find the derivative of the solution map $\\theta ^ { * } ( y )$ w.r.t. $y$ (Bengio, 2000). Under the smoothness assumption, the optimality condition of the lower level (2) is $\\nabla _ { \\boldsymbol { \\theta } } f ( y , \\boldsymbol { \\theta } ) = \\mathbf { 0 }$ , which defines an implicit function $\\theta ^ { * } ( y )$ . With the assumption that $\\mathrm { m i n } _ { \\boldsymbol { \\theta } } f ( \\boldsymbol { y } , \\boldsymbol { \\theta } )$ has a solution, there exists a $( y , \\theta ^ { * } )$ such that $\\nabla _ { \\theta } f ( y , \\theta ^ { * } ) = \\mathbf { 0 }$ . Under the condition that $\\nabla _ { \\theta } f ( y , \\theta ^ { * } ) = 0$ is continuously differentiable and that $\\theta ^ { * } ( y )$ is continuously differentiable at $y$ , implicitly differentiating the last equality from both sides w.r.t. y and applying the chain rule, yields: ", + "bbox": [ + 173, + 231, + 825, + 358 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/e976f45c7578ae4782542919e5dc97409d499f2aabc3598506b1243968dc6c1a.jpg", + "text": "$$\n\\frac { \\partial ( \\nabla _ { \\theta } f ) } { \\partial \\theta } ( y , \\theta ^ { * } ) \\cdot \\frac { \\partial \\theta ^ { * } } { \\partial y } ( y ) + \\frac { \\partial ( \\nabla _ { \\theta } f ) } { \\partial y } ( y , \\theta ^ { * } ) = { \\bf 0 } .\n$$", + "text_format": "latex", + "bbox": [ + 338, + 363, + 661, + 397 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Assuming that the Hessian $\\nabla _ { { \\theta } } ^ { 2 } f ( y , { \\theta } ^ { * } )$ is invertible, we can rewrite (3) as follows: ", + "bbox": [ + 176, + 410, + 710, + 426 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/7e28e610db647a1a3b7ab6aabdc38fa5a4e3752da59bfd39401e6974a8aeb56f.jpg", + "text": "$$\n\\frac { \\partial \\theta ^ { * } } { \\partial y } ( y ) = - \\Big ( \\nabla _ { \\theta } ^ { 2 } f ( y , \\theta ^ { * } ) \\Big ) ^ { - 1 } \\cdot \\frac { \\partial ( \\nabla _ { \\theta } f ) } { \\partial y } ( y , \\theta ^ { * } ) .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 433, + 658, + 465 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Applying the chain rule to (1) for computing the total derivative of $F$ with respect to $y$ yields: ", + "bbox": [ + 171, + 478, + 784, + 493 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/3d02a29e2a6f02120b40286ab737582bc35369d5151ba635a0e85827dad0f982.jpg", + "text": "$$\n\\frac { d F } { d y } = \\frac { \\partial F } { \\partial \\theta } \\cdot \\frac { \\partial \\theta ^ { * } } { \\partial y } + \\frac { \\partial F } { \\partial y } ,\n$$", + "text_format": "latex", + "bbox": [ + 415, + 497, + 583, + 531 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where we have omitted the evaluation at $( y , \\theta ^ { * } )$ . Substituting (4) into (5) and reordering yields: ", + "bbox": [ + 173, + 537, + 795, + 553 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/7cb20d047af1de4be5229d062ca46134ced72aec6b0a68e41bb6199d6363bdea.jpg", + "text": "$$\n\\frac { d F } { d y } = \\frac { \\partial F } { \\partial y } - \\frac { \\partial F } { \\partial \\theta } \\cdot \\left( \\nabla _ { \\theta } ^ { 2 } f \\right) ^ { - 1 } \\cdot \\frac { \\partial ^ { 2 } f } { \\partial \\theta \\partial y } .\n$$", + "text_format": "latex", + "bbox": [ + 369, + 559, + 629, + 593 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "equation 6 computes the gradient of $F$ , given the function $\\theta ^ { * } ( y )$ , which maps outer variables to the inner variables minimizing the inner problem. However, in most of the cases obtaining such a mapping is computationally expensive, therefore different heuristics have been proposed to approximate $d F / d y$ (Maclaurin et al., 2015; Pedregosa, 2016; Franceschi et al., 2017; 2018). ", + "bbox": [ + 174, + 606, + 825, + 662 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.2 DARTS ARCHITECTURAL GRADIENT COMPUTATION ", + "text_level": 1, + "bbox": [ + 176, + 679, + 576, + 694 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "DARTS optimization procedure is defined as a bi-level optimization problem where $\\mathcal { L } _ { v a l i d }$ is the outer objective (1) and $\\mathcal { L } _ { t r a i n }$ is the inner objective (2): ", + "bbox": [ + 169, + 704, + 823, + 734 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/cdbf4fc11dd269a1000bb27b169fa1404d1bd3d09816513b3118bd5734f44d65.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { \\alpha } { \\operatorname* { m i n } } \\mathcal { L } _ { v a l i d } ( \\alpha , w ^ { * } ( \\alpha ) ) } \\\\ & { s . t . \\quad w ^ { * } ( \\alpha ) = \\underset { w } { \\arg \\operatorname* { m i n } } \\mathcal { L } _ { t r a i n } ( \\alpha , w ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 372, + 738, + 625, + 787 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where both losses are determined by both the architecture parameters $\\alpha$ (outer variables) and the network weights $w$ (inner variables). Based on Appendix A.1, under some conditions, the total derivative of $\\mathcal { L } _ { v a l i d }$ w.r.t. $\\alpha$ evaluated on $( \\alpha , w ^ { * } ( \\alpha ) )$ would be: ", + "bbox": [ + 174, + 794, + 825, + 835 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/9701ab8e4a8afc6e465e6625740af3d499eacdbed70c2bb247118d80d85f6ba6.jpg", + "text": "$$\n\\frac { d \\mathcal { L } _ { v a l i d } } { d \\alpha } = \\nabla _ { \\alpha } \\mathcal { L } _ { v a l i d } - \\nabla _ { w } \\mathcal { L } _ { v a l i d } \\big ( \\nabla _ { w } ^ { 2 } \\mathcal { L } _ { t r a i n } \\big ) ^ { - 1 } \\nabla _ { \\alpha , w } ^ { 2 } \\mathcal { L } _ { t r a i n } ,\n$$", + "text_format": "latex", + "bbox": [ + 292, + 842, + 704, + 872 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where ∇α = , $\\begin{array} { r } { \\nabla _ { w } = \\frac { \\partial } { \\partial w } } \\\\ { . } \\end{array}$ and $\\begin{array} { r } { \\nabla _ { \\alpha , w } ^ { 2 } = \\frac { \\partial ^ { 2 } } { \\partial \\alpha \\partial w } } \\end{array}$ = ∂2∂α∂w . Computing the inverse of the Hessian is in general not possible considering the high dimensionality of the model parameters $w$ , therefore resolving to gradient-based iterative algorithms for finding $w ^ { * }$ is necessary. However, this would also require to optimize the model parameters $w$ till convergence each time $\\alpha$ is updated. If our model is a deep neural network it is clear that this computation is expensive, therefore Liu et al. (2019) propose to approximate $w ^ { * } ( \\alpha )$ by updating the current model parameters $w$ using a single gradient descent step: ", + "bbox": [ + 176, + 878, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/246a3f11013fedaa9d2f7f3b8780ec400a20f8e680d00999dd7a209cd365f36f.jpg", + "text": "$$\n\\begin{array} { r } { w ^ { * } ( \\alpha ) \\approx w - \\xi \\nabla _ { w } \\mathcal { L } _ { t r a i n } ( \\alpha , w ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 387, + 165, + 609, + 183 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\xi$ is the learning rate for the virtual gradient step DARTS takes with respect to the model weights $w$ . From equation 10 the gradient of $w ^ { * } ( \\alpha )$ with respect to $\\alpha$ is ", + "bbox": [ + 173, + 188, + 823, + 217 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/4bf2ae52f91a188dec04bb05175cd60926915b2dd24b037badfff543ba0eff7b.jpg", + "text": "$$\n\\frac { \\partial w ^ { * } } { \\partial \\alpha } ( \\alpha ) = - \\xi \\nabla _ { \\alpha , w } ^ { 2 } \\mathcal { L } _ { t r a i n } ( \\alpha , w ) ,\n$$", + "text_format": "latex", + "bbox": [ + 383, + 222, + 612, + 253 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "By setting the evaluation point $\\boldsymbol { w ^ { * } } = \\boldsymbol { w } - \\xi \\nabla _ { \\boldsymbol { w } } \\mathcal { L } _ { t r a i n } ( \\alpha , \\boldsymbol { w } )$ and following the same derivation as in Appendix A.1, we obtain the DARTS architectural gradient approximation: ", + "bbox": [ + 171, + 265, + 825, + 295 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/3edebd9c03954bd0cc7f5dee601ed66d388f8d69404dc124c3c1a53dcda14012.jpg", + "text": "$$\n\\frac { d \\mathcal { L } _ { v a l i d } } { d \\alpha } ( \\alpha ) = \\nabla _ { \\alpha } \\mathcal { L } _ { v a l i d } ( \\alpha , w ^ { * } ) - \\xi \\nabla _ { w } \\mathcal { L } _ { v a l i d } ( \\alpha , w ^ { * } ) \\nabla _ { \\alpha , w } ^ { 2 } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { * } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 251, + 300, + 746, + 330 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where the inverse Hessian however contains again an e $\\nabla _ { w } ^ { 2 } \\mathcal { L } _ { t r a i n } ^ { - 1 }$ in (9) is replaced by the learning rate ector-matrix product. Liu et al. (2019) re $\\xi$ . This expressionuce the complexity by using the finite difference approximation around $w ^ { \\pm } = w \\pm \\epsilon \\nabla _ { w } \\mathcal { L } _ { v a l i d } ( \\alpha , w ^ { * } )$ for some small $\\epsilon = 0 . 0 1 / \\left. \\nabla _ { w } \\mathcal { L } _ { v a l i d } ( \\alpha , w ^ { * } ) \\right. _ { 2 }$ to compute the gradient of $\\nabla _ { \\alpha } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { * } )$ with respect to $w$ as ", + "bbox": [ + 174, + 335, + 825, + 395 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/0766b4f21d822b150145424d238c405233db02c94c33da5bffd6a780f604d0f4.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { \\nabla _ { \\alpha , w } ^ { 2 } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { * } ) \\approx \\frac { \\nabla _ { \\alpha } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { + } ) - \\nabla _ { \\alpha } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { - } ) } { 2 \\epsilon \\nabla _ { w } \\mathcal { L } _ { v a l i d } ( \\alpha , w ^ { * } ) } \\qquad \\Leftrightarrow } \\\\ & { } & { \\nabla _ { w } \\mathcal { L } _ { v a l i d } ( \\alpha , w ^ { * } ) \\nabla _ { \\alpha , w } ^ { 2 } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { * } ) \\approx \\frac { \\nabla _ { \\alpha } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { + } ) - \\nabla _ { \\alpha } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { - } ) } { 2 \\epsilon } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 228, + 400, + 767, + 469 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In the end, combining equation 12 and equation 13 gives the gradient to compute the architectural updates in DARTS: ", + "bbox": [ + 171, + 472, + 828, + 500 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/8ef227ccc3eb03ba486bd5ec0843f803768304d23cdc4e5b0bcbbeebf2c61b89.jpg", + "text": "$$\n\\frac { d \\mathcal { L } _ { v a l i d } } { d \\alpha } ( \\alpha ) = \\nabla _ { \\alpha } \\mathcal { L } _ { v a l i d } ( \\alpha , w ^ { * } ) - \\frac { \\xi } { 2 \\epsilon } \\big ( \\nabla _ { \\alpha } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { + } ) - \\nabla _ { \\alpha } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { - } ) \\big )\n$$", + "text_format": "latex", + "bbox": [ + 233, + 505, + 763, + 536 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In all our experiments we always use $\\xi = \\eta$ (also called second order approximation in Liu et al. \n(2019)), where $\\eta$ is the learning rate used in SGD for updating the parameters $w$ . ", + "bbox": [ + 171, + 549, + 826, + 577 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B CONSTRUCTION OF S1 FROM SECTION 3", + "text_level": 1, + "bbox": [ + 174, + 597, + 547, + 613 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We ran DARTS two times on the default search space to find the two most important operations per mixed operation. Initially, every mixed operation consists of 8 operations. After the first DARTS run, we drop the 4 (out of 8) least important ones. In the second DARTS run, we drop the 2 (out of the remaining 4) least important ones. S1 is then defined to contain only the two remaining most important operations per mixed op. Refer to Figure 9 for an illustration of this pre-optimized space. ", + "bbox": [ + 173, + 627, + 825, + 699 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C FINAL ARCHITECTURE EVALUATION ", + "text_level": 1, + "bbox": [ + 174, + 718, + 514, + 734 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Similar to the original DARTS paper (Liu et al., 2019), the architecture found during the search are scaled up by increasing the number of filters and cells and retrained from scratch to obtain the final test performance. For CIFAR-100 and SVHN we use 16 number of initial filters and 8 cells when training architectures from scratch for all the experiments we conduct. The rest of the settings is the same as in Liu et al. (2019). ", + "bbox": [ + 174, + 748, + 825, + 819 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "On CIFAR-10, when scaling the ScheduledDropPath drop probability, we use the same settings for training from scratch the found architectures as in the original DARTS paper, i.e. 36 initial filters and 20 stacked cells. However, for search space S2 and S4 we reduce the number of initial filters to 16 in order to avoid memory issues, since the cells found with more regularization usually are composed only with separable convolutions. When scaling the $L _ { 2 }$ factor on CIFAR-10 experiments we use 16 initial filters and 8 stacked cells, except the experiments on S1, where the settings are the same as in Liu et al. (2019), i.e. 36 initial filters and 20 stacked cells. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/ce86ca890ed3cbc8354ebd6d805b89b039a842003ed876447bfb906d9ff4525b.jpg", + "image_caption": [ + "(a) Normal cell space " + ], + "image_footnote": [], + "bbox": [ + 199, + 114, + 799, + 390 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/a5e9654b804ac972433196284b72635adb5df60dff44aee7a1e0f5c83a924f18.jpg", + "image_caption": [ + "Figure 9: Search space S1. " + ], + "image_footnote": [], + "bbox": [ + 196, + 436, + 802, + 752 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Note that although altering the regularization factors during DARTS search, when training the final architectures from scratch we always use the same values for them as in Liu et al. (2019), i.e. ScheduledDropPath maximum drop probability linearly increases from 0 towards 0.2 throughout training, Cutout is always enabled with cutout probability 1.0, and the $L _ { 2 }$ regularization factor is set to $3 \\cdot 1 \\bar { 0 } ^ { - 4 }$ . ", + "bbox": [ + 173, + 810, + 825, + 881 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D ADDITIONAL EMPIRICAL RESULTS ", + "text_level": 1, + "bbox": [ + 173, + 103, + 493, + 117 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/b7331d112ac5bc032e8b79382b010517d1b6290a6d78e201ab163ee77534f5b6.jpg", + "image_caption": [ + "Figure 10: Test regret and validation error of the search (one-shot) model when running DARTS on S5 and CIFAR-10 with different $L _ { 2 }$ regularization values. The architectural parameters’ overfit reduces as we increase the $L _ { 2 }$ factor and successfully finds the global minimum. However, we notice that the architectural parameters start underfitting as we increase to much the $L _ { 2 }$ factor, i.e. both validation and test error increase. " + ], + "image_footnote": [], + "bbox": [ + 176, + 137, + 820, + 392 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/56c16261c8c2960ae8b3cb54f6c87423c6ced15a230818c251f68defc8b5abeb.jpg", + "table_caption": [ + "Table 5: Validation (train) and test accuracy on CIFAR-10 of the search and final evaluation models, respectively. The values in the last column show the maximum eigenvalue $\\lambda _ { m a x } ^ { \\alpha }$ (computed on a random sampled mini-batch) of the Hessian, at the end of search for different maximum drop path probability). The four blocks in the table state results for the search spaces S1-S4, respectively. " + ], + "table_footnote": [], + "table_body": "
Drop Prob.Valid acc.Test acc.ParamsXmax
seed 1seed2seed 3seed 1seed 2seed 3seed 1seed 2seed 3seed 1seed 2seed 3
S10.087.2287.0186.9896.1694.4395.432.241.932.031.0230.8350.698
0.284.2484.3284.2296.3996.6696.202.632.842.480.1480.2640.228
0.482.2882.1882.7996.4496.9496.762.632.993.170.1920.1990.149
0.679.1779.1878.8496.8996.9396.963.383.023.170.3000.2550.256
S20.088.4988.4088.3595.1595.4896.110.930.860.970.6840.4090.268
0.285.2984.8185.3695.1595.4096.141.281.441.360.2700.2170.145
0.482.0382.6683.2096.3496.5096.441.281.281.360.3040.4110.282
0.679.8680.1979.7096.5296.3596.291.211.281.360.2920.2950.281
S30.088.7889.1588.6794.7096.2796.662.212.432.850.4960.5350.446
0.285.6185.6085.5096.7896.8496.743.624.042.990.1790.1850.202
0.483.0383.2483.4397.0796.8596.484.103.743.380.1560.3700.184
0.679.8680.0379.6896.9194.5696.444.462.302.660.2390.2750.280
S40.086.3386.7286.4692.8093.2293.141.051.131.050.4000.4420.314
0.281.0182.4382.0395.8496.0896.151.441.441.440.0700.0540.079
0.479.4979.6778.9696.1196.3096.281.441.441.440.0640.0570.049
0.674.5474.7474.3796.4296.3696.641.441.441.440.0570.0600.066
", + "bbox": [ + 176, + 566, + 826, + 770 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "D.1 ADAPTIVE DARTS DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 804, + 415, + 819 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We evaluated DARTS-ADA (Section 5.3) with $R = 3 \\cdot 1 0 ^ { - 4 }$ (DARTS default), $R _ { m a x } = 3 \\cdot 1 0 ^ { - 2 }$ and $\\eta = 1 0$ on all the search spaces and datasets we use for image classification. The results are shown in Table 3 (DARTS-ADA). The function train and eval conducts the normal DARTS search for one epoch and returns the architecture at the end of that epoch’s updates and the stop value if a decision was made to stop the search and rollback to stop epoch. ", + "bbox": [ + 174, + 829, + 825, + 900 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Algorithm 1: DARTS ADA ", + "text_level": 1, + "bbox": [ + 174, + 125, + 346, + 138 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "/\\* E: epochs to search; $R$ : initial regularization value; $R _ { m a x }$ : maximal regularization value; stop criter: stopping criterion; η: regularization increase factor \\*/ \nInput : E, $R$ , $R _ { m a x }$ , stop criter, η \n$/ \\star$ start search for E epochs \\*/ \nfor epoch in $E$ do $/ \\star$ run DARTS for one epoch and return stop $^ { \\prime = }$ True together with the stop epoch \\*/ $/ \\star$ and the architecture at stop epoch if the criterion is met \\*/ stop, stop epoch, arch train and eval(stop criter); if stop & $R \\leq R _ { m a x }$ then /\\* start DARTS from stop epoch with a larger R \\*/ arch DARTS ADA(E - stop epoch, $\\eta \\cdot R$ , $R _ { m a x }$ , stop criter, η); break end \nend ", + "bbox": [ + 171, + 141, + 825, + 348 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/f0a0895953ddcdd46fb30b08746b7b72292a9beed0d927ed27dd155abe25308d.jpg", + "image_caption": [ + "Output: arch ", + "Figure 11: Test errors of architectures along with the validation error of the search (one-shot) model for each dataset and space when scaling the ScheduledDropPath drop probability. Note that these results (blue lines) are the same as the ones in Figure 8. " + ], + "image_footnote": [], + "bbox": [ + 176, + 416, + 821, + 845 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/b9aa25ab092484d336484395f65efd065feeff22d4645f14b4efdef49086e3d6.jpg", + "image_caption": [ + "Figure 12: Test errors of architectures along with the validation error of the search (one-shot) model for each dataset and space when scaling the $L _ { 2 }$ factor. Note that these results (blue lines) are the same as the ones in Figure 7. " + ], + "image_footnote": [], + "bbox": [ + 174, + 268, + 821, + 695 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/7b24bf051f9ab95335af0577fc8ce3ab713a7df590d547f94c17ed9aa84d1ef7.jpg", + "image_caption": [ + "Figure 13: Local average of the dominant EV $\\lambda _ { m a x } ^ { \\alpha }$ throughout DARTS search (for different drop path prob. values). Markers denote the early stopping point based on the criterion in Section 4.3. " + ], + "image_footnote": [], + "bbox": [ + 174, + 102, + 818, + 469 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/f6e2cb8af3ff642c0a7635b1e698cf0276ab7989535f7d5fa86f472276b49889.jpg", + "image_caption": [ + "Figure 14: Effect of $L _ { 2 }$ regularization no the EV trajectory. The figure is analogous to Figure 13. " + ], + "image_footnote": [], + "bbox": [ + 174, + 526, + 818, + 891 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/af9a3cc0a9a9f1fd276ef24fd923c38fefc410ade2e8c61a48b55e7b7f81feea.jpg", + "image_caption": [ + "Figure 15: Effect of ScheduledDropPath and Cutout on the full eigenspectrum of the Hessian at the end of architecture search for each of the search spaces. Since most of the eigenvalues after the 30-th largest one are almost zero, we plot only the largest (based on magnitude) 30 eigenvalues here. We also provide the eigenvalue distribution for these 30 eigenvalues. Notice that not only the dominant eigenvalue is larger when $d p = 0$ but in general also the others. " + ], + "image_footnote": [], + "bbox": [ + 178, + 29, + 820, + 848 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/2d016717a63594c416f146c80f2aaf320529f86b2416c6a34563deb2a3f22f66.jpg", + "image_caption": [ + "Figure 16: Effect of $L _ { 2 }$ regularization on the full eigenspectrum of the Hessian at the end of architecture search for each of the search spaces. Since most of the eigenvalues after the 30-th largest one are almost zero, we plot only the largest (based on magnitude) 30 eigenvalues here. We also provide the eigenvalue distribution for these 30 eigenvalues. Notice that not only the dominant eigenvalue is larger when $L _ { 2 } = 3 \\cdot 1 0 ^ { - 4 }$ but in general also the others. " + ], + "image_footnote": [], + "bbox": [ + 178, + 39, + 820, + 847 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/9527133a63f756f7d48e5e25490a96e5822d103e13bd78b213a2da457e61a4f1.jpg", + "image_caption": [ + "Figure 17: Drop in accuracy after discretizing the search model for different spaces, datasets and drop path regularization strengths.. Example of some of the settings from Section 5. " + ], + "image_footnote": [], + "bbox": [ + 279, + 88, + 714, + 306 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/7b67e03e4688728a0465362fc8a588136fd41fb25d64c359054fd5106996e48b.jpg", + "image_caption": [ + "Figure 18: Effect of more regularization on the performance of found architectures by DARTS. " + ], + "image_footnote": [], + "bbox": [ + 212, + 381, + 784, + 597 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Table 6: Performance of architectures found by DARTS (-ES / -ADA) vs. RandomNAS with weight sharing. For each of the settings we repeat the search 3 times and report the mean $\\pm$ std of the 3 found architectures retrained from scratch. ", + "bbox": [ + 176, + 683, + 825, + 724 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/9df9b2853eabbf14a3064c71bc11090c243cefaadfa326e0571da7950173cf75.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
SettingRandomNASDARTSDARTS-ESDARTS-ADA
C10S13.17 ± 0.154.66 ± 0.713.05 ± 0.073.03 ±0.08
S23.46 ± 0.154.42 ± 0.403.41 ± 0.143.59 ± 0.31
S32.92 ± 0.044.12 ± 0.853.71 ± 1.142.99 ± 0.34
S489.39 ± 0.846.95±0.184.17 ± 0.213.89 ± 0.67
C100S125.81 ± 0.3929.93 ± 0.4128.90 ± 0.8124.94 ± 0.81
S222.88 ± 0.1628.75 ± 0.9224.68 ± 1.4326.88 ± 1.11
S324.58 ± 0.6129.01 ± 0.2426.99 ± 1.7924.55± 0.63
S430.01 ± 1.5224.77 ± 1.5123.90 ± 2.0123.66 ± 0.90
SVHNS12.64±0.099.88 ± 5.502.80± 0.092.59± 0.07
S22.57 ± 0.043.69 ± 0.122.68 ± 0.182.79 ± 0.22
S32.89±0.094.00 ± 1.012.78± 0.292.58 ± 0.07
S43.42 ± 0.042.90± 0.022.55 ± 0.152.52 ± 0.06
", + "bbox": [ + 235, + 736, + 753, + 921 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "D.2 A CLOSER LOOK AT THE EIGENVALUES", + "text_level": 1, + "bbox": [ + 176, + 104, + 486, + 117 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Over the course of all experiments from the paper, we tracked the largest eigenvalue across all configuration and datasets to see how they evolve during the search. Figures 13 and 14 shows the results across all the settings for image classification. It can be clearly seen that increasing the inner objective regularization, both in terms of $L _ { 2 }$ or data augmentation, helps controlling the largest eigenvalue and keeping it to a small value, which again helps explaining why the architectures found with stronger regularization generalize better. The markers on each line highlight the epochs where DARTS is early stopped. As one can see from Figure 4, there is indeed some correlation between the average dominant eigenvalue throughout the search and the test performance of the found architectures by DARTS. ", + "bbox": [ + 174, + 131, + 825, + 256 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Figures 15 and 16 (top 3 rows) show the full spectrum (sorted based on eigenvalue absolute values) at the end of search, whilst bottom 3 rows plot the distribution of eigenvalues in the eigenspectrum. As one can see, not only the dominant eigenvalue is larger compared to the cases when the regularization is stronger and the generalization of architectures is better, but also the other eigenvalues in the spectrum have larger absolute value, indicating a sharper objective landscape towards many dimensions. Furthermore, from the distribution plots note the presence of more negative eigenvalues whenever the architectures are degenerate (lower regularization value) indicating that DARTS gets stuck in a point with larger positive and negative curvature of the validation loss objective, associated with a more degenerate Hessian matrix. ", + "bbox": [ + 174, + 262, + 825, + 387 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "E DISPARITY ESTIMATION ", + "text_level": 1, + "bbox": [ + 176, + 409, + 410, + 425 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "E.1 DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 443, + 289, + 457 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We use the FlyingThings3D dataset (Mayer et al., 2016) for training AutoDispNet. It consists of rendered stereo image pairs and their ground truth disparity maps. The dataset provides a training and testing split consisting of 21, 818 and 4248 samples respectively with an image resolution of $9 6 0 \\times 5 4 0$ . We use the Sintel dataset ( Butler et al. (2012)) for testing our networks. Sintel is another synthetic dataset from derived from an animated movie which also provides ground truth disparity maps (1064 samples) with a resolution of $1 0 2 4 \\times 4 3 6$ . ", + "bbox": [ + 174, + 469, + 825, + 553 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "E.2 TRAINING ", + "text_level": 1, + "bbox": [ + 174, + 571, + 287, + 585 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We use the AutoDispNet-C architecture as described in Saikia et al. (2019). However, we use the smaller search which consists of three operations: $M a x P o o l 3 \\times 3$ , $S e p C o n v 3 \\times 3$ , and SkipConnect. For training the search network, images are downsampled by a factor of two and trained for $3 0 0 k$ mini-batch iterations. During search, we use SGD and ADAM to optimize the inner and outer objectives respectively. Differently from the original AutoDispNet we do not warmstart the search model weights before starting the architectural parameter updates. The extracted network is also trained for $3 0 0 k$ mini-batch iterations but full resolution images are used. Here, ADAM is used for optimization and the learning rate is annealed to 0 from $1 e - 4$ , using a cosine decay schedule. ", + "bbox": [ + 174, + 598, + 825, + 724 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "E.3 EFFECT OF REGULARIZATION ON THE INNER OBJECTIVE ", + "text_level": 1, + "bbox": [ + 174, + 743, + 606, + 757 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "To study the effect of regularization on the inner objective for AutoDispNet-C we use experiment with two types of regularization: data augmentation and of $L 2$ regularization on network weights. ", + "bbox": [ + 176, + 768, + 821, + 797 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We note that we could not test the early stopping method on AutoDispNet since AutoDispNet relies on custom operations to compute feature map correlation (Dosovitskiy et al., 2015) and resampling, for which second order derivatives are currently not available (which are required to compute the Hessian). ", + "bbox": [ + 174, + 804, + 825, + 861 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Data augmentation. Inspite of fairly large number of training samples in FlyingThings3D, data augmentation is crucial for good generalization performance. Disparity estimation networks employ spatial transformations such as translation, cropping, shearing and scaling. Additionally, appearance transformations such as additive Gaussian noise, changes in brightness, contrast, gamma and color are also applied. Parameters for such transformations are sampled from a uniform or Gaussian distribution (parameterized by a mean and variance). In our experiments, we vary the data augmentation strength by multiplying the variance of these parameter distributions by a fixed factor, which we dub the augmentation scaling factor. The extracted networks are evaluated with the same augmentation parameters. The results of increasing the augmentation strength of the inner objective can be seen in Table 2. We observe that as augmentation strength increases DARTS finds networks with more number of parameters and better test performance. The best test performance is obtained for the network with maximum augmentation for the inner objective. At the same time the search model validation error increases when scaling up the augmentation factor, which again enforces the argument that the overfitting of architectural parameters is reduced by this implicit regularizer. ", + "bbox": [ + 176, + 867, + 823, + 922 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 242 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "L2 regularization. We study the effect of increasing regularization strength on the weights of the network. The results are shown in Table 2. Also in this case best test performance is obtained with the maximum regularization strength. ", + "bbox": [ + 174, + 242, + 823, + 284 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "F RESULTS ON PENN TREEBANK ", + "text_level": 1, + "bbox": [ + 176, + 304, + 460, + 320 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Here we investigate the effect of more $L _ { 2 }$ regularization on the inner objective for searching recurrent cells on Penn Treebank (PTB). We again used a reduced search space with only $R e L U$ and identity mapping as possible operations. The rest of the settings is the same as in (Liu et al., 2019). ", + "bbox": [ + 174, + 335, + 825, + 377 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We run DARTS search four independent times with different random seeds, each with four $L _ { 2 }$ regularization factors, namely $5 \\times 1 0 ^ { - 7 }$ (DARTS default), $1 5 \\times 1 0 ^ { - 7 }$ , $4 5 \\times 1 0 ^ { - 7 }$ and $1 3 5 \\times 1 0 ^ { - 7 }$ . Figure 19 shows the test perplexity of the architectures found by DARTS with the aforementioned $L _ { 2 }$ regularization values. As we can see, a stronger regularization factor on the inner objective makes the search procedure more robust. The median perplexity of the discovered architectures gets better as we increase the $L _ { 2 }$ factor from $5 \\times 1 0 ^ { - 7 }$ to $4 5 \\times 1 { \\bar { 0 } } ^ { - 7 }$ , while the search model (one-shot) validation mean perplexity increases. This observation is similar to the ones on image classification shown in Figure 10, showing again that properly regularizing the inner objective helps reduce overfitting the architectural parameters. ", + "bbox": [ + 174, + 385, + 825, + 511 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/19e3fbc500fb56ed5e804cdea7b10e36efc91d5ea8816f800626992b249fccbd.jpg", + "image_caption": [ + "Figure 19: Performance of recurrent cells found with different $L _ { 2 }$ regularization factors on the inner objective on PTB. We run DARTS 4 independent times with different random seeds, train each of them from scratch with the evaluation settings for 1600 epochs and report the median test perplexity. The blue dashed line denotes the validation perplexity of the search model. " + ], + "image_footnote": [], + "bbox": [ + 279, + 531, + 718, + 777 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "G DISCOVERED CELLS ON SEARCH SPACES S1-S4 FROM SECTION 3 ON OTHER DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 782, + 136 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/e5c15f36c9fa9942c509897788209cd3863b9fd6c00780c9c18d3d3dc06bf91a.jpg", + "image_caption": [ + "Figure 20: Reduction cells found by DARTS when ran on CIFAR-10 with its default hyperparameters on spaces S1-S4. These cells correspond with the normal ones in Figure 1. " + ], + "image_footnote": [], + "bbox": [ + 179, + 161, + 816, + 261 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/75b26c0638f345e827b81254c06ccdc067c56131d4dbdcff0c891a1cb3b3d83b.jpg", + "image_caption": [ + "Figure 21: Normal cells found by DARTS on CIFAR-100 and SVHN when ran with its default hyperparameters on spaces S1-S4. Notice the dominance of parameter-less operations such as skip connection and pooling ops. " + ], + "image_footnote": [], + "bbox": [ + 181, + 330, + 818, + 535 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/4f69ba60c7b0b915a16593f64b6ffced642774d088cd3858a8b0fb6b6eb88349.jpg", + "image_caption": [ + "Figure 22: Reduction cells found by DARTS on CIFAR-100 and SVHN when ran with its default hyperparameters on spaces S1-S4. " + ], + "image_footnote": [], + "bbox": [ + 181, + 617, + 816, + 823 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/d413bd1ced5e4fb8943b53b4f8050e43fd78a03e2ace84f2ce891cd15a69b2a2.jpg", + "image_caption": [ + "Figure 23: Normal cells found by DARTS-ES when ran with DARTS default hyperparameters on spaces S1-S4. " + ], + "image_footnote": [], + "bbox": [ + 179, + 138, + 818, + 419 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/42e9631125fbc3e9ea247caffd1295f6db3a68a426482be78c847ac31ab02ecc.jpg", + "image_caption": [ + "Figure 24: Reduction cells found by DARTS-ES when ran with DARTS default hyperparameters on spaces S1-S4. " + ], + "image_footnote": [], + "bbox": [ + 179, + 545, + 816, + 847 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/6577dd764de4c8c2b9d5784a3ecbd72913a296ed638b02c0dac98b6e25ecd97e.jpg", + "image_caption": [ + "Figure 25: Cells found by AutoDispNet when ran on S6-d. These cells correspond to the results for augmentation scale 0.0 of Table 2. " + ], + "image_footnote": [], + "bbox": [ + 192, + 145, + 808, + 232 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/fd14d0b51347b7b2184445cbcbb994a15a3919e1935a795fb0fc97c0ddc82238.jpg", + "image_caption": [ + "Figure 26: Cells found by AutoDispNet when ran on S6-d. These cells correspond to the results for augmentation scale 2.0 of Table 2. " + ], + "image_footnote": [], + "bbox": [ + 192, + 296, + 812, + 438 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/7b03cd9ea3a7ff99dead7df7e0f9171f3fb7adcd69e77572679afbdc83b4ff66.jpg", + "image_caption": [ + "Figure 27: Cells found by AutoDispNet when ran on S6-d. These cells correspond to the results for $L _ { 2 } ^ { - } = 3 \\cdot 1 0 ^ { - 4 }$ of Table 2. " + ], + "image_footnote": [], + "bbox": [ + 187, + 498, + 812, + 652 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/da3988a460bed8c275043491a290288f8b60fe0c87afb59f601d990a1bc88c99.jpg", + "image_caption": [ + "Figure 28: Cells found by AutoDispNet when ran on S6-d. These cells correspond to the results for $L _ { 2 } ^ { - } = 8 1 \\cdot 1 0 ^ { - 4 }$ of Table 2. " + ], + "image_footnote": [], + "bbox": [ + 187, + 713, + 812, + 844 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/db916f855d2c879b8897361343c9c6f5adcc6983c96909caa7de440d66fc25c0.jpg", + "image_caption": [ + "Figure 29: Normal (top row) and reduction (bottom) cells found by DARTS on CIFAR-10 when ran with its default hyperparameters on spaces S1-S4. Same as Figure 1 but with different random seed (seed 2). " + ], + "image_footnote": [], + "bbox": [ + 181, + 108, + 815, + 314 + ], + "page_idx": 27 + }, + { + "type": "image", + "img_path": "images/64a2ba721bb2305cc825e265f46d0bce30d59baf8dd30f97bfe16998f3bb0cb1.jpg", + "image_caption": [ + "Figure 30: Normal (top row) and reduction (bottom) cells found by DARTS on CIFAR-10 when ran with its default hyperparameters on spaces S1-S4. Same as Figure 1 but with different random seed (seed 3). 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However, DARTS", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 469, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 469, + 299 + ], + "score": 1.0, + "content": "does not work robustly for new problems: we identify a wide range of search", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "score": 1.0, + "content": "spaces for which DARTS yields degenerate architectures with very poor test per-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "score": 1.0, + "content": "formance. We study this failure mode and show that, while DARTS successfully", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 469, + 332 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 469, + 332 + ], + "score": 1.0, + "content": "minimizes validation loss, the found solutions generalize poorly when they coin-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 331, + 470, + 343 + ], + "spans": [ + { + "bbox": [ + 142, + 331, + 470, + 343 + ], + "score": 1.0, + "content": "cide with high validation loss curvature in the architecture space. We show that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 342, + 470, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 470, + 354 + ], + "score": 1.0, + "content": "by adding one of various types of regularization we can robustify DARTS to find", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 353, + 470, + 365 + ], + "spans": [ + { + "bbox": [ + 142, + 353, + 470, + 365 + ], + "score": 1.0, + "content": "solutions with less curvature and better generalization properties. Based on these", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 365, + 469, + 375 + ], + "spans": [ + { + "bbox": [ + 142, + 365, + 469, + 375 + ], + "score": 1.0, + "content": "observations, we propose several simple variations of DARTS that perform sub-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 375, + 469, + 387 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 469, + 387 + ], + "score": 1.0, + "content": "stantially more robustly in practice. Our observations are robust across five search", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 385, + 469, + 398 + ], + "spans": [ + { + "bbox": [ + 141, + 385, + 469, + 398 + ], + "score": 1.0, + "content": "spaces on three image classification tasks and also hold for the very different do-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 397, + 462, + 410 + ], + "spans": [ + { + "bbox": [ + 141, + 397, + 462, + 410 + ], + "score": 1.0, + "content": "mains of disparity estimation (a dense regression task) and language modelling.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 426, + 206, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 208, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 208, + 442 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "score": 1.0, + "content": "Neural Architecture Search (NAS), the process of automatically designing neural network archi-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "tectures, has recently attracted attention by achieving state-of-the-art performance on a variety of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "tasks (Zoph & Le, 2017; Real et al., 2019). Differentiable architecture search (DARTS) (Liu et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 482, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 497 + ], + "score": 1.0, + "content": "2019) significantly improved the efficiency of NAS over prior work, reducing its costs to the same or-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "der of magnitude as training a single neural network. This expanded the scope of NAS substantially,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "allowing it to also be applied on more expensive problems, such as semantic segmentation (Chenxi", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 332, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 332, + 528 + ], + "score": 1.0, + "content": "et al., 2019) or disparity estimation (Saikia et al., 2019).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "However, several researchers have also reported DARTS to not work well, in some cases even no", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 555 + ], + "score": 1.0, + "content": "better than random search (Li & Talwalkar, 2019; Sciuto et al., 2019). Why is this? How can these", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "seemingly contradicting results be explained? The overall goal of this paper is to understand and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 565, + 473, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 473, + 578 + ], + "score": 1.0, + "content": "overcome such failure modes of DARTS. To this end, we make the following contributions:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 585, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "1. We identify 12 NAS benchmarks based on four search spaces in which standard DARTS yields", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 118, + 597, + 468, + 609 + ], + "spans": [ + { + "bbox": [ + 118, + 597, + 468, + 609 + ], + "score": 1.0, + "content": "degenerate architectures with poor test performance across several datasets (Section 3).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "2. By computing the eigenspectrum of the Hessian of the validation loss with respect to the archi-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 118, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 118, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "tectural parameters, we show that there is a strong correlation between its dominant eigenvalue", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 118, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 118, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "and the architecture’s generalization error. Based on this finding, we propose a simple variation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 118, + 643, + 459, + 657 + ], + "spans": [ + { + "bbox": [ + 118, + 643, + 459, + 657 + ], + "score": 1.0, + "content": "of DARTS with early stopping that performs substantially more robustly (Section 4).", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "score": 1.0, + "content": "3. We show that, related to previous work on sharp/flat local minima, regularizing the inner objective", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 118, + 669, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 118, + 669, + 506, + 681 + ], + "score": 1.0, + "content": "of DARTS more strongly allows it to find solutions with smaller Hessian spectrum and better", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 117, + 678, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 117, + 678, + 506, + 692 + ], + "score": 1.0, + "content": "generalization properties. Based on these insights, we propose two practical robustifications of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 118, + 690, + 453, + 703 + ], + "spans": [ + { + "bbox": [ + 118, + 690, + 453, + 703 + ], + "score": 1.0, + "content": "DARTS that overcome its failure modes in all our 12 NAS benchmarks (Section 5).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Our findings are robust across a wide range of NAS benchmarks based on image recognition and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "also hold for the very different domains of language modelling (PTB) and disparity estimation. 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However, DARTS", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 469, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 469, + 299 + ], + "score": 1.0, + "content": "does not work robustly for new problems: we identify a wide range of search", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "score": 1.0, + "content": "spaces for which DARTS yields degenerate architectures with very poor test per-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "score": 1.0, + "content": "formance. We study this failure mode and show that, while DARTS successfully", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 469, + 332 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 469, + 332 + ], + "score": 1.0, + "content": "minimizes validation loss, the found solutions generalize poorly when they coin-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 331, + 470, + 343 + ], + "spans": [ + { + "bbox": [ + 142, + 331, + 470, + 343 + ], + "score": 1.0, + "content": "cide with high validation loss curvature in the architecture space. We show that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 342, + 470, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 470, + 354 + ], + "score": 1.0, + "content": "by adding one of various types of regularization we can robustify DARTS to find", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 353, + 470, + 365 + ], + "spans": [ + { + "bbox": [ + 142, + 353, + 470, + 365 + ], + "score": 1.0, + "content": "solutions with less curvature and better generalization properties. Based on these", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 365, + 469, + 375 + ], + "spans": [ + { + "bbox": [ + 142, + 365, + 469, + 375 + ], + "score": 1.0, + "content": "observations, we propose several simple variations of DARTS that perform sub-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 375, + 469, + 387 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 469, + 387 + ], + "score": 1.0, + "content": "stantially more robustly in practice. Our observations are robust across five search", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 385, + 469, + 398 + ], + "spans": [ + { + "bbox": [ + 141, + 385, + 469, + 398 + ], + "score": 1.0, + "content": "spaces on three image classification tasks and also hold for the very different do-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 397, + 462, + 410 + ], + "spans": [ + { + "bbox": [ + 141, + 397, + 462, + 410 + ], + "score": 1.0, + "content": "mains of disparity estimation (a dense regression task) and language modelling.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15.5, + "bbox_fs": [ + 141, + 254, + 470, + 410 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 426, + 206, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 208, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 208, + 442 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "score": 1.0, + "content": "Neural Architecture Search (NAS), the process of automatically designing neural network archi-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "tectures, has recently attracted attention by achieving state-of-the-art performance on a variety of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "tasks (Zoph & Le, 2017; Real et al., 2019). Differentiable architecture search (DARTS) (Liu et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 482, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 497 + ], + "score": 1.0, + "content": "2019) significantly improved the efficiency of NAS over prior work, reducing its costs to the same or-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "der of magnitude as training a single neural network. This expanded the scope of NAS substantially,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "allowing it to also be applied on more expensive problems, such as semantic segmentation (Chenxi", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 332, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 332, + 528 + ], + "score": 1.0, + "content": "et al., 2019) or disparity estimation (Saikia et al., 2019).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 449, + 506, + 528 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "However, several researchers have also reported DARTS to not work well, in some cases even no", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 555 + ], + "score": 1.0, + "content": "better than random search (Li & Talwalkar, 2019; Sciuto et al., 2019). Why is this? How can these", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "seemingly contradicting results be explained? The overall goal of this paper is to understand and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 565, + 473, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 473, + 578 + ], + "score": 1.0, + "content": "overcome such failure modes of DARTS. To this end, we make the following contributions:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 533, + 505, + 578 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 585, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "1. We identify 12 NAS benchmarks based on four search spaces in which standard DARTS yields", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 118, + 597, + 468, + 609 + ], + "spans": [ + { + "bbox": [ + 118, + 597, + 468, + 609 + ], + "score": 1.0, + "content": "degenerate architectures with poor test performance across several datasets (Section 3).", + "type": "text" + } + ], + "index": 36, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "2. By computing the eigenspectrum of the Hessian of the validation loss with respect to the archi-", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 118, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 118, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "tectural parameters, we show that there is a strong correlation between its dominant eigenvalue", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 118, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 118, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "and the architecture’s generalization error. Based on this finding, we propose a simple variation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 118, + 643, + 459, + 657 + ], + "spans": [ + { + "bbox": [ + 118, + 643, + 459, + 657 + ], + "score": 1.0, + "content": "of DARTS with early stopping that performs substantially more robustly (Section 4).", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "score": 1.0, + "content": "3. We show that, related to previous work on sharp/flat local minima, regularizing the inner objective", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 118, + 669, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 118, + 669, + 506, + 681 + ], + "score": 1.0, + "content": "of DARTS more strongly allows it to find solutions with smaller Hessian spectrum and better", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 117, + 678, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 117, + 678, + 506, + 692 + ], + "score": 1.0, + "content": "generalization properties. Based on these insights, we propose two practical robustifications of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 118, + 690, + 453, + 703 + ], + "spans": [ + { + "bbox": [ + 118, + 690, + 453, + 703 + ], + "score": 1.0, + "content": "DARTS that overcome its failure modes in all our 12 NAS benchmarks (Section 5).", + "type": "text" + } + ], + "index": 44, + "is_list_end_line": true + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 586, + 506, + 703 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Our findings are robust across a wide range of NAS benchmarks based on image recognition and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "also hold for the very different domains of language modelling (PTB) and disparity estimation. They", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "consolidate the findings of the various results in the literature and lead to a substantially more robust", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 474, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 474, + 105 + ], + "score": 1.0, + "content": "version of DARTS. We provide our implementation and scripts to facilitate reproducibility1.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 709, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "consolidate the findings of the various results in the literature and lead to a substantially more robust", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 474, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 474, + 105 + ], + "score": 1.0, + "content": "version of DARTS. We provide our implementation and scripts to facilitate reproducibility1.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 107, + 123, + 313, + 136 + ], + "lines": [ + { + "bbox": [ + 104, + 122, + 315, + 138 + ], + "spans": [ + { + "bbox": [ + 104, + 122, + 315, + 138 + ], + "score": 1.0, + "content": "2 BACKGROUND AND RELATED WORK", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 106, + 150, + 477, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 150, + 478, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 478, + 162 + ], + "score": 1.0, + "content": "2.1 RELATION BETWEEN FLAT/SHARP MINIMA AND GENERALIZATION PERFORMANCE", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 171, + 505, + 260 + ], + "lines": [ + { + "bbox": [ + 106, + 171, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 506, + 184 + ], + "score": 1.0, + "content": "Already Hochreiter & Schmidhuber (1997) observed that flat minima of the training loss yield better", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 183, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 195 + ], + "score": 1.0, + "content": "generalization performance than sharp minima. Recent work (Keskar et al., 2016; Yao et al., 2018)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "focuses more on the settings of large/small batch size training, where observations show that small", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "batch training tends to get attracted to flatter minima and generalizes better. Similarly, Nguyen et al.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "score": 1.0, + "content": "(2018) observed that this phenomenon manifests also in the hyperparameter space. They showed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "that whenever the hyperparameters overfit the validation data, the minima lie in a sharper region of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 238, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 505, + 249 + ], + "score": 1.0, + "content": "the space. This motivated us to conduct a similar analysis in the context of differentiable architecture", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "search later in Section 4.1, where we see the same effect in the space of neural network architectures.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 276, + 240, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 241, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 241, + 289 + ], + "score": 1.0, + "content": "2.2 BI-LEVEL OPTIMIZATION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 503, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 310 + ], + "score": 1.0, + "content": "We start by a short introduction of the bi-level optimization problem (Colson et al., 2007). These", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 308, + 416, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 416, + 321 + ], + "score": 1.0, + "content": "are problems which contain two optimization tasks, nested within each other.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 504, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 316, + 338 + ], + "score": 1.0, + "content": "Definition 2.1. Given the outer objective function", + "type": "text" + }, + { + "bbox": [ + 316, + 323, + 407, + 335 + ], + "score": 0.92, + "content": "F : \\mathbb { R } ^ { P } \\times \\mathbb { R } ^ { N } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 320, + 506, + 338 + ], + "score": 1.0, + "content": "and the inner objective", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 332, + 407, + 350 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 142, + 350 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 335, + 221, + 347 + ], + "score": 0.92, + "content": "f : \\mathbb { R } ^ { P } \\times \\mathbb { R } ^ { N } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 332, + 407, + 350 + ], + "score": 1.0, + "content": ", the bi-level optimization problem is given by", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 353, + 368, + 397 + ], + "lines": [ + { + "bbox": [ + 242, + 353, + 368, + 397 + ], + "spans": [ + { + "bbox": [ + 242, + 353, + 368, + 397 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\underset { y \\in \\mathbb { R } ^ { P } } { \\operatorname* { m i n } } F ( y , \\theta ^ { * } ( y ) ) } \\\\ & { s . t . \\quad \\theta ^ { * } ( y ) \\in \\underset { \\theta \\in \\mathbb { R } ^ { N } } { \\arg \\operatorname* { m i n } } f ( y , \\theta ) , } \\end{array}", + "type": "interline_equation", + "image_path": "b761278795671c43fe1527e1d6f6a0b6c9b4613ca324f66d39f1681c3bab5662.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 242, + 353, + 368, + 375.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 242, + 375.0, + 368, + 397.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 504, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 133, + 425 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 411, + 168, + 423 + ], + "score": 0.93, + "content": "y \\in \\mathbb { R } ^ { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 410, + 187, + 425 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 188, + 411, + 222, + 422 + ], + "score": 0.92, + "content": "\\boldsymbol { \\theta } \\in \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 410, + 506, + 425 + ], + "score": 1.0, + "content": "are the outer and inner variables, respectively. One may also see the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 421, + 487, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 487, + 435 + ], + "score": 1.0, + "content": "bi-level problem as a constrained optimization problem, with the inner problem as a constraint.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 504, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 504, + 451 + ], + "score": 1.0, + "content": "In general, even in the case when the inner objective (2) is strongly convex and has an unique", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 150, + 464 + ], + "score": 1.0, + "content": "minimizer", + "type": "text" + }, + { + "bbox": [ + 151, + 451, + 272, + 463 + ], + "score": 0.9, + "content": "\\theta ^ { * } ( y ) = \\arg \\operatorname* { m i n } _ { \\theta \\in \\mathbb { R } ^ { N } } f ( y , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 450, + 506, + 464 + ], + "score": 1.0, + "content": ", it is not possible to directly optimize the outer objective", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "(1). A possible method around this issue is to use the implicit function theorem to retrieve the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 470, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 298, + 487 + ], + "score": 1.0, + "content": "derivative of the solution map (or response map)", + "type": "text" + }, + { + "bbox": [ + 298, + 472, + 367, + 485 + ], + "score": 0.92, + "content": "\\theta ^ { * } ( y ) \\in \\mathbb { F } \\subseteq \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 470, + 390, + 487 + ], + "score": 1.0, + "content": "w.r.t.", + "type": "text" + }, + { + "bbox": [ + 390, + 474, + 398, + 484 + ], + "score": 0.78, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 470, + 506, + 487 + ], + "score": 1.0, + "content": "(Bengio, 2000; Pedregosa,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "2016; Beirami et al., 2017). Another strategy is to approximate the inner problem with a dynamical", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "system (Domke, 2012; Maclaurin et al., 2015; Franceschi et al., 2017; 2018), where the optimization", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "dynamics could, e.g., describe gradient descent. In the case that the minimizer of the inner problem", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "is unique, under some conditions the set of minimizers of this approximate problem will indeed", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 528, + 442, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 442, + 540 + ], + "score": 1.0, + "content": "converge to the minimizers of the bilevel problem (1) (see Franceschi et al. (2018)).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 108, + 554, + 276, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 278, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 278, + 567 + ], + "score": 1.0, + "content": "2.3 NEURAL ARCHITECTURE SEARCH", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 576, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "Neural Architecture Search (NAS) denotes the process of automatically designing neural network", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "architectures in order to overcome the cumbersome trial-and-error process when designing archi-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 599, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 610 + ], + "score": 1.0, + "content": "tectures manually. We briefly review NAS here and refer to the recent survey by Elsken et al.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "score": 1.0, + "content": "(2019b) for a more thorough overview. Prior work mostly employs either reinforcement learning", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "techniques (Baker et al., 2017a; Zoph & Le, 2017; Zhong et al., 2018; Zoph et al., 2018) or evo-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 632, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 643 + ], + "score": 1.0, + "content": "lutionary algorithms (Stanley & Miikkulainen, 2002; Liu et al., 2018b; Miikkulainen et al., 2017;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "Real et al., 2017; 2019) to optimize the discrete architecture space. As these methods are often", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "very expensive, various works focus on reducing the search costs by, e.g., employing network mor-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 664, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 676 + ], + "score": 1.0, + "content": "phisms (Cai et al., 2018a;b; Elsken et al., 2017; 2019a), weight sharing within search models (Sax-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "ena & Verbeek, 2016; Bender et al., 2018; Pham et al., 2018) or multi-fidelity optimization (Baker", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 685, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 104, + 685, + 505, + 699 + ], + "score": 1.0, + "content": "et al., 2017b; Falkner et al., 2018; Li et al., 2017; Zela et al., 2018), but their applicability still often", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 698, + 345, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 345, + 709 + ], + "score": 1.0, + "content": "remains restricted to rather simple tasks and small datasets.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 36.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 722, + 325, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 326, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 326, + 733 + ], + "score": 1.0, + "content": "1 https://github.com/automl/RobustDARTS", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 106, + 82, + 505, + 105 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 123, + 313, + 136 + ], + "lines": [ + { + "bbox": [ + 104, + 122, + 315, + 138 + ], + "spans": [ + { + "bbox": [ + 104, + 122, + 315, + 138 + ], + "score": 1.0, + "content": "2 BACKGROUND AND RELATED WORK", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 106, + 150, + 477, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 150, + 478, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 478, + 162 + ], + "score": 1.0, + "content": "2.1 RELATION BETWEEN FLAT/SHARP MINIMA AND GENERALIZATION PERFORMANCE", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 171, + 505, + 260 + ], + "lines": [ + { + "bbox": [ + 106, + 171, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 506, + 184 + ], + "score": 1.0, + "content": "Already Hochreiter & Schmidhuber (1997) observed that flat minima of the training loss yield better", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 183, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 195 + ], + "score": 1.0, + "content": "generalization performance than sharp minima. Recent work (Keskar et al., 2016; Yao et al., 2018)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "focuses more on the settings of large/small batch size training, where observations show that small", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "batch training tends to get attracted to flatter minima and generalizes better. Similarly, Nguyen et al.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "score": 1.0, + "content": "(2018) observed that this phenomenon manifests also in the hyperparameter space. They showed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "that whenever the hyperparameters overfit the validation data, the minima lie in a sharper region of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 238, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 505, + 249 + ], + "score": 1.0, + "content": "the space. This motivated us to conduct a similar analysis in the context of differentiable architecture", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "search later in Section 4.1, where we see the same effect in the space of neural network architectures.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 171, + 506, + 261 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 276, + 240, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 241, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 241, + 289 + ], + "score": 1.0, + "content": "2.2 BI-LEVEL OPTIMIZATION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 503, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 310 + ], + "score": 1.0, + "content": "We start by a short introduction of the bi-level optimization problem (Colson et al., 2007). These", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 308, + 416, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 416, + 321 + ], + "score": 1.0, + "content": "are problems which contain two optimization tasks, nested within each other.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 297, + 505, + 321 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 504, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 316, + 338 + ], + "score": 1.0, + "content": "Definition 2.1. Given the outer objective function", + "type": "text" + }, + { + "bbox": [ + 316, + 323, + 407, + 335 + ], + "score": 0.92, + "content": "F : \\mathbb { R } ^ { P } \\times \\mathbb { R } ^ { N } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 320, + 506, + 338 + ], + "score": 1.0, + "content": "and the inner objective", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 332, + 407, + 350 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 142, + 350 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 335, + 221, + 347 + ], + "score": 0.92, + "content": "f : \\mathbb { R } ^ { P } \\times \\mathbb { R } ^ { N } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 332, + 407, + 350 + ], + "score": 1.0, + "content": ", the bi-level optimization problem is given by", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 320, + 506, + 350 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 353, + 368, + 397 + ], + "lines": [ + { + "bbox": [ + 242, + 353, + 368, + 397 + ], + "spans": [ + { + "bbox": [ + 242, + 353, + 368, + 397 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\underset { y \\in \\mathbb { R } ^ { P } } { \\operatorname* { m i n } } F ( y , \\theta ^ { * } ( y ) ) } \\\\ & { s . t . \\quad \\theta ^ { * } ( y ) \\in \\underset { \\theta \\in \\mathbb { R } ^ { N } } { \\arg \\operatorname* { m i n } } f ( y , \\theta ) , } \\end{array}", + "type": "interline_equation", + "image_path": "b761278795671c43fe1527e1d6f6a0b6c9b4613ca324f66d39f1681c3bab5662.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 242, + 353, + 368, + 375.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 242, + 375.0, + 368, + 397.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 504, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 133, + 425 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 411, + 168, + 423 + ], + "score": 0.93, + "content": "y \\in \\mathbb { R } ^ { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 410, + 187, + 425 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 188, + 411, + 222, + 422 + ], + "score": 0.92, + "content": "\\boldsymbol { \\theta } \\in \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 410, + 506, + 425 + ], + "score": 1.0, + "content": "are the outer and inner variables, respectively. One may also see the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 421, + 487, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 487, + 435 + ], + "score": 1.0, + "content": "bi-level problem as a constrained optimization problem, with the inner problem as a constraint.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 410, + 506, + 435 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 504, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 504, + 451 + ], + "score": 1.0, + "content": "In general, even in the case when the inner objective (2) is strongly convex and has an unique", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 150, + 464 + ], + "score": 1.0, + "content": "minimizer", + "type": "text" + }, + { + "bbox": [ + 151, + 451, + 272, + 463 + ], + "score": 0.9, + "content": "\\theta ^ { * } ( y ) = \\arg \\operatorname* { m i n } _ { \\theta \\in \\mathbb { R } ^ { N } } f ( y , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 450, + 506, + 464 + ], + "score": 1.0, + "content": ", it is not possible to directly optimize the outer objective", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "(1). A possible method around this issue is to use the implicit function theorem to retrieve the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 470, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 298, + 487 + ], + "score": 1.0, + "content": "derivative of the solution map (or response map)", + "type": "text" + }, + { + "bbox": [ + 298, + 472, + 367, + 485 + ], + "score": 0.92, + "content": "\\theta ^ { * } ( y ) \\in \\mathbb { F } \\subseteq \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 470, + 390, + 487 + ], + "score": 1.0, + "content": "w.r.t.", + "type": "text" + }, + { + "bbox": [ + 390, + 474, + 398, + 484 + ], + "score": 0.78, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 470, + 506, + 487 + ], + "score": 1.0, + "content": "(Bengio, 2000; Pedregosa,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "2016; Beirami et al., 2017). Another strategy is to approximate the inner problem with a dynamical", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "system (Domke, 2012; Maclaurin et al., 2015; Franceschi et al., 2017; 2018), where the optimization", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "dynamics could, e.g., describe gradient descent. In the case that the minimizer of the inner problem", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "is unique, under some conditions the set of minimizers of this approximate problem will indeed", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 528, + 442, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 442, + 540 + ], + "score": 1.0, + "content": "converge to the minimizers of the bilevel problem (1) (see Franceschi et al. (2018)).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 439, + 506, + 540 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 554, + 276, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 278, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 278, + 567 + ], + "score": 1.0, + "content": "2.3 NEURAL ARCHITECTURE SEARCH", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 576, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "Neural Architecture Search (NAS) denotes the process of automatically designing neural network", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "architectures in order to overcome the cumbersome trial-and-error process when designing archi-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 599, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 610 + ], + "score": 1.0, + "content": "tectures manually. We briefly review NAS here and refer to the recent survey by Elsken et al.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "score": 1.0, + "content": "(2019b) for a more thorough overview. Prior work mostly employs either reinforcement learning", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "techniques (Baker et al., 2017a; Zoph & Le, 2017; Zhong et al., 2018; Zoph et al., 2018) or evo-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 632, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 643 + ], + "score": 1.0, + "content": "lutionary algorithms (Stanley & Miikkulainen, 2002; Liu et al., 2018b; Miikkulainen et al., 2017;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "Real et al., 2017; 2019) to optimize the discrete architecture space. As these methods are often", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "very expensive, various works focus on reducing the search costs by, e.g., employing network mor-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 664, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 676 + ], + "score": 1.0, + "content": "phisms (Cai et al., 2018a;b; Elsken et al., 2017; 2019a), weight sharing within search models (Sax-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "ena & Verbeek, 2016; Bender et al., 2018; Pham et al., 2018) or multi-fidelity optimization (Baker", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 685, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 104, + 685, + 505, + 699 + ], + "score": 1.0, + "content": "et al., 2017b; Falkner et al., 2018; Li et al., 2017; Zela et al., 2018), but their applicability still often", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 698, + 345, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 345, + 709 + ], + "score": 1.0, + "content": "remains restricted to rather simple tasks and small datasets.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 576, + 506, + 709 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 359, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 360, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 360, + 96 + ], + "score": 1.0, + "content": "2.4 DIFFERENTIABLE ARCHITECTURE SEARCH (DARTS)", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "A recent line of work focuses on relaxing the discrete neural architecture search problem to a con-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 127 + ], + "score": 1.0, + "content": "tinuous one that can be solved by gradient descent (Liu et al., 2019; Xie et al., 2019; Casale et al.,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 138 + ], + "score": 1.0, + "content": "2019; Cai et al., 2019). In DARTS (Liu et al., 2019), this is achieved by simply using a weighted", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 137, + 504, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 504, + 148 + ], + "score": 1.0, + "content": "sum of possible candidate operations for each layer, whereas the real-valued weights then effec-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 160 + ], + "score": 1.0, + "content": "tively parametrize the network’s architecture. We will now review DARTS in more detail, as our", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 159, + 222, + 170 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 222, + 170 + ], + "score": 1.0, + "content": "work builds directly upon it.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 182, + 505, + 313 + ], + "lines": [ + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "Continuous relaxation of the search space. In agreement with prior work (Zoph et al., 2018;", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "Real et al., 2019), DARTS optimizes only substructures called cells that are stacked to define the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 291, + 216 + ], + "score": 1.0, + "content": "full network architecture. Each cell contains", + "type": "text" + }, + { + "bbox": [ + 291, + 204, + 301, + 214 + ], + "score": 0.78, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "nodes organized in a directed acyclic graph. The", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 227 + ], + "score": 1.0, + "content": "graph contains two inputs nodes (given by the outputs of the previous two cells), a set of intermediate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "nodes, and one output node (given by concatenating all intermediate nodes). Each intermediate node", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 235, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 124, + 248 + ], + "score": 0.9, + "content": "x ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 235, + 506, + 252 + ], + "score": 1.0, + "content": "represents a feature map. See Figure 1 for an illustration of such a cell. Instead of applying", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "a single operation to a specific node during architecture search, Liu et al. (2019) relax the decision", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 259, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 273 + ], + "score": 1.0, + "content": "which operation to choose by computing the intermediate node as a mixture of candidate operations,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 102, + 269, + 504, + 294 + ], + "spans": [ + { + "bbox": [ + 102, + 269, + 435, + 294 + ], + "score": 1.0, + "content": "applied to predecessor nodes x(i), i < j, x(j) = PiBenchmarkDARTSDARTS-ESC10S14.66 ± 0.713.05± 0.07S24.42 ± 0.403.41 ± 0.14S34.12 ± 0.853.71 ± 1.14S46.95±0.184.17 ± 0.21C100S129.93 ± 0.4128.90±0.81S228.75±0.9224.68±1.43S329.01 ± 0.2426.99 ± 1.79S424.77 ± 1.5123.90±2.01SVHNS19.88±5.502.80±0.09S23.69 ±0.122.68± 0.18S34.00 ± 1.012.78± 0.29S42.90±0.022.55±0.15", + "type": "table", + "image_path": "888cc870183c9284910dc4cfcbdd7b8a7ce0be6bdffc14bc4a0d7b2457e1a49b.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 384, + 297, + 502, + 349.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 384, + 349.0, + 502, + 401.0 + ], + "spans": [], + "index": 21 + } + ] + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 407, + 359, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 331, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 331, + 421 + ], + "score": 1.0, + "content": "5 REGULARIZATION OF INNER OBJECTIVE", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 422, + 360, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 360, + 435 + ], + "score": 1.0, + "content": "IMPROVES GENERALIZATION OF ARCHITECTURES", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "As we saw in Section 4.1, sharper minima (by means of large eigenvalues) of the validation loss", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 457, + 504, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 504, + 470 + ], + "score": 1.0, + "content": "lead to poor generalization performance. In our bi-level optimization setting, the outer variables’", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 467, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 104, + 467, + 506, + 483 + ], + "score": 1.0, + "content": "trajectory depends on the inner optimization procedure. 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We study two ways of regularization (data augmentation in Section 5.1 and", + "type": "text" + }, + { + "bbox": [ + 491, + 491, + 504, + 501 + ], + "score": 0.87, + "content": "L _ { 2 }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "regularization in Section 5.2) and find that both, along with the early stopping criterion from Section", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "4.3, make DARTS more robust in practice. We emphasize that we do not alter the regularization of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 523, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 535 + ], + "score": 1.0, + "content": "the final training and evaluation phase, but solely that of the search phase. The setting we use for all", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 534, + 465, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 465, + 547 + ], + "score": 1.0, + "content": "experiments in this paper to obtain the final test performance is described in Appendix C.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 107, + 562, + 323, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 324, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 324, + 575 + ], + "score": 1.0, + "content": "5.1 REGULARIZATION VIA DATA AUGMENTATION", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "We first investigate the effect of regularizing via data augmentation, namely masking out parts of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "the input and intermediate feature maps via Cutout (CO, DeVries & Taylor (2017)) and Scheduled-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "DropPath (DP, Zoph et al. 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We ran", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "DARTS with CO and DP (with and without our early stopping criterion, DARTS-ES) with different", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 639, + 482, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 482, + 650 + ], + "score": 1.0, + "content": "maximum DP probabilities on all three image classification datasets and search spaces S1-S4.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "Figure 7 summarizes the results: regularization improves the test performance of DARTS and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "DARTS-ES in all cases, sometimes very substantially, and at the same time kept the dominant eigen-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 676, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 691 + ], + "score": 1.0, + "content": "value relatively low (Figure 13). 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BenchmarkDARTSDARTS-ES
C10S14.66 ± 0.713.05± 0.07
S24.42 ± 0.403.41 ± 0.14
S34.12 ± 0.853.71 ± 1.14
S46.95±0.184.17 ± 0.21
C100S129.93 ± 0.4128.90±0.81
S228.75±0.9224.68±1.43
S329.01 ± 0.2426.99 ± 1.79
S424.77 ± 1.5123.90±2.01
SVHNS19.88±5.502.80±0.09
S23.69 ±0.122.68± 0.18
S34.00 ± 1.012.78± 0.29
S42.90±0.022.55±0.15
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The simplest off-the-shelf procedure towards this desiderata", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 629, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 644 + ], + "score": 1.0, + "content": "would be to increase the regularization strength whenever the dominant eigenvalue starts increasing", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "score": 1.0, + "content": "rapidly. Algorithm 1 (DARTS-ADA, Appendix D.1) shows such a procedure. 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This procedure is repeated whenever the criterion is met, unless the regularization", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 682, + 329, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 299, + 699 + ], + "score": 1.0, + "content": "value exceeds some maximum predefined value", + "type": "text" + }, + { + "bbox": [ + 299, + 685, + 324, + 696 + ], + "score": 0.91, + "content": "R _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 682, + 329, + 699 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 596, + 506, + 699 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 707, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 505, + 723 + ], + "score": 1.0, + "content": "Multiple DARTS runs with different regularization strength Liu et al. (2019) already sug-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "gested to run the search phase of DARTS four times, resulting in four architectures, and to return", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 83, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 506, + 95 + ], + "score": 1.0, + "content": "the best of these four architectures w.r.t. validation performance when retrained from scratch for a", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "limited number of epochs. 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BenchmarkRS-wsDARTSR-DARTS(DP)R-DARTS(L2)DARTS-ESDARTS-ADA
C10S13.233.843.112.783.013.10
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S221.2126.0522.3022.2423.2523.52
S323.7528.9022.3623.9923.7323.37
S428.1922.8522.1821.9421.2623.20
SVHNS12.594.582.554.792.722.53
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As Table 4", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 434, + 336, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 336, + 446 + ], + "score": 1.0, + "content": "shows, RobustDARTS performed similarly to DARTS", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 444, + 337, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 337, + 457 + ], + "score": 1.0, + "content": "for the two original benchmarks from the DARTS paper", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 456, + 337, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 337, + 468 + ], + "score": 1.0, + "content": "(PTB and CIFAR-10), on which DARTS was developed", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 467, + 337, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 337, + 479 + ], + "score": 1.0, + "content": "and is well tuned; however, even when only changing the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 477, + 337, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 337, + 490 + ], + "score": 1.0, + "content": "dataset to CIFAR-100 or SVHN, RobustDARTS already", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 488, + 337, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 337, + 502 + ], + "score": 1.0, + "content": "performed significantly better than DARTS, underlining", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 500, + 165, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 165, + 512 + ], + "score": 1.0, + "content": "its robustness.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 344, + 391, + 504, + 456 + ], + "lines": [ + { + "bbox": [ + 343, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 343, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "Table 4: DARTS vs. RobustDARTS on", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 344, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 344, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "the original DARTS search spaces. 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BenchmarkDARTSR-DARTS(L2)
C102.91± 0.252.95 ± 0.21
C10020.58 ± 0.4418.01 ± 0.26
SVHN2.46±0.092.17 ± 0.09
PTB58.6457.59
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They are consistent across many different search spaces on image recognition tasks and", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "also for the very different domains of language modelling and disparity estimation. 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BenchmarkRS-wsDARTSR-DARTS(DP)R-DARTS(L2)DARTS-ESDARTS-ADA
C10S13.233.843.112.783.013.10
S23.664.853.483.313.263.35
S32.953.342.932.512.742.59
S48.077.203.583.563.714.84
C100S123.3029.4625.9324.2528.3724.03
S221.2126.0522.3022.2423.2523.52
S323.7528.9022.3623.9923.7323.37
S428.1922.8522.1821.9421.2623.20
SVHNS12.594.582.554.792.722.53
S22.723.532.522.512.602.54
S32.873.412.492.482.502.50
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Finally, RobustDARTS yielded the best performance and since it", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "is also easier to implement than DARTS-ES and DARTS-ADA, it is the method that we recommend", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 374, + 196, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 196, + 386 + ], + "score": 1.0, + "content": "to be used in practice.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 307, + 506, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 335, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 337, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 337, + 402 + ], + "score": 1.0, + "content": "Finally, since the evaluations in this paper have so far", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 401, + 336, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 336, + 414 + ], + "score": 1.0, + "content": "focussed on smaller subspaces of the original DARTS", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 412, + 337, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 337, + 424 + ], + "score": 1.0, + "content": "search space, the reader may wonder how well Robust-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 423, + 337, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 337, + 435 + ], + "score": 1.0, + "content": "DARTS works on the full search spaces. As Table 4", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 434, + 336, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 336, + 446 + ], + "score": 1.0, + "content": "shows, RobustDARTS performed similarly to DARTS", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 444, + 337, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 337, + 457 + ], + "score": 1.0, + "content": "for the two original benchmarks from the DARTS paper", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 456, + 337, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 337, + 468 + ], + "score": 1.0, + "content": "(PTB and CIFAR-10), on which DARTS was developed", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 467, + 337, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 337, + 479 + ], + "score": 1.0, + "content": "and is well tuned; however, even when only changing the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 477, + 337, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 337, + 490 + ], + "score": 1.0, + "content": "dataset to CIFAR-100 or SVHN, RobustDARTS already", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 488, + 337, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 337, + 502 + ], + "score": 1.0, + "content": "performed significantly better than DARTS, underlining", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 500, + 165, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 165, + 512 + ], + "score": 1.0, + "content": "its robustness.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 390, + 337, + 512 + ] + }, + { + "type": "text", + "bbox": [ + 344, + 391, + 504, + 456 + ], + "lines": [ + { + "bbox": [ + 343, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 343, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "Table 4: DARTS vs. RobustDARTS on", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 344, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 344, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "the original DARTS search spaces. We", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 343, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 343, + 412, + 393, + 424 + ], + "score": 1.0, + "content": "show mean", + "type": "text" + }, + { + "bbox": [ + 394, + 414, + 405, + 423 + ], + "score": 0.51, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "stddev for 5 repetitions", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 344, + 424, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 344, + 424, + 505, + 435 + ], + "score": 1.0, + "content": "(based on 4 fresh subruns each as in Ta-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 343, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 343, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "ble 3); for the more expensive PTB we", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 344, + 446, + 487, + 457 + ], + "spans": [ + { + "bbox": [ + 344, + 446, + 487, + 457 + ], + "score": 1.0, + "content": "could only afford 1 such repetition.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 52.5, + "bbox_fs": [ + 343, + 390, + 505, + 457 + ] + }, + { + "type": "table", + "bbox": [ + 346, + 466, + 502, + 520 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 346, + 466, + 502, + 520 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 346, + 466, + 502, + 520 + ], + "spans": [ + { + "bbox": [ + 346, + 466, + 502, + 520 + ], + "score": 0.975, + "html": "
BenchmarkDARTSR-DARTS(L2)
C102.91± 0.252.95 ± 0.21
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In Interna-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 115, + 561, + 336, + 574 + ], + "spans": [ + { + "bbox": [ + 115, + 561, + 336, + 574 + ], + "score": 1.0, + "content": "tional Conference on Learning Representations, 2017.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 549, + 504, + 574 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 503, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 580, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 593 + ], + "score": 1.0, + "content": "Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V. Le. Learning transferable architectures", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 116, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 116, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "for scalable image recognition. In Conference on Computer Vision and Pattern Recognition, 2018.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 106, + 580, + 505, + 604 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 263, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 264, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 264, + 96 + ], + "score": 1.0, + "content": "A MORE DETAIL ON DARTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 504, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 504, + 118 + ], + "score": 1.0, + "content": "Here we present a detailed description of DARTS architectural update steps. We firstly provide", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "the general formalism which computes the gradient of the outer level problem in (1) by means of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "score": 1.0, + "content": "the implicit function theorem. Afterwards, we present how DARTS computes the gradient used to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 261, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 249, + 152 + ], + "score": 1.0, + "content": "update the architectural parameters", + "type": "text" + }, + { + "bbox": [ + 249, + 141, + 256, + 149 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 139, + 261, + 152 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 164, + 443, + 175 + ], + "lines": [ + { + "bbox": [ + 106, + 164, + 445, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 445, + 176 + ], + "score": 1.0, + "content": "A.1 DERIVATIVE WITH SMOOTHED NON-QUADRATIC LOWER LEVEL PROBLEM", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 183, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 107, + 185, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 107, + 185, + 505, + 196 + ], + "score": 1.0, + "content": "Consider the general definition of the bi-level optimization problem as given by (1) and (2). Given", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 125, + 208 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 196, + 133, + 207 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 194, + 506, + 208 + ], + "score": 1.0, + "content": "is twice continuously differentiable and that all stationary points are local minimas, one", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 206, + 504, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 480, + 219 + ], + "score": 1.0, + "content": "can make use of the implicit function theorem to find the derivative of the solution map", + "type": "text" + }, + { + "bbox": [ + 480, + 206, + 504, + 218 + ], + "score": 0.91, + "content": "\\theta ^ { * } ( y )", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 104, + 216, + 131, + 231 + ], + "score": 1.0, + "content": "w.r.t.", + "type": "text" + }, + { + "bbox": [ + 131, + 219, + 138, + 229 + ], + "score": 0.73, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 216, + 506, + 231 + ], + "score": 1.0, + "content": "(Bengio, 2000). Under the smoothness assumption, the optimality condition of the lower", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 228, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 154, + 241 + ], + "score": 1.0, + "content": "level (2) is", + "type": "text" + }, + { + "bbox": [ + 155, + 228, + 219, + 240 + ], + "score": 0.92, + "content": "\\nabla _ { \\boldsymbol { \\theta } } f ( y , \\boldsymbol { \\theta } ) = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 228, + 368, + 241 + ], + "score": 1.0, + "content": ", which defines an implicit function", + "type": "text" + }, + { + "bbox": [ + 369, + 228, + 392, + 240 + ], + "score": 0.88, + "content": "\\theta ^ { * } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 228, + 506, + 241 + ], + "score": 1.0, + "content": ". With the assumption that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 238, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 158, + 251 + ], + "score": 0.67, + "content": "\\mathrm { m i n } _ { \\boldsymbol { \\theta } } f ( \\boldsymbol { y } , \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 238, + 279, + 252 + ], + "score": 1.0, + "content": "has a solution, there exists a", + "type": "text" + }, + { + "bbox": [ + 279, + 239, + 306, + 251 + ], + "score": 0.92, + "content": "( y , \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 238, + 348, + 252 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 348, + 240, + 416, + 251 + ], + "score": 0.92, + "content": "\\nabla _ { \\theta } f ( y , \\theta ^ { * } ) = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 238, + 505, + 252 + ], + "score": 1.0, + "content": ". 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However, this would also require to", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "13", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 263, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 264, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 264, + 96 + ], + "score": 1.0, + "content": "A MORE DETAIL ON DARTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 504, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 504, + 118 + ], + "score": 1.0, + "content": "Here we present a detailed description of DARTS architectural update steps. 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\\frac { \\xi } { 2 \\epsilon } \\big ( \\nabla _ { \\alpha } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { + } ) - \\nabla _ { \\alpha } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { - } ) \\big )", + "type": "interline_equation", + "image_path": "8ef227ccc3eb03ba486bd5ec0843f803768304d23cdc4e5b0bcbbeebf2c61b89.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 143, + 400, + 467, + 425 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 435, + 506, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 260, + 447 + ], + "score": 1.0, + "content": "In all our experiments we always use", + "type": "text" + }, + { + "bbox": [ + 261, + 435, + 288, + 447 + ], + "score": 0.91, + "content": "\\xi = \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "(also called second order approximation in Liu et al.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 446, + 430, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 168, + 459 + ], + "score": 1.0, + "content": "(2019)), where", + "type": "text" + }, + { + "bbox": [ + 168, + 448, + 175, + 457 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 446, + 418, + 459 + ], + "score": 1.0, + "content": "is the learning rate used in SGD for updating the parameters", + "type": "text" + }, + { + "bbox": [ + 419, + 448, + 426, + 456 + ], + "score": 0.77, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 446, + 430, + 459 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 473, + 335, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 472, + 335, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 335, + 488 + ], + "score": 1.0, + "content": "B CONSTRUCTION OF S1 FROM SECTION 3", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "We ran DARTS two times on the default search space to find the two most important operations per", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "mixed operation. Initially, every mixed operation consists of 8 operations. After the first DARTS", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "score": 1.0, + "content": "run, we drop the 4 (out of 8) least important ones. In the second DARTS run, we drop the 2 (out", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "of the remaining 4) least important ones. S1 is then defined to contain only the two remaining most", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 541, + 504, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 504, + 555 + ], + "score": 1.0, + "content": "important operations per mixed op. Refer to Figure 9 for an illustration of this pre-optimized space.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 107, + 569, + 315, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 317, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 317, + 584 + ], + "score": 1.0, + "content": "C FINAL ARCHITECTURE EVALUATION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 593, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "Similar to the original DARTS paper (Liu et al., 2019), the architecture found during the search are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "scaled up by increasing the number of filters and cells and retrained from scratch to obtain the final", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "test performance. For CIFAR-100 and SVHN we use 16 number of initial filters and 8 cells when", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 627, + 504, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 504, + 639 + ], + "score": 1.0, + "content": "training architectures from scratch for all the experiments we conduct. The rest of the settings is the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 637, + 221, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 221, + 651 + ], + "score": 1.0, + "content": "same as in Liu et al. (2019).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "On CIFAR-10, when scaling the ScheduledDropPath drop probability, we use the same settings for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "training from scratch the found architectures as in the original DARTS paper, i.e. 36 initial filters", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "and 20 stacked cells. 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\\frac { \\xi } { 2 \\epsilon } \\big ( \\nabla _ { \\alpha } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { + } ) - \\nabla _ { \\alpha } \\mathcal { L } _ { t r a i n } ( \\alpha , w ^ { - } ) \\big )", + "type": "interline_equation", + "image_path": "8ef227ccc3eb03ba486bd5ec0843f803768304d23cdc4e5b0bcbbeebf2c61b89.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 143, + 400, + 467, + 425 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "list", + "bbox": [ + 105, + 435, + 506, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 260, + 447 + ], + "score": 1.0, + "content": "In all our experiments we always use", + "type": "text" + }, + { + "bbox": [ + 261, + 435, + 288, + 447 + ], + "score": 0.91, + "content": "\\xi = \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "(also called second order approximation in Liu et al.", + "type": "text" + } + ], + "index": 20, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 446, + 430, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 168, + 459 + ], + "score": 1.0, + "content": "(2019)), where", + "type": "text" + }, + { + "bbox": [ + 168, + 448, + 175, + 457 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 446, + 418, + 459 + ], + "score": 1.0, + "content": "is the learning rate used in SGD for updating the parameters", + "type": "text" + }, + { + "bbox": [ + 419, + 448, + 426, + 456 + ], + "score": 0.77, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 446, + 430, + 459 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 435, + 505, + 459 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 473, + 335, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 472, + 335, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 335, + 488 + ], + "score": 1.0, + "content": "B CONSTRUCTION OF S1 FROM SECTION 3", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "We ran DARTS two times on the default search space to find the two most important operations per", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "mixed operation. 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Refer to Figure 9 for an illustration of this pre-optimized space.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 497, + 506, + 555 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 569, + 315, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 317, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 317, + 584 + ], + "score": 1.0, + "content": "C FINAL ARCHITECTURE EVALUATION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 593, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "Similar to the original DARTS paper (Liu et al., 2019), the architecture found during the search are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "scaled up by increasing the number of filters and cells and retrained from scratch to obtain the final", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "test performance. For CIFAR-100 and SVHN we use 16 number of initial filters and 8 cells when", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 627, + 504, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 504, + 639 + ], + "score": 1.0, + "content": "training architectures from scratch for all the experiments we conduct. The rest of the settings is the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 637, + 221, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 221, + 651 + ], + "score": 1.0, + "content": "same as in Liu et al. (2019).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 594, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "On CIFAR-10, when scaling the ScheduledDropPath drop probability, we use the same settings for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "training from scratch the found architectures as in the original DARTS paper, i.e. 36 initial filters", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "and 20 stacked cells. 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SettingRandomNASDARTSDARTS-ESDARTS-ADA
C10S13.17 ± 0.154.66 ± 0.713.05 ± 0.073.03 ±0.08
S23.46 ± 0.154.42 ± 0.403.41 ± 0.143.59 ± 0.31
S32.92 ± 0.044.12 ± 0.853.71 ± 1.142.99 ± 0.34
S489.39 ± 0.846.95±0.184.17 ± 0.213.89 ± 0.67
C100S125.81 ± 0.3929.93 ± 0.4128.90 ± 0.8124.94 ± 0.81
S222.88 ± 0.1628.75 ± 0.9224.68 ± 1.4326.88 ± 1.11
S324.58 ± 0.6129.01 ± 0.2426.99 ± 1.7924.55± 0.63
S430.01 ± 1.5224.77 ± 1.5123.90 ± 2.0123.66 ± 0.90
SVHNS12.64±0.099.88 ± 5.502.80± 0.092.59± 0.07
S22.57 ± 0.043.69 ± 0.122.68 ± 0.182.79 ± 0.22
S32.89±0.094.00 ± 1.012.78± 0.292.58 ± 0.07
S43.42 ± 0.042.90± 0.022.55 ± 0.152.52 ± 0.06
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SettingRandomNASDARTSDARTS-ESDARTS-ADA
C10S13.17 ± 0.154.66 ± 0.713.05 ± 0.073.03 ±0.08
S23.46 ± 0.154.42 ± 0.403.41 ± 0.143.59 ± 0.31
S32.92 ± 0.044.12 ± 0.853.71 ± 1.142.99 ± 0.34
S489.39 ± 0.846.95±0.184.17 ± 0.213.89 ± 0.67
C100S125.81 ± 0.3929.93 ± 0.4128.90 ± 0.8124.94 ± 0.81
S222.88 ± 0.1628.75 ± 0.9224.68 ± 1.4326.88 ± 1.11
S324.58 ± 0.6129.01 ± 0.2426.99 ± 1.7924.55± 0.63
S430.01 ± 1.5224.77 ± 1.5123.90 ± 2.0123.66 ± 0.90
SVHNS12.64±0.099.88 ± 5.502.80± 0.092.59± 0.07
S22.57 ± 0.043.69 ± 0.122.68 ± 0.182.79 ± 0.22
S32.89±0.094.00 ± 1.012.78± 0.292.58 ± 0.07
S43.42 ± 0.042.90± 0.022.55 ± 0.152.52 ± 0.06
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The markers on each line highlight the epochs", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "where DARTS is early stopped. As one can see from Figure 4, there is indeed some correlation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "score": 1.0, + "content": "between the average dominant eigenvalue throughout the search and the test performance of the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 191, + 235, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 235, + 204 + ], + "score": 1.0, + "content": "found architectures by DARTS.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 104, + 505, + 204 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 208, + 505, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "Figures 15 and 16 (top 3 rows) show the full spectrum (sorted based on eigenvalue absolute values)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 232 + ], + "score": 1.0, + "content": "at the end of search, whilst bottom 3 rows plot the distribution of eigenvalues in the eigenspectrum.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "As one can see, not only the dominant eigenvalue is larger compared to the cases when the regu-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "larization is stronger and the generalization of architectures is better, but also the other eigenvalues", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 251, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 266 + ], + "score": 1.0, + "content": "in the spectrum have larger absolute value, indicating a sharper objective landscape towards many", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "dimensions. Furthermore, from the distribution plots note the presence of more negative eigenvalues", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 272, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 288 + ], + "score": 1.0, + "content": "whenever the architectures are degenerate (lower regularization value) indicating that DARTS gets", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "stuck in a point with larger positive and negative curvature of the validation loss objective, associated", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 296, + 266, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 266, + 308 + ], + "score": 1.0, + "content": "with a more degenerate Hessian matrix.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 208, + 506, + 308 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 324, + 251, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 323, + 253, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 253, + 340 + ], + "score": 1.0, + "content": "E DISPARITY ESTIMATION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 107, + 351, + 177, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 350, + 178, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 178, + 363 + ], + "score": 1.0, + "content": "E.1 DATASETS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "We use the FlyingThings3D dataset (Mayer et al., 2016) for training AutoDispNet. It consists of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "rendered stereo image pairs and their ground truth disparity maps. The dataset provides a training", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "and testing split consisting of 21, 818 and 4248 samples respectively with an image resolution of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 148, + 416 + ], + "score": 0.9, + "content": "9 6 0 \\times 5 4 0", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 405, + 505, + 417 + ], + "score": 1.0, + "content": ". We use the Sintel dataset ( Butler et al. (2012)) for testing our networks. Sintel is another", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "synthetic dataset from derived from an animated movie which also provides ground truth disparity", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 427, + 326, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 274, + 439 + ], + "score": 1.0, + "content": "maps (1064 samples) with a resolution of", + "type": "text" + }, + { + "bbox": [ + 274, + 427, + 322, + 438 + ], + "score": 0.89, + "content": "1 0 2 4 \\times 4 3 6", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 427, + 326, + 439 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 371, + 506, + 439 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 453, + 176, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 178, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 178, + 466 + ], + "score": 1.0, + "content": "E.2 TRAINING", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "We use the AutoDispNet-C architecture as described in Saikia et al. (2019). However, we use", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 486, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 340, + 497 + ], + "score": 1.0, + "content": "the smaller search which consists of three operations:", + "type": "text" + }, + { + "bbox": [ + 340, + 486, + 409, + 496 + ], + "score": 0.52, + "content": "M a x P o o l 3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 486, + 416, + 497 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 416, + 486, + 482, + 497 + ], + "score": 0.26, + "content": "S e p C o n v 3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 486, + 505, + 497 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "SkipConnect. For training the search network, images are downsampled by a factor of two and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 508, + 504, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 153, + 520 + ], + "score": 1.0, + "content": "trained for", + "type": "text" + }, + { + "bbox": [ + 153, + 508, + 175, + 518 + ], + "score": 0.84, + "content": "3 0 0 k", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 508, + 504, + 520 + ], + "score": 1.0, + "content": "mini-batch iterations. During search, we use SGD and ADAM to optimize the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "inner and outer objectives respectively. Differently from the original AutoDispNet we do not warm-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "start the search model weights before starting the architectural parameter updates. The extracted", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 218, + 554 + ], + "score": 1.0, + "content": "network is also trained for", + "type": "text" + }, + { + "bbox": [ + 218, + 541, + 240, + 551 + ], + "score": 0.85, + "content": "3 0 0 k", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 540, + 505, + 554 + ], + "score": 1.0, + "content": "mini-batch iterations but full resolution images are used. Here,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 412, + 564 + ], + "score": 1.0, + "content": "ADAM is used for optimization and the learning rate is annealed to 0 from", + "type": "text" + }, + { + "bbox": [ + 412, + 552, + 441, + 562 + ], + "score": 0.83, + "content": "1 e - 4", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 550, + 505, + 564 + ], + "score": 1.0, + "content": ", using a cosine", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 563, + 171, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 171, + 575 + ], + "score": 1.0, + "content": "decay schedule.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 474, + 506, + 575 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 589, + 371, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 372, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 372, + 601 + ], + "score": 1.0, + "content": "E.3 EFFECT OF REGULARIZATION ON THE INNER OBJECTIVE", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 108, + 609, + 503, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "To study the effect of regularization on the inner objective for AutoDispNet-C we use experiment", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 621, + 498, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 343, + 634 + ], + "score": 1.0, + "content": "with two types of regularization: data augmentation and of", + "type": "text" + }, + { + "bbox": [ + 343, + 622, + 356, + 631 + ], + "score": 0.78, + "content": "L 2", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 621, + 498, + 634 + ], + "score": 1.0, + "content": "regularization on network weights.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 106, + 609, + 505, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "We note that we could not test the early stopping method on AutoDispNet since AutoDispNet relies", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "on custom operations to compute feature map correlation (Dosovitskiy et al., 2015) and resampling,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "for which second order derivatives are currently not available (which are required to compute the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 669, + 147, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 147, + 683 + ], + "score": 1.0, + "content": "Hessian).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 638, + 505, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Data augmentation. Inspite of fairly large number of training samples in FlyingThings3D, data", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "augmentation is crucial for good generalization performance. Disparity estimation networks employ", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "spatial transformations such as translation, cropping, shearing and scaling. Additionally, appearance", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "transformations such as additive Gaussian noise, changes in brightness, contrast, gamma and color", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "are also applied. Parameters for such transformations are sampled from a uniform or Gaussian distri-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "bution (parameterized by a mean and variance). In our experiments, we vary the data augmentation", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "strength by multiplying the variance of these parameter distributions by a fixed factor, which we dub", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "the augmentation scaling factor. The extracted networks are evaluated with the same augmentation", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "parameters. The results of increasing the augmentation strength of the inner objective can be seen", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "in Table 2. We observe that as augmentation strength increases DARTS finds networks with more", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "number of parameters and better test performance. The best test performance is obtained for the", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "score": 1.0, + "content": "network with maximum augmentation for the inner objective. At the same time the search model", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "validation error increases when scaling up the augmentation factor, which again enforces the argu-", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 468, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 468, + 194 + ], + "score": 1.0, + "content": "ment that the overfitting of architectural parameters is reduced by this implicit regularizer.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "are also applied. Parameters for such transformations are sampled from a uniform or Gaussian distri-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "bution (parameterized by a mean and variance). In our experiments, we vary the data augmentation", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "strength by multiplying the variance of these parameter distributions by a fixed factor, which we dub", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "the augmentation scaling factor. The extracted networks are evaluated with the same augmentation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "parameters. The results of increasing the augmentation strength of the inner objective can be seen", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "in Table 2. We observe that as augmentation strength increases DARTS finds networks with more", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "number of parameters and better test performance. The best test performance is obtained for the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "score": 1.0, + "content": "network with maximum augmentation for the inner objective. At the same time the search model", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "validation error increases when scaling up the augmentation factor, which again enforces the argu-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 468, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 468, + 194 + ], + "score": 1.0, + "content": "ment that the overfitting of architectural parameters is reduced by this implicit regularizer.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 192, + 504, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "L2 regularization. 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BenchmarkDARTSDARTS-ES
C10S14.66 ± 0.713.05± 0.07
S24.42 ± 0.403.41 ± 0.14
S34.12 ± 0.853.71 ± 1.14
S46.95±0.184.17 ± 0.21
C100S129.93 ± 0.4128.90±0.81
S228.75±0.9224.68±1.43
S329.01 ± 0.2426.99 ± 1.79
S424.77 ± 1.5123.90±2.01
SVHNS19.88±5.502.80±0.09
S23.69 ±0.122.68± 0.18
S34.00 ± 1.012.78± 0.29
S42.90±0.022.55±0.15
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9×10-45.972.304.1213.92
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BenchmarkRS-wsDARTSR-DARTS(DP)R-DARTS(L2)DARTS-ESDARTS-ADA
C10S13.233.843.112.783.013.10
S23.664.853.483.313.263.35
S32.953.342.932.512.742.59
S48.077.203.583.563.714.84
C100S123.3029.4625.9324.2528.3724.03
S221.2126.0522.3022.2423.2523.52
S323.7528.9022.3623.9923.7323.37
S428.1922.8522.1821.9421.2623.20
SVHNS12.594.582.554.792.722.53
S22.723.532.522.512.602.54
S32.873.412.492.482.502.50
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BenchmarkDARTSR-DARTS(L2)
C102.91± 0.252.95 ± 0.21
C10020.58 ± 0.4418.01 ± 0.26
SVHN2.46±0.092.17 ± 0.09
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C10S13.17 ± 0.154.66 ± 0.713.05 ± 0.073.03 ±0.08
S23.46 ± 0.154.42 ± 0.403.41 ± 0.143.59 ± 0.31
S32.92 ± 0.044.12 ± 0.853.71 ± 1.142.99 ± 0.34
S489.39 ± 0.846.95±0.184.17 ± 0.213.89 ± 0.67
C100S125.81 ± 0.3929.93 ± 0.4128.90 ± 0.8124.94 ± 0.81
S222.88 ± 0.1628.75 ± 0.9224.68 ± 1.4326.88 ± 1.11
S324.58 ± 0.6129.01 ± 0.2426.99 ± 1.7924.55± 0.63
S430.01 ± 1.5224.77 ± 1.5123.90 ± 2.0123.66 ± 0.90
SVHNS12.64±0.099.88 ± 5.502.80± 0.092.59± 0.07
S22.57 ± 0.043.69 ± 0.122.68 ± 0.182.79 ± 0.22
S32.89±0.094.00 ± 1.012.78± 0.292.58 ± 0.07
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0000000000000000000000000000000000000000..3248a34fe9c599224660d5ddd96c4c90deeeec5d --- /dev/null +++ b/parse/train/S1xh5sYgx/S1xh5sYgx.md @@ -0,0 +1,264 @@ +# SQUEEZENET: ALEXNET-LEVEL ACCURACY WITH 50X FEWER PARAMETERS AND $< 0$ .5MB MODEL SIZE + +Forrest N. Iandola1, Song $\mathbf { H a n } ^ { 2 }$ , Matthew W. Moskewicz1, Khalid Ashraf1, +William J. Dally2, Kurt Keutzer1 +1DeepScale∗ & UC Berkeley 2Stanford University +{forresti, moskewcz, kashraf, keutzer}@eecs.berkeley.edu +{songhan, dally}@stanford.edu + +# ABSTRACT + +Recent research on deep convolutional neural networks (CNNs) has focused primarily on improving accuracy. For a given accuracy level, it is typically possible to identify multiple CNN architectures that achieve that accuracy level. With equivalent accuracy, smaller CNN architectures offer at least three advantages: (1) Smaller CNNs require less communication across servers during distributed training. (2) Smaller CNNs require less bandwidth to export a new model from the cloud to an autonomous car. (3) Smaller CNNs are more feasible to deploy on FPGAs and other hardware with limited memory. To provide all of these advantages, we propose a small CNN architecture called SqueezeNet. SqueezeNet achieves AlexNet-level accuracy on ImageNet with $5 0 \mathrm { x }$ fewer parameters. Additionally, with model compression techniques, we are able to compress SqueezeNet to less than 0.5MB $5 1 0 \times$ smaller than AlexNet). + +The SqueezeNet architecture is available for download here: https://github.com/DeepScale/SqueezeNet + +# 1 INTRODUCTION AND MOTIVATION + +Much of the recent research on deep convolutional neural networks (CNNs) has focused on increasing accuracy on computer vision datasets. For a given accuracy level, there typically exist multiple CNN architectures that achieve that accuracy level. Given equivalent accuracy, a CNN architecture with fewer parameters has several advantages: + +• More efficient distributed training. Communication among servers is the limiting factor to the scalability of distributed CNN training. For distributed data-parallel training, communication overhead is directly proportional to the number of parameters in the model (Iandola et al., 2016). In short, small models train faster due to requiring less communication. + +• Less overhead when exporting new models to clients. For autonomous driving, companies such as Tesla periodically copy new models from their servers to customers’ cars. This practice is often referred to as an over-the-air update. Consumer Reports has found that the safety of Tesla’s Autopilot semi-autonomous driving functionality has incrementally improved with recent over-the-air updates (Consumer Reports, 2016). However, over-theair updates of today’s typical CNN/DNN models can require large data transfers. With AlexNet, this would require 240MB of communication from the server to the car. Smaller models require less communication, making frequent updates more feasible. + +• Feasible FPGA and embedded deployment. FPGAs often have less than $1 0 \mathbf { M B } ^ { 1 }$ of onchip memory and no off-chip memory or storage. For inference, a sufficiently small model could be stored directly on the FPGA instead of being bottlenecked by memory bandwidth (Qiu et al., 2016), while video frames stream through the FPGA in real time. Further, when deploying CNNs on Application-Specific Integrated Circuits (ASICs), a sufficiently small model could be stored directly on-chip, and smaller models may enable the ASIC to fit on a smaller die. + +As you can see, there are several advantages of smaller CNN architectures. With this in mind, we focus directly on the problem of identifying a CNN architecture with fewer parameters but equivalent accuracy compared to a well-known model. We have discovered such an architecture, which we call SqueezeNet. In addition, we present our attempt at a more disciplined approach to searching the design space for novel CNN architectures. + +The rest of the paper is organized as follows. In Section 2 we review the related work. Then, in Sections 3 and 4 we describe and evaluate the SqueezeNet architecture. After that, we turn our attention to understanding how CNN architectural design choices impact model size and accuracy. We gain this understanding by exploring the design space of SqueezeNet-like architectures. In Section 5, we do design space exploration on the CNN microarchitecture, which we define as the organization and dimensionality of individual layers and modules. In Section 6, we do design space exploration on the CNN macroarchitecture, which we define as high-level organization of layers in a CNN. Finally, we conclude in Section 7. In short, Sections 3 and 4 are useful for CNN researchers as well as practitioners who simply want to apply SqueezeNet to a new application. The remaining sections are aimed at advanced researchers who intend to design their own CNN architectures. + +# 2 RELATED WORK + +# 2.1 MODEL COMPRESSION + +The overarching goal of our work is to identify a model that has very few parameters while preserving accuracy. To address this problem, a sensible approach is to take an existing CNN model and compress it in a lossy fashion. In fact, a research community has emerged around the topic of model compression, and several approaches have been reported. A fairly straightforward approach by Denton et al. is to apply singular value decomposition (SVD) to a pretrained CNN model (Denton et al., 2014). Han et al. developed Network Pruning, which begins with a pretrained model, then replaces parameters that are below a certain threshold with zeros to form a sparse matrix, and finally performs a few iterations of training on the sparse CNN (Han et al., 2015b). Recently, Han et al. extended their work by combining Network Pruning with quantization (to 8 bits or less) and huffman encoding to create an approach called Deep Compression (Han et al., 2015a), and further designed a hardware accelerator called EIE (Han et al., 2016a) that operates directly on the compressed model, achieving substantial speedups and energy savings. + +# 2.2 CNN MICROARCHITECTURE + +Convolutions have been used in artificial neural networks for at least 25 years; LeCun et al. helped to popularize CNNs for digit recognition applications in the late 1980s (LeCun et al., 1989). In neural networks, convolution filters are typically 3D, with height, width, and channels as the key dimensions. When applied to images, CNN filters typically have 3 channels in their first layer (i.e. RGB), and in each subsequent layer $L _ { i }$ the filters have the same number of channels as $L _ { i - 1 }$ has filters. The early work by LeCun et al. (LeCun et al., 1989) uses 5x5xChannels2 filters, and the recent VGG (Simonyan & Zisserman, 2014) architectures extensively use 3x3 filters. Models such as Network-in-Network (Lin et al., 2013) and the GoogLeNet family of architectures (Szegedy et al., 2014; Ioffe & Szegedy, 2015; Szegedy et al., 2015; 2016) use 1x1 filters in some layers. + +With the trend of designing very deep CNNs, it becomes cumbersome to manually select filter dimensions for each layer. To address this, various higher level building blocks, or modules, comprised of multiple convolution layers with a specific fixed organization have been proposed. For example, the GoogLeNet papers propose Inception modules, which are comprised of a number of different dimensionalities of filters, usually including 1x1 and 3x3, plus sometimes 5x5 (Szegedy et al., 2014) and sometimes 1x3 and 3x1 (Szegedy et al., 2015). Many such modules are then combined, perhaps with additional ad-hoc layers, to form a complete network. We use the term CNN microarchitecture to refer to the particular organization and dimensions of the individual modules. + +# 2.3 CNN MACROARCHITECTURE + +While the CNN microarchitecture refers to individual layers and modules, we define the CNN macroarchitecture as the system-level organization of multiple modules into an end-to-end CNN architecture. + +Perhaps the mostly widely studied CNN macroarchitecture topic in the recent literature is the impact of depth (i.e. number of layers) in networks. Simoyan and Zisserman proposed the VGG (Simonyan & Zisserman, 2014) family of CNNs with 12 to 19 layers and reported that deeper networks produce higher accuracy on the ImageNet-1k dataset (Deng et al., 2009). K. He et al. proposed deeper CNNs with up to 30 layers that deliver even higher ImageNet accuracy (He et al., 2015a). + +The choice of connections across multiple layers or modules is an emerging area of CNN macroarchitectural research. Residual Networks (ResNet) (He et al., 2015b) and Highway Networks (Srivastava et al., 2015) each propose the use of connections that skip over multiple layers, for example additively connecting the activations from layer 3 to the activations from layer 6. We refer to these connections as bypass connections. The authors of ResNet provide an A/B comparison of a 34-layer CNN with and without bypass connections; adding bypass connections delivers a 2 percentage-point improvement on Top-5 ImageNet accuracy. + +# 2.4 NEURAL NETWORK DESIGN SPACE EXPLORATION + +Neural networks (including deep and convolutional NNs) have a large design space, with numerous options for microarchitectures, macroarchitectures, solvers, and other hyperparameters. It seems natural that the community would want to gain intuition about how these factors impact a NN’s accuracy (i.e. the shape of the design space). Much of the work on design space exploration (DSE) of NNs has focused on developing automated approaches for finding NN architectures that deliver higher accuracy. These automated DSE approaches include bayesian optimization (Snoek et al., 2012), simulated annealing (Ludermir et al., 2006), randomized search (Bergstra & Bengio, 2012), and genetic algorithms (Stanley & Miikkulainen, 2002). To their credit, each of these papers provides a case in which the proposed DSE approach produces a NN architecture that achieves higher accuracy compared to a representative baseline. However, these papers make no attempt to provide intuition about the shape of the NN design space. Later in this paper, we eschew automated approaches – instead, we refactor CNNs in such a way that we can do principled A/B comparisons to investigate how CNN architectural decisions influence model size and accuracy. + +In the following sections, we first propose and evaluate the SqueezeNet architecture with and without model compression. Then, we explore the impact of design choices in microarchitecture and macroarchitecture for SqueezeNet-like CNN architectures. + +# 3 SQUEEZENET: PRESERVING ACCURACY WITH FEW PARAMETERS + +In this section, we begin by outlining our design strategies for CNN architectures with few parameters. Then, we introduce the Fire module, our new building block out of which to build CNN architectures. Finally, we use our design strategies to construct SqueezeNet, which is comprised mainly of Fire modules. + +# 3.1 ARCHITECTURAL DESIGN STRATEGIES + +Our overarching objective in this paper is to identify CNN architectures that have few parameters while maintaining competitive accuracy. To achieve this, we employ three main strategies when designing CNN architectures: + +Strategy 1. Replace 3x3 filters with 1x1 filters. Given a budget of a certain number of convolution filters, we will choose to make the majority of these filters 1x1, since a 1x1 filter has 9X fewer parameters than a $3 { \tt X } 3$ filter. + +Strategy 2. Decrease the number of input channels to 3x3 filters. Consider a convolution layer that is comprised entirely of 3x3 filters. The total quantity of parameters in this layer is (number of input channels) \* (number of filters) $^ { * } \left( 3 ^ { * } 3 \right)$ . So, to maintain a small total number of parameters in a CNN, it is important not only to decrease the number of 3x3 filters (see Strategy 1 above), but also to decrease the number of input channels to the 3x3 filters. We decrease the number of input channels to $3 { \tt X } 3$ filters using squeeze layers, which we describe in the next section. + +Strategy 3. Downsample late in the network so that convolution layers have large activation maps. In a convolutional network, each convolution layer produces an output activation map with a spatial resolution that is at least 1x1 and often much larger than 1x1. The height and width of these activation maps are controlled by: (1) the size of the input data (e.g. $2 5 6 \times 2 5 6$ images) and (2) + +![](images/193086b73f62c7db727e7c35fc266ac0e5235c6a3b0ff8631d85977ada06922f.jpg) +Figure 1: Microarchitectural view: Organization of convolution filters in the Fire module. In this example, $s _ { 1 x 1 } ~ = ~ 3$ , $e _ { 1 x 1 } = 4$ , and $e _ { 3 x 3 } ~ = ~ 4$ . We illustrate the convolution filters but not the activations. + +the choice of layers in which to downsample in the CNN architecture. Most commonly, downsampling is engineered into CNN architectures by setting the (stride $> 1$ ) in some of the convolution or pooling layers (e.g. (Szegedy et al., 2014; Simonyan & Zisserman, 2014; Krizhevsky et al., 2012)). If early3 layers in the network have large strides, then most layers will have small activation maps. Conversely, if most layers in the network have a stride of 1, and the strides greater than 1 are concentrated toward the end4 of the network, then many layers in the network will have large activation maps. Our intuition is that large activation maps (due to delayed downsampling) can lead to higher classification accuracy, with all else held equal. Indeed, K. He and H. Sun applied delayed downsampling to four different CNN architectures, and in each case delayed downsampling led to higher classification accuracy (He & Sun, 2015). + +Strategies 1 and 2 are about judiciously decreasing the quantity of parameters in a CNN while attempting to preserve accuracy. Strategy 3 is about maximizing accuracy on a limited budget of parameters. Next, we describe the Fire module, which is our building block for CNN architectures that enables us to successfully employ Strategies 1, 2, and 3. + +# 3.2 THE FIRE MODULE + +We define the Fire module as follows. A Fire module is comprised of: a squeeze convolution layer (which has only 1x1 filters), feeding into an expand layer that has a mix of 1x1 and 3x3 convolution filters; we illustrate this in Figure 1. The liberal use of 1x1 filters in Fire modules is an application of Strategy 1 from Section 3.1. We expose three tunable dimensions (hyperparameters) in a Fire module: $s _ { 1 x 1 }$ , $e _ { 1 x 1 }$ , and $e _ { 3 x 3 }$ . In a Fire module, $s _ { 1 x 1 }$ is the number of filters in the squeeze layer (all 1x1), $e _ { 1 x 1 }$ is the number of 1x1 filters in the expand layer, and $e _ { 3 x 3 }$ is the number of $3 { \tt X } 3$ filters in the expand layer. When we use Fire modules we set $s _ { 1 x 1 }$ to be less than $( e _ { 1 x 1 } + e _ { 3 x 3 } )$ ), so the squeeze layer helps to limit the number of input channels to the $3 { \tt X } 3$ filters, as per Strategy 2 from Section 3.1. + +# 3.3 THE SQUEEZENET ARCHITECTURE + +We now describe the SqueezeNet CNN architecture. We illustrate in Figure 2 that SqueezeNet begins with a standalone convolution layer (conv1), followed by 8 Fire modules (fire2-9), ending with a final conv layer (conv10). We gradually increase the number of filters per fire module from the beginning to the end of the network. SqueezeNet performs max-pooling with a stride of 2 after layers conv1, fire4, fire8, and conv10; these relatively late placements of pooling are per Strategy 3 from Section 3.1. We present the full SqueezeNet architecture in Table 1. + +![](images/d6987d491bfc37cf5ff3c407418d791d44145aa3a732ae68f4cc7632f3c8e1b3.jpg) +Figure 2: Macroarchitectural view of our SqueezeNet architecture. Left: SqueezeNet (Section 3.3); Middle: SqueezeNet with simple bypass (Section 6); Right: SqueezeNet with complex bypass (Section 6). + +# 3.3.1 OTHER SQUEEZENET DETAILS + +For brevity, we have omitted number of details and design choices about SqueezeNet from Table 1 and Figure 2. We provide these design choices in the following. The intuition behind these choices may be found in the papers cited below. + +• So that the output activations from 1x1 and 3x3 filters have the same height and width, we add a 1-pixel border of zero-padding in the input data to $3 { \tt X } 3$ filters of expand modules. +• ReLU (Nair & Hinton, 2010) is applied to activations from squeeze and expand layers. +• Dropout (Srivastava et al., 2014) with a ratio of $50 \%$ is applied after the fire9 module. +• Note the lack of fully-connected layers in SqueezeNet; this design choice was inspired by the NiN (Lin et al., 2013) architecture. When training SqueezeNet, we begin with a learning rate of 0.04, and we linearly decrease the learning rate throughout training, as described in (Mishkin et al., 2016). For details on the training protocol (e.g. batch size, learning rate, parameter initialization), please refer to our Caffe-compatible configuration files located here: https://github.com/DeepScale/SqueezeNet. The Caffe framework does not natively support a convolution layer that contains multiple filter resolutions (e.g. 1x1 and $3 { \bf x } 3$ ) (Jia et al., 2014). To get around this, we implement our expand layer with two separate convolution layers: a layer with 1x1 filters, and a layer with 3x3 filters. Then, we concatenate the outputs of these layers together in the channel dimension. This is numerically equivalent to implementing one layer that contains both 1x1 and 3x3 filters. + +We released the SqueezeNet configuration files in the format defined by the Caffe CNN framework. However, in addition to Caffe, several other CNN frameworks have emerged, including MXNet (Chen et al., 2015a), Chainer (Tokui et al., 2015), Keras (Chollet, 2016), and Torch (Collobert et al., 2011). Each of these has its own native format for representing a CNN architecture. That said, most of these libraries use the same underlying computational back-ends such as cuDNN (Chetlur et al., 2014) and MKL-DNN (Das et al., 2016). The research community has ported the SqueezeNet CNN architecture for compatibility with a number of other CNN software frameworks: + +• MXNet (Chen et al., 2015a) port of SqueezeNet: (Haria, 2016) +• Chainer (Tokui et al., 2015) port of SqueezeNet: (Bell, 2016) +• Keras (Chollet, 2016) port of SqueezeNet: (DT42, 2016) +• Torch (Collobert et al., 2011) port of SqueezeNet’s Fire Modules: (Waghmare, 2016) + +# 4 EVALUATION OF SQUEEZENET + +We now turn our attention to evaluating SqueezeNet. In each of the CNN model compression papers reviewed in Section 2.1, the goal was to compress an AlexNet (Krizhevsky et al., 2012) model that was trained to classify images using the ImageNet (Deng et al., 2009) (ILSVRC 2012) dataset. Therefore, we use AlexNet5 and the associated model compression results as a basis for comparison when evaluating SqueezeNet. + +Table 1: SqueezeNet architectural dimensions. (The formatting of this table was inspired by the Inception2 paper (Ioffe & Szegedy, 2015).) + +
layername/typeoutput sizefilter size/stride(if not a firelayer)depthS1x1(#1x1squeeze)e1x1(#1x1expand)e3x3(#3x3expand)S1x1sparsitye1x1sparsitye3x3sparsity# bits#parameterbefore pruning#parameterafter pruning
input image224x224x3--
conv1111x111x967×7/2 (x96)1100%(7×7)6bit14,20814,208
maxpool155x55x963x3/2。
fire255x55x1282166464100%100%33%6bit11,9205,746
fire355x55x1282166464100%100%33%6bit12,4326,258
fire455x55×256232128128100%100%33%6bit45,34420,646
maxpool427x27x2563x3/20
fire527×27×256232128128100%100%33%6bit49,44024,742
fire627x27x384248192192100%50%33%6bit104,88044,700
fire727×27x38424819219250%100%33%6bit111,02446,236
fire827×27×512264256256100%50%33%6bit188,99277,581
maxpool813x12x5123x3/20
fire913x13x51226425625650%100%30%6bit197,18477,581
conv1013x13x10001x1/1 (x1000)120%(3x3)6bit513,000103,400
avgpool101x1x100013x13/1。
1 Jactivations parameters compression info1,248,424(total)421,098(total)
+ +In Table 2, we review SqueezeNet in the context of recent model compression results. The SVDbased approach is able to compress a pretrained AlexNet model by a factor of ${ 5 } \mathbf { x }$ , while diminishing top-1 accuracy to $5 6 . 0 \%$ (Denton et al., 2014). Network Pruning achieves a $9 \mathbf { x }$ reduction in model size while maintaining the baseline of $5 7 . 2 \%$ top-1 and $8 0 . 3 \%$ top-5 accuracy on ImageNet (Han et al., 2015b). Deep Compression achieves a $3 5 \mathrm { x }$ reduction in model size while still maintaining the baseline accuracy level (Han et al., 2015a). Now, with SqueezeNet, we achieve a 50X reduction in model size compared to AlexNet, while meeting or exceeding the top-1 and top-5 accuracy of AlexNet. We summarize all of the aforementioned results in Table 2. + +It appears that we have surpassed the state-of-the-art results from the model compression community: even when using uncompressed 32-bit values to represent the model, SqueezeNet has a $1 . 4 \times$ smaller model size than the best efforts from the model compression community while maintaining or exceeding the baseline accuracy. Until now, an open question has been: are small models amenable to compression, or do small models “need” all of the representational power afforded by dense floating-point values? To find out, we applied Deep Compression (Han et al., 2015a) + +Table 2: Comparing SqueezeNet to model compression approaches. By model size, we mean the number of bytes required to store all of the parameters in the trained model. + +
CNN architectureCompression ApproachDataTypeOriginal→Compressed ModelSizeReduction inModel Sizevs.AlexNetTop-1ImageNetAccuracyTop-5ImageNetAccuracy
AlexNetNone (baseline)32 bit240MB1x57.2%80.3%
AlexNetSVD (Denton et al.,2014)32 bit240MB→48MB5x56.0%79.4%
AlexNetNetwork Pruning (Hanet al.,2015b)32 bit240MB→27MB9x57.2%80.3%
AlexNetDeepCompression (Hanet al.,2015a)5-8bit240MB→6.9MB35x57.2%80.3%
SqueezeNet (ours)None32 bit4.8MB50x57.5%80.3%
SqueezeNet (ours)Deep Compression8bit4.8MB→0.66MB363x57.5%80.3%
SqueezeNet (ours)Deep Compression6bit4.8MB→0.47MB510x57.5%80.3%
+ +to SqueezeNet, using $33 \%$ sparsity6 and 8-bit quantization. This yields a $0 . 6 6 ~ \mathrm { M B }$ model $( 3 6 3 \times$ smaller than 32-bit AlexNet) with equivalent accuracy to AlexNet. Further, applying Deep Compression with 6-bit quantization and $33 \%$ sparsity on SqueezeNet, we produce a 0.47MB model $( 5 1 0 \times$ smaller than 32-bit AlexNet) with equivalent accuracy. Our small model is indeed amenable to compression. + +In addition, these results demonstrate that Deep Compression (Han et al., 2015a) not only works well on CNN architectures with many parameters (e.g. AlexNet and VGG), but it is also able to compress the already compact, fully convolutional SqueezeNet architecture. Deep Compression compressed SqueezeNet by $1 0 \times$ while preserving the baseline accuracy. In summary: by combining CNN architectural innovation (SqueezeNet) with state-of-the-art compression techniques (Deep Compression), we achieved a $5 1 0 \times$ reduction in model size with no decrease in accuracy compared to the baseline. + +Finally, note that Deep Compression (Han et al., 2015b) uses a codebook as part of its scheme for quantizing CNN parameters to 6- or 8-bits of precision. Therefore, on most commodity processors, it is not trivial to achieve a speedup of $\begin{array} { r } { \frac { 3 2 } { 8 } = 4 x } \end{array}$ with 8-bit quantization or $\begin{array} { r } { \frac { 3 2 } { 6 } = 5 . \dot { 3 } x } \end{array}$ with 6-bit quantization using the scheme developed in Deep Compression. However, Han et al. developed custom hardware – Efficient Inference Engine (EIE) – that can compute codebook-quantized CNNs more efficiently (Han et al., 2016a). In addition, in the months since we released SqueezeNet, P. Gysel developed a strategy called Ristretto for linearly quantizing SqueezeNet to 8 bits (Gysel, 2016). Specifically, Ristretto does computation in 8 bits, and it stores parameters and activations in 8-bit data types. Using the Ristretto strategy for 8-bit computation in SqueezeNet inference, Gysel observed less than 1 percentage-point of drop in accuracy when using 8-bit instead of 32-bit data types. + +# 5 CNN MICROARCHITECTURE DESIGN SPACE EXPLORATION + +So far, we have proposed architectural design strategies for small models, followed these principles to create SqueezeNet, and discovered that SqueezeNet is 50x smaller than AlexNet with equivalent accuracy. However, SqueezeNet and other models reside in a broad and largely unexplored design space of CNN architectures. Now, in Sections 5 and 6, we explore several aspects of the design space. We divide this architectural exploration into two main topics: microarchitectural exploration (per-module layer dimensions and configurations) and macroarchitectural exploration (high-level end-to-end organization of modules and other layers). + +In this section, we design and execute experiments with the goal of providing intuition about the shape of the microarchitectural design space with respect to the design strategies that we proposed in Section 3.1. Note that our goal here is not to maximize accuracy in every experiment, but rather to understand the impact of CNN architectural choices on model size and accuracy. + +![](images/680240d6400757ba8a40ea928e6c20c6e26ac1d8fd8de2eeb0707082c24d3aee.jpg) +Figure 3: Microarchitectural design space exploration. + +# 5.1 CNN MICROARCHITECTURE METAPARAMETERS + +In SqueezeNet, each Fire module has three dimensional hyperparameters that we defined in Section 3.2: $s _ { 1 x 1 }$ , $e _ { 1 x 1 }$ , and $e _ { 3 x 3 }$ . SqueezeNet has 8 Fire modules with a total of 24 dimensional hyperparameters. To do broad sweeps of the design space of SqueezeNet-like architectures, we define the following set of higher level metaparameters which control the dimensions of all Fire modules in a CNN. We define $b a s e _ { e }$ as the number of expand filters in the first Fire module in a CNN. After every freq Fire modules, we increase the number of expand filters by $i n c r _ { e }$ . In other words, for Fire module $i$ , the number of expand filters is $\begin{array} { r } { e _ { i } = b a s e _ { e } + ( i n c r _ { e } * \left\lfloor \frac { i } { f r e q } \right\rfloor ) } \end{array}$ . In the expand layer of a Fire module, some filters are 1x1 and some are $3 { \bf x } 3$ ; we define $e _ { i } = e _ { i , 1 x 1 } + e _ { i , 3 x 3 }$ with $p c t _ { 3 x 3 }$ (in the range [0, 1], shared over all Fire modules) as the percentage of expand filters that are $3 { \bf x } 3$ . In other words, $e _ { i , 3 x 3 } = e _ { i } * p c t _ { 3 x 3 }$ , and $e _ { i , 1 x 1 } = e _ { i } * ( 1 - p c t _ { 3 x 3 } )$ . Finally, we define the number of filters in the squeeze layer of a Fire module using a metaparameter called the squeeze ratio (SR) (again, in the range $[ 0 , 1 ]$ , shared by all Fire modules): $s _ { i , 1 x 1 } = S R * e _ { i }$ (or equivalently $s _ { i , 1 x 1 } = S R * ( e _ { i , 1 x 1 } + e _ { i , 3 x \bar { 3 } } ) \}$ ). SqueezeNet (Table 1) is an example architecture that we generated with the aforementioned set of metaparameters. Specifically, SqueezeNet has the following metaparameters: $b a s e _ { e } = 1 2 8$ , $i n c r _ { e } = 1 2 8$ , $p c t _ { 3 x 3 } = 0 . 5$ , $f r e q = 2$ , and $S R = 0 . 1 2 5$ . + +# 5.2 SQUEEZE RATIO + +In Section 3.1, we proposed decreasing the number of parameters by using squeeze layers to decrease the number of input channels seen by $3 \mathrm { x } 3$ filters. We defined the squeeze ratio $( S R )$ as the ratio between the number of filters in squeeze layers and the number of filters in expand layers. We now design an experiment to investigate the effect of the squeeze ratio on model size and accuracy. + +In these experiments, we use SqueezeNet (Figure 2) as a starting point. As in SqueezeNet, these experiments use the following metaparameters: $b a s e _ { e } = 1 2 8$ , $i n c r _ { e } = 1 2 8$ , $p c t _ { 3 x 3 } = 0 . 5$ , and $f r e q = 2$ . We train multiple models, where each model has a different squeeze ratio $( \mathrm { S R } ) ^ { 7 }$ in the range [0.125, 1.0]. In Figure 3(a), we show the results of this experiment, where each point on the graph is an independent model that was trained from scratch. SqueezeNet is the $\mathrm { S R } { = } 0 . 1 2 5$ point in this figure.8 From this figure, we learn that increasing SR beyond 0.125 can further increase ImageNet top-5 accuracy from $8 0 . 3 \%$ (i.e. AlexNet-level) with a 4.8MB model to $8 6 . 0 \%$ with a 19MB model. Accuracy plateaus at $8 6 . 0 \%$ with $\mathrm { S R } { = } 0 . 7 5$ (a 19MB model), and setting $\mathrm { S R } { = } 1 . 0$ further increases model size without improving accuracy. + +# 5.3 TRADING OFF 1X1 AND 3X3 FILTERS + +In Section 3.1, we proposed decreasing the number of parameters in a CNN by replacing some $3 { \tt X } 3$ filters with 1x1 filters. An open question is, how important is spatial resolution in CNNs? The + +VGG (Simonyan & Zisserman, 2014) architectures have 3x3 spatial resolution in most layers’ filters; GoogLeNet (Szegedy et al., 2014) and Network-in-Network (NiN) (Lin et al., 2013) have 1x1 filters in some layers. In GoogLeNet and NiN, the authors simply propose a specific quantity of 1x1 and $3 { \tt X } 3$ filters without further analysis.9 Here, we attempt to shed light on how the proportion of 1x1 and $3 { \tt X } 3$ filters affects model size and accuracy. + +We use the following metaparameters in this experiment: $b a s e _ { e } = i n c r _ { e } = 1 2 8$ , $f r e q = 2$ , $S R =$ 0.500, and we vary $p c t _ { 3 x 3 }$ from $1 \%$ to $9 9 \%$ . In other words, each Fire module’s expand layer has a predefined number of filters partitioned between 1x1 and $3 { \tt X } 3$ , and here we turn the knob on these filters from “mostly 1x1” to “mostly $3 \mathrm { x } 3 ^ { \circ }$ . As in the previous experiment, these models have 8 Fire modules, following the same organization of layers as in Figure 2. We show the results of this experiment in Figure 3(b). Note that the 13MB models in Figure 3(a) and Figure 3(b) are the same architecture: $S R = 0 . 5 0 0$ and $p c t _ { 3 x 3 } = 5 0 \%$ . We see in Figure 3(b) that the top-5 accuracy plateaus at $8 5 . 6 \%$ using $50 \%$ 3x3 filters, and further increasing the percentage of 3x3 filters leads to a larger model size but provides no improvement in accuracy on ImageNet. + +# 6 CNN MACROARCHITECTURE DESIGN SPACE EXPLORATION + +So far we have explored the design space at the microarchitecture level, i.e. the contents of individual modules of the CNN. Now, we explore design decisions at the macroarchitecture level concerning the high-level connections among Fire modules. Inspired by ResNet (He et al., 2015b), we explored three different architectures: + +• Vanilla SqueezeNet (as per the prior sections). +• SqueezeNet with simple bypass connections between some Fire modules. (Inspired by (Srivastava et al., 2015; He et al., 2015b).) +• SqueezeNet with complex bypass connections between the remaining Fire modules. + +We illustrate these three variants of SqueezeNet in Figure 2. + +Our simple bypass architecture adds bypass connections around Fire modules 3, 5, 7, and 9, requiring these modules to learn a residual function between input and output. As in ResNet, to implement a bypass connection around Fire3, we set the input to Fire4 equal to (output of ${ \mathrm { F i r e } } 2 +$ output of Fire3), where the $^ +$ operator is elementwise addition. This changes the regularization applied to the parameters of these Fire modules, and, as per ResNet, can improve the final accuracy and/or ability to train the full model. + +One limitation is that, in the straightforward case, the number of input channels and number of output channels has to be the same; as a result, only half of the Fire modules can have simple bypass connections, as shown in the middle diagram of Fig 2. When the “same number of channels” requirement can’t be met, we use a complex bypass connection, as illustrated on the right of Figure 2. While a simple bypass is “just a wire,” we define a complex bypass as a bypass that includes a 1x1 convolution layer with the number of filters set equal to the number of output channels that are needed. Note that complex bypass connections add extra parameters to the model, while simple bypass connections do not. + +In addition to changing the regularization, it is intuitive to us that adding bypass connections would help to alleviate the representational bottleneck introduced by squeeze layers. In SqueezeNet, the squeeze ratio (SR) is 0.125, meaning that every squeeze layer has 8x fewer output channels than the accompanying expand layer. Due to this severe dimensionality reduction, a limited amount of information can pass through squeeze layers. However, by adding bypass connections to SqueezeNet, we open up avenues for information to flow around the squeeze layers. + +We trained SqueezeNet with the three macroarchitectures in Figure 2 and compared the accuracy and model size in Table 3. We fixed the microarchitecture to match SqueezeNet as described in Table 1 throughout the macroarchitecture exploration. Complex and simple bypass connections both yielded an accuracy improvement over the vanilla SqueezeNet architecture. Interestingly, the simple bypass enabled a higher accuracy accuracy improvement than complex bypass. Adding the simple bypass connections yielded an increase of 2.9 percentage-points in top-1 accuracy and 2.2 percentage-points in top-5 accuracy without increasing model size. + +Table 3: SqueezeNet accuracy and model size using different macroarchitecture configurations + +
ArchitectureTop-1 AccuracyTop-5 AccuracyModel Size
Vanilla SqueezeNet57.5%80.3%4.8MB
SqueezeNet+SimpleBypass60.4%82.5%4.8MB
SqueezeNet+ComplexBypass58.8%82.0%7.7MB
+ +# 7 CONCLUSIONS + +In this paper, we have proposed steps toward a more disciplined approach to the design-space exploration of convolutional neural networks. Toward this goal we have presented SqueezeNet, a CNN architecture that has $5 0 \times$ fewer parameters than AlexNet and maintains AlexNet-level accuracy on ImageNet. We also compressed SqueezeNet to less than 0.5MB, or $5 1 0 \times$ smaller than AlexNet without compression. Since we released this paper as a technical report in 2016, Song Han and his collaborators have experimented further with SqueezeNet and model compression. Using a new approach called Dense-Sparse-Dense (DSD) (Han et al., 2016b), Han et al. use model compression during training as a regularizer to further improve accuracy, producing a compressed set of SqueezeNet parameters that is 1.2 percentage-points more accurate on ImageNet-1k, and also producing an uncompressed set of SqueezeNet parameters that is 4.3 percentage-points more accurate, compared to our results in Table 2. + +We mentioned near the beginning of this paper that small models are more amenable to on-chip implementations on FPGAs. Since we released the SqueezeNet model, Gschwend has developed a variant of SqueezeNet and implemented it on an FPGA (Gschwend, 2016). As we anticipated, Gschwend was able to able to store the parameters of a SqueezeNet-like model entirely within the FPGA and eliminate the need for off-chip memory accesses to load model parameters. + +In the context of this paper, we focused on ImageNet as a target dataset. However, it has become common practice to apply ImageNet-trained CNN representations to a variety of applications such as fine-grained object recognition (Zhang et al., 2013; Donahue et al., 2013), logo identification in images (Iandola et al., 2015), and generating sentences about images (Fang et al., 2015). ImageNettrained CNNs have also been applied to a number of applications pertaining to autonomous driving, including pedestrian and vehicle detection in images (Iandola et al., 2014; Girshick et al., 2015; Ashraf et al., 2016) and videos (Chen et al., 2015b), as well as segmenting the shape of the road (Badrinarayanan et al., 2015). We think SqueezeNet will be a good candidate CNN architecture for a variety of applications, especially those in which small model size is of importance. + +SqueezeNet is one of several new CNNs that we have discovered while broadly exploring the design space of CNN architectures. 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Deformable part descriptors for fine-grained recognition and attribute prediction. In ICCV, 2013. \ No newline at end of file diff --git a/parse/train/S1xh5sYgx/S1xh5sYgx_content_list.json b/parse/train/S1xh5sYgx/S1xh5sYgx_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..87780257a5ff9569936a798337e2733ab41880a0 --- /dev/null +++ b/parse/train/S1xh5sYgx/S1xh5sYgx_content_list.json @@ -0,0 +1,1304 @@ +[ + { + "type": "text", + "text": "SQUEEZENET: ALEXNET-LEVEL ACCURACY WITH 50X FEWER PARAMETERS AND $< 0$ .5MB MODEL SIZE ", + "text_level": 1, + "bbox": [ + 176, + 99, + 820, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Forrest N. Iandola1, Song $\\mathbf { H a n } ^ { 2 }$ , Matthew W. Moskewicz1, Khalid Ashraf1, \nWilliam J. Dally2, Kurt Keutzer1 \n1DeepScale∗ & UC Berkeley 2Stanford University \n{forresti, moskewcz, kashraf, keutzer}@eecs.berkeley.edu \n{songhan, dally}@stanford.edu ", + "bbox": [ + 184, + 161, + 710, + 234 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 271, + 544, + 286 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent research on deep convolutional neural networks (CNNs) has focused primarily on improving accuracy. For a given accuracy level, it is typically possible to identify multiple CNN architectures that achieve that accuracy level. With equivalent accuracy, smaller CNN architectures offer at least three advantages: (1) Smaller CNNs require less communication across servers during distributed training. (2) Smaller CNNs require less bandwidth to export a new model from the cloud to an autonomous car. (3) Smaller CNNs are more feasible to deploy on FPGAs and other hardware with limited memory. To provide all of these advantages, we propose a small CNN architecture called SqueezeNet. SqueezeNet achieves AlexNet-level accuracy on ImageNet with $5 0 \\mathrm { x }$ fewer parameters. Additionally, with model compression techniques, we are able to compress SqueezeNet to less than 0.5MB $5 1 0 \\times$ smaller than AlexNet). ", + "bbox": [ + 233, + 301, + 764, + 468 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The SqueezeNet architecture is available for download here: https://github.com/DeepScale/SqueezeNet ", + "bbox": [ + 232, + 470, + 764, + 498 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION AND MOTIVATION ", + "text_level": 1, + "bbox": [ + 176, + 525, + 493, + 540 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Much of the recent research on deep convolutional neural networks (CNNs) has focused on increasing accuracy on computer vision datasets. For a given accuracy level, there typically exist multiple CNN architectures that achieve that accuracy level. Given equivalent accuracy, a CNN architecture with fewer parameters has several advantages: ", + "bbox": [ + 176, + 542, + 825, + 598 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• More efficient distributed training. Communication among servers is the limiting factor to the scalability of distributed CNN training. For distributed data-parallel training, communication overhead is directly proportional to the number of parameters in the model (Iandola et al., 2016). In short, small models train faster due to requiring less communication. ", + "bbox": [ + 215, + 609, + 823, + 665 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• Less overhead when exporting new models to clients. For autonomous driving, companies such as Tesla periodically copy new models from their servers to customers’ cars. This practice is often referred to as an over-the-air update. Consumer Reports has found that the safety of Tesla’s Autopilot semi-autonomous driving functionality has incrementally improved with recent over-the-air updates (Consumer Reports, 2016). However, over-theair updates of today’s typical CNN/DNN models can require large data transfers. With AlexNet, this would require 240MB of communication from the server to the car. Smaller models require less communication, making frequent updates more feasible. ", + "bbox": [ + 217, + 667, + 825, + 779 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• Feasible FPGA and embedded deployment. FPGAs often have less than $1 0 \\mathbf { M B } ^ { 1 }$ of onchip memory and no off-chip memory or storage. For inference, a sufficiently small model could be stored directly on the FPGA instead of being bottlenecked by memory bandwidth (Qiu et al., 2016), while video frames stream through the FPGA in real time. Further, when deploying CNNs on Application-Specific Integrated Circuits (ASICs), a sufficiently small model could be stored directly on-chip, and smaller models may enable the ASIC to fit on a smaller die. ", + "bbox": [ + 217, + 781, + 823, + 877 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "As you can see, there are several advantages of smaller CNN architectures. With this in mind, we focus directly on the problem of identifying a CNN architecture with fewer parameters but equivalent accuracy compared to a well-known model. We have discovered such an architecture, which we call SqueezeNet. In addition, we present our attempt at a more disciplined approach to searching the design space for novel CNN architectures. ", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The rest of the paper is organized as follows. In Section 2 we review the related work. Then, in Sections 3 and 4 we describe and evaluate the SqueezeNet architecture. After that, we turn our attention to understanding how CNN architectural design choices impact model size and accuracy. We gain this understanding by exploring the design space of SqueezeNet-like architectures. In Section 5, we do design space exploration on the CNN microarchitecture, which we define as the organization and dimensionality of individual layers and modules. In Section 6, we do design space exploration on the CNN macroarchitecture, which we define as high-level organization of layers in a CNN. Finally, we conclude in Section 7. In short, Sections 3 and 4 are useful for CNN researchers as well as practitioners who simply want to apply SqueezeNet to a new application. The remaining sections are aimed at advanced researchers who intend to design their own CNN architectures. ", + "bbox": [ + 174, + 181, + 825, + 319 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 343, + 344, + 358 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 MODEL COMPRESSION ", + "text_level": 1, + "bbox": [ + 176, + 363, + 375, + 376 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The overarching goal of our work is to identify a model that has very few parameters while preserving accuracy. To address this problem, a sensible approach is to take an existing CNN model and compress it in a lossy fashion. In fact, a research community has emerged around the topic of model compression, and several approaches have been reported. A fairly straightforward approach by Denton et al. is to apply singular value decomposition (SVD) to a pretrained CNN model (Denton et al., 2014). Han et al. developed Network Pruning, which begins with a pretrained model, then replaces parameters that are below a certain threshold with zeros to form a sparse matrix, and finally performs a few iterations of training on the sparse CNN (Han et al., 2015b). Recently, Han et al. extended their work by combining Network Pruning with quantization (to 8 bits or less) and huffman encoding to create an approach called Deep Compression (Han et al., 2015a), and further designed a hardware accelerator called EIE (Han et al., 2016a) that operates directly on the compressed model, achieving substantial speedups and energy savings. ", + "bbox": [ + 173, + 377, + 825, + 542 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 CNN MICROARCHITECTURE", + "text_level": 1, + "bbox": [ + 176, + 564, + 413, + 577 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Convolutions have been used in artificial neural networks for at least 25 years; LeCun et al. helped to popularize CNNs for digit recognition applications in the late 1980s (LeCun et al., 1989). In neural networks, convolution filters are typically 3D, with height, width, and channels as the key dimensions. When applied to images, CNN filters typically have 3 channels in their first layer (i.e. RGB), and in each subsequent layer $L _ { i }$ the filters have the same number of channels as $L _ { i - 1 }$ has filters. The early work by LeCun et al. (LeCun et al., 1989) uses 5x5xChannels2 filters, and the recent VGG (Simonyan & Zisserman, 2014) architectures extensively use 3x3 filters. Models such as Network-in-Network (Lin et al., 2013) and the GoogLeNet family of architectures (Szegedy et al., 2014; Ioffe & Szegedy, 2015; Szegedy et al., 2015; 2016) use 1x1 filters in some layers. ", + "bbox": [ + 174, + 578, + 825, + 702 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "With the trend of designing very deep CNNs, it becomes cumbersome to manually select filter dimensions for each layer. To address this, various higher level building blocks, or modules, comprised of multiple convolution layers with a specific fixed organization have been proposed. For example, the GoogLeNet papers propose Inception modules, which are comprised of a number of different dimensionalities of filters, usually including 1x1 and 3x3, plus sometimes 5x5 (Szegedy et al., 2014) and sometimes 1x3 and 3x1 (Szegedy et al., 2015). Many such modules are then combined, perhaps with additional ad-hoc layers, to form a complete network. We use the term CNN microarchitecture to refer to the particular organization and dimensions of the individual modules. ", + "bbox": [ + 174, + 709, + 825, + 820 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.3 CNN MACROARCHITECTURE", + "text_level": 1, + "bbox": [ + 174, + 842, + 418, + 854 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While the CNN microarchitecture refers to individual layers and modules, we define the CNN macroarchitecture as the system-level organization of multiple modules into an end-to-end CNN architecture. ", + "bbox": [ + 176, + 856, + 825, + 896 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Perhaps the mostly widely studied CNN macroarchitecture topic in the recent literature is the impact of depth (i.e. number of layers) in networks. Simoyan and Zisserman proposed the VGG (Simonyan & Zisserman, 2014) family of CNNs with 12 to 19 layers and reported that deeper networks produce higher accuracy on the ImageNet-1k dataset (Deng et al., 2009). K. He et al. proposed deeper CNNs with up to 30 layers that deliver even higher ImageNet accuracy (He et al., 2015a). ", + "bbox": [ + 174, + 103, + 823, + 174 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The choice of connections across multiple layers or modules is an emerging area of CNN macroarchitectural research. Residual Networks (ResNet) (He et al., 2015b) and Highway Networks (Srivastava et al., 2015) each propose the use of connections that skip over multiple layers, for example additively connecting the activations from layer 3 to the activations from layer 6. We refer to these connections as bypass connections. The authors of ResNet provide an A/B comparison of a 34-layer CNN with and without bypass connections; adding bypass connections delivers a 2 percentage-point improvement on Top-5 ImageNet accuracy. ", + "bbox": [ + 174, + 180, + 825, + 279 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.4 NEURAL NETWORK DESIGN SPACE EXPLORATION ", + "text_level": 1, + "bbox": [ + 174, + 296, + 565, + 309 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Neural networks (including deep and convolutional NNs) have a large design space, with numerous options for microarchitectures, macroarchitectures, solvers, and other hyperparameters. It seems natural that the community would want to gain intuition about how these factors impact a NN’s accuracy (i.e. the shape of the design space). Much of the work on design space exploration (DSE) of NNs has focused on developing automated approaches for finding NN architectures that deliver higher accuracy. These automated DSE approaches include bayesian optimization (Snoek et al., 2012), simulated annealing (Ludermir et al., 2006), randomized search (Bergstra & Bengio, 2012), and genetic algorithms (Stanley & Miikkulainen, 2002). To their credit, each of these papers provides a case in which the proposed DSE approach produces a NN architecture that achieves higher accuracy compared to a representative baseline. However, these papers make no attempt to provide intuition about the shape of the NN design space. Later in this paper, we eschew automated approaches – instead, we refactor CNNs in such a way that we can do principled A/B comparisons to investigate how CNN architectural decisions influence model size and accuracy. ", + "bbox": [ + 174, + 309, + 825, + 489 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the following sections, we first propose and evaluate the SqueezeNet architecture with and without model compression. Then, we explore the impact of design choices in microarchitecture and macroarchitecture for SqueezeNet-like CNN architectures. ", + "bbox": [ + 176, + 496, + 825, + 537 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 SQUEEZENET: PRESERVING ACCURACY WITH FEW PARAMETERS ", + "text_level": 1, + "bbox": [ + 176, + 560, + 741, + 575 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we begin by outlining our design strategies for CNN architectures with few parameters. Then, we introduce the Fire module, our new building block out of which to build CNN architectures. Finally, we use our design strategies to construct SqueezeNet, which is comprised mainly of Fire modules. ", + "bbox": [ + 174, + 579, + 825, + 633 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 ARCHITECTURAL DESIGN STRATEGIES ", + "text_level": 1, + "bbox": [ + 178, + 654, + 485, + 666 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our overarching objective in this paper is to identify CNN architectures that have few parameters while maintaining competitive accuracy. To achieve this, we employ three main strategies when designing CNN architectures: ", + "bbox": [ + 176, + 666, + 823, + 707 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Strategy 1. Replace 3x3 filters with 1x1 filters. Given a budget of a certain number of convolution filters, we will choose to make the majority of these filters 1x1, since a 1x1 filter has 9X fewer parameters than a $3 { \\tt X } 3$ filter. ", + "bbox": [ + 176, + 714, + 823, + 756 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Strategy 2. Decrease the number of input channels to 3x3 filters. Consider a convolution layer that is comprised entirely of 3x3 filters. The total quantity of parameters in this layer is (number of input channels) \\* (number of filters) $^ { * } \\left( 3 ^ { * } 3 \\right)$ . So, to maintain a small total number of parameters in a CNN, it is important not only to decrease the number of 3x3 filters (see Strategy 1 above), but also to decrease the number of input channels to the 3x3 filters. We decrease the number of input channels to $3 { \\tt X } 3$ filters using squeeze layers, which we describe in the next section. ", + "bbox": [ + 174, + 772, + 825, + 856 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Strategy 3. Downsample late in the network so that convolution layers have large activation maps. In a convolutional network, each convolution layer produces an output activation map with a spatial resolution that is at least 1x1 and often much larger than 1x1. The height and width of these activation maps are controlled by: (1) the size of the input data (e.g. $2 5 6 \\times 2 5 6$ images) and (2) ", + "bbox": [ + 174, + 868, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/193086b73f62c7db727e7c35fc266ac0e5235c6a3b0ff8631d85977ada06922f.jpg", + "image_caption": [ + "Figure 1: Microarchitectural view: Organization of convolution filters in the Fire module. In this example, $s _ { 1 x 1 } ~ = ~ 3$ , $e _ { 1 x 1 } = 4$ , and $e _ { 3 x 3 } ~ = ~ 4$ . We illustrate the convolution filters but not the activations. " + ], + "image_footnote": [], + "bbox": [ + 282, + 99, + 715, + 309 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "the choice of layers in which to downsample in the CNN architecture. Most commonly, downsampling is engineered into CNN architectures by setting the (stride $> 1$ ) in some of the convolution or pooling layers (e.g. (Szegedy et al., 2014; Simonyan & Zisserman, 2014; Krizhevsky et al., 2012)). If early3 layers in the network have large strides, then most layers will have small activation maps. Conversely, if most layers in the network have a stride of 1, and the strides greater than 1 are concentrated toward the end4 of the network, then many layers in the network will have large activation maps. Our intuition is that large activation maps (due to delayed downsampling) can lead to higher classification accuracy, with all else held equal. Indeed, K. He and H. Sun applied delayed downsampling to four different CNN architectures, and in each case delayed downsampling led to higher classification accuracy (He & Sun, 2015). ", + "bbox": [ + 174, + 390, + 825, + 530 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Strategies 1 and 2 are about judiciously decreasing the quantity of parameters in a CNN while attempting to preserve accuracy. Strategy 3 is about maximizing accuracy on a limited budget of parameters. Next, we describe the Fire module, which is our building block for CNN architectures that enables us to successfully employ Strategies 1, 2, and 3. ", + "bbox": [ + 174, + 537, + 825, + 593 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 THE FIRE MODULE ", + "text_level": 1, + "bbox": [ + 176, + 616, + 349, + 628 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We define the Fire module as follows. A Fire module is comprised of: a squeeze convolution layer (which has only 1x1 filters), feeding into an expand layer that has a mix of 1x1 and 3x3 convolution filters; we illustrate this in Figure 1. The liberal use of 1x1 filters in Fire modules is an application of Strategy 1 from Section 3.1. We expose three tunable dimensions (hyperparameters) in a Fire module: $s _ { 1 x 1 }$ , $e _ { 1 x 1 }$ , and $e _ { 3 x 3 }$ . In a Fire module, $s _ { 1 x 1 }$ is the number of filters in the squeeze layer (all 1x1), $e _ { 1 x 1 }$ is the number of 1x1 filters in the expand layer, and $e _ { 3 x 3 }$ is the number of $3 { \\tt X } 3$ filters in the expand layer. When we use Fire modules we set $s _ { 1 x 1 }$ to be less than $( e _ { 1 x 1 } + e _ { 3 x 3 } )$ ), so the squeeze layer helps to limit the number of input channels to the $3 { \\tt X } 3$ filters, as per Strategy 2 from Section 3.1. ", + "bbox": [ + 174, + 631, + 825, + 755 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 THE SQUEEZENET ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 779, + 457, + 791 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We now describe the SqueezeNet CNN architecture. We illustrate in Figure 2 that SqueezeNet begins with a standalone convolution layer (conv1), followed by 8 Fire modules (fire2-9), ending with a final conv layer (conv10). We gradually increase the number of filters per fire module from the beginning to the end of the network. SqueezeNet performs max-pooling with a stride of 2 after layers conv1, fire4, fire8, and conv10; these relatively late placements of pooling are per Strategy 3 from Section 3.1. We present the full SqueezeNet architecture in Table 1. ", + "bbox": [ + 174, + 792, + 825, + 876 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/d6987d491bfc37cf5ff3c407418d791d44145aa3a732ae68f4cc7632f3c8e1b3.jpg", + "image_caption": [ + "Figure 2: Macroarchitectural view of our SqueezeNet architecture. Left: SqueezeNet (Section 3.3); Middle: SqueezeNet with simple bypass (Section 6); Right: SqueezeNet with complex bypass (Section 6). " + ], + "image_footnote": [], + "bbox": [ + 232, + 104, + 769, + 410 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3.1 OTHER SQUEEZENET DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 496, + 439, + 510 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For brevity, we have omitted number of details and design choices about SqueezeNet from Table 1 and Figure 2. We provide these design choices in the following. The intuition behind these choices may be found in the papers cited below. ", + "bbox": [ + 176, + 512, + 825, + 554 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• So that the output activations from 1x1 and 3x3 filters have the same height and width, we add a 1-pixel border of zero-padding in the input data to $3 { \\tt X } 3$ filters of expand modules. \n• ReLU (Nair & Hinton, 2010) is applied to activations from squeeze and expand layers. \n• Dropout (Srivastava et al., 2014) with a ratio of $50 \\%$ is applied after the fire9 module. \n• Note the lack of fully-connected layers in SqueezeNet; this design choice was inspired by the NiN (Lin et al., 2013) architecture. When training SqueezeNet, we begin with a learning rate of 0.04, and we linearly decrease the learning rate throughout training, as described in (Mishkin et al., 2016). For details on the training protocol (e.g. batch size, learning rate, parameter initialization), please refer to our Caffe-compatible configuration files located here: https://github.com/DeepScale/SqueezeNet. The Caffe framework does not natively support a convolution layer that contains multiple filter resolutions (e.g. 1x1 and $3 { \\bf x } 3$ ) (Jia et al., 2014). To get around this, we implement our expand layer with two separate convolution layers: a layer with 1x1 filters, and a layer with 3x3 filters. Then, we concatenate the outputs of these layers together in the channel dimension. This is numerically equivalent to implementing one layer that contains both 1x1 and 3x3 filters. ", + "bbox": [ + 215, + 569, + 825, + 825 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We released the SqueezeNet configuration files in the format defined by the Caffe CNN framework. However, in addition to Caffe, several other CNN frameworks have emerged, including MXNet (Chen et al., 2015a), Chainer (Tokui et al., 2015), Keras (Chollet, 2016), and Torch (Collobert et al., 2011). Each of these has its own native format for representing a CNN architecture. That said, most of these libraries use the same underlying computational back-ends such as cuDNN (Chetlur et al., 2014) and MKL-DNN (Das et al., 2016). The research community has ported the SqueezeNet CNN architecture for compatibility with a number of other CNN software frameworks: ", + "bbox": [ + 173, + 840, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 103, + 823, + 132 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "• MXNet (Chen et al., 2015a) port of SqueezeNet: (Haria, 2016) \n• Chainer (Tokui et al., 2015) port of SqueezeNet: (Bell, 2016) \n• Keras (Chollet, 2016) port of SqueezeNet: (DT42, 2016) \n• Torch (Collobert et al., 2011) port of SqueezeNet’s Fire Modules: (Waghmare, 2016) ", + "bbox": [ + 215, + 143, + 789, + 219 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EVALUATION OF SQUEEZENET ", + "text_level": 1, + "bbox": [ + 176, + 239, + 460, + 256 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We now turn our attention to evaluating SqueezeNet. In each of the CNN model compression papers reviewed in Section 2.1, the goal was to compress an AlexNet (Krizhevsky et al., 2012) model that was trained to classify images using the ImageNet (Deng et al., 2009) (ILSVRC 2012) dataset. Therefore, we use AlexNet5 and the associated model compression results as a basis for comparison when evaluating SqueezeNet. ", + "bbox": [ + 174, + 260, + 825, + 329 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/9b5479672bc826dd062e7b49fab2cd62ed9127784630f2286d4f07b2cf61c8aa.jpg", + "table_caption": [ + "Table 1: SqueezeNet architectural dimensions. (The formatting of this table was inspired by the Inception2 paper (Ioffe & Szegedy, 2015).) " + ], + "table_footnote": [], + "table_body": "
layername/typeoutput sizefilter size/stride(if not a firelayer)depthS1x1(#1x1squeeze)e1x1(#1x1expand)e3x3(#3x3expand)S1x1sparsitye1x1sparsitye3x3sparsity# bits#parameterbefore pruning#parameterafter pruning
input image224x224x3--
conv1111x111x967×7/2 (x96)1100%(7×7)6bit14,20814,208
maxpool155x55x963x3/2。
fire255x55x1282166464100%100%33%6bit11,9205,746
fire355x55x1282166464100%100%33%6bit12,4326,258
fire455x55×256232128128100%100%33%6bit45,34420,646
maxpool427x27x2563x3/20
fire527×27×256232128128100%100%33%6bit49,44024,742
fire627x27x384248192192100%50%33%6bit104,88044,700
fire727×27x38424819219250%100%33%6bit111,02446,236
fire827×27×512264256256100%50%33%6bit188,99277,581
maxpool813x12x5123x3/20
fire913x13x51226425625650%100%30%6bit197,18477,581
conv1013x13x10001x1/1 (x1000)120%(3x3)6bit513,000103,400
avgpool101x1x100013x13/1。
1 Jactivations parameters compression info1,248,424(total)421,098(total)
", + "bbox": [ + 176, + 383, + 821, + 676 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Table 2, we review SqueezeNet in the context of recent model compression results. The SVDbased approach is able to compress a pretrained AlexNet model by a factor of ${ 5 } \\mathbf { x }$ , while diminishing top-1 accuracy to $5 6 . 0 \\%$ (Denton et al., 2014). Network Pruning achieves a $9 \\mathbf { x }$ reduction in model size while maintaining the baseline of $5 7 . 2 \\%$ top-1 and $8 0 . 3 \\%$ top-5 accuracy on ImageNet (Han et al., 2015b). Deep Compression achieves a $3 5 \\mathrm { x }$ reduction in model size while still maintaining the baseline accuracy level (Han et al., 2015a). Now, with SqueezeNet, we achieve a 50X reduction in model size compared to AlexNet, while meeting or exceeding the top-1 and top-5 accuracy of AlexNet. We summarize all of the aforementioned results in Table 2. ", + "bbox": [ + 173, + 695, + 825, + 808 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "It appears that we have surpassed the state-of-the-art results from the model compression community: even when using uncompressed 32-bit values to represent the model, SqueezeNet has a $1 . 4 \\times$ smaller model size than the best efforts from the model compression community while maintaining or exceeding the baseline accuracy. Until now, an open question has been: are small models amenable to compression, or do small models “need” all of the representational power afforded by dense floating-point values? To find out, we applied Deep Compression (Han et al., 2015a) ", + "bbox": [ + 173, + 814, + 825, + 897 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/e23e5c79de27ffffb8b0c9cce3b2850b233e601040ab742fbae58b87f6bde616.jpg", + "table_caption": [ + "Table 2: Comparing SqueezeNet to model compression approaches. By model size, we mean the number of bytes required to store all of the parameters in the trained model. " + ], + "table_footnote": [], + "table_body": "
CNN architectureCompression ApproachDataTypeOriginal→Compressed ModelSizeReduction inModel Sizevs.AlexNetTop-1ImageNetAccuracyTop-5ImageNetAccuracy
AlexNetNone (baseline)32 bit240MB1x57.2%80.3%
AlexNetSVD (Denton et al.,2014)32 bit240MB→48MB5x56.0%79.4%
AlexNetNetwork Pruning (Hanet al.,2015b)32 bit240MB→27MB9x57.2%80.3%
AlexNetDeepCompression (Hanet al.,2015a)5-8bit240MB→6.9MB35x57.2%80.3%
SqueezeNet (ours)None32 bit4.8MB50x57.5%80.3%
SqueezeNet (ours)Deep Compression8bit4.8MB→0.66MB363x57.5%80.3%
SqueezeNet (ours)Deep Compression6bit4.8MB→0.47MB510x57.5%80.3%
", + "bbox": [ + 178, + 140, + 820, + 287 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "to SqueezeNet, using $33 \\%$ sparsity6 and 8-bit quantization. This yields a $0 . 6 6 ~ \\mathrm { M B }$ model $( 3 6 3 \\times$ smaller than 32-bit AlexNet) with equivalent accuracy to AlexNet. Further, applying Deep Compression with 6-bit quantization and $33 \\%$ sparsity on SqueezeNet, we produce a 0.47MB model $( 5 1 0 \\times$ smaller than 32-bit AlexNet) with equivalent accuracy. Our small model is indeed amenable to compression. ", + "bbox": [ + 174, + 320, + 823, + 391 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In addition, these results demonstrate that Deep Compression (Han et al., 2015a) not only works well on CNN architectures with many parameters (e.g. AlexNet and VGG), but it is also able to compress the already compact, fully convolutional SqueezeNet architecture. Deep Compression compressed SqueezeNet by $1 0 \\times$ while preserving the baseline accuracy. In summary: by combining CNN architectural innovation (SqueezeNet) with state-of-the-art compression techniques (Deep Compression), we achieved a $5 1 0 \\times$ reduction in model size with no decrease in accuracy compared to the baseline. ", + "bbox": [ + 173, + 398, + 825, + 496 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Finally, note that Deep Compression (Han et al., 2015b) uses a codebook as part of its scheme for quantizing CNN parameters to 6- or 8-bits of precision. Therefore, on most commodity processors, it is not trivial to achieve a speedup of $\\begin{array} { r } { \\frac { 3 2 } { 8 } = 4 x } \\end{array}$ with 8-bit quantization or $\\begin{array} { r } { \\frac { 3 2 } { 6 } = 5 . \\dot { 3 } x } \\end{array}$ with 6-bit quantization using the scheme developed in Deep Compression. However, Han et al. developed custom hardware – Efficient Inference Engine (EIE) – that can compute codebook-quantized CNNs more efficiently (Han et al., 2016a). In addition, in the months since we released SqueezeNet, P. Gysel developed a strategy called Ristretto for linearly quantizing SqueezeNet to 8 bits (Gysel, 2016). Specifically, Ristretto does computation in 8 bits, and it stores parameters and activations in 8-bit data types. Using the Ristretto strategy for 8-bit computation in SqueezeNet inference, Gysel observed less than 1 percentage-point of drop in accuracy when using 8-bit instead of 32-bit data types. ", + "bbox": [ + 173, + 503, + 825, + 656 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 CNN MICROARCHITECTURE DESIGN SPACE EXPLORATION", + "text_level": 1, + "bbox": [ + 176, + 685, + 699, + 702 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "So far, we have proposed architectural design strategies for small models, followed these principles to create SqueezeNet, and discovered that SqueezeNet is 50x smaller than AlexNet with equivalent accuracy. However, SqueezeNet and other models reside in a broad and largely unexplored design space of CNN architectures. Now, in Sections 5 and 6, we explore several aspects of the design space. We divide this architectural exploration into two main topics: microarchitectural exploration (per-module layer dimensions and configurations) and macroarchitectural exploration (high-level end-to-end organization of modules and other layers). ", + "bbox": [ + 174, + 709, + 825, + 808 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we design and execute experiments with the goal of providing intuition about the shape of the microarchitectural design space with respect to the design strategies that we proposed in Section 3.1. Note that our goal here is not to maximize accuracy in every experiment, but rather to understand the impact of CNN architectural choices on model size and accuracy. ", + "bbox": [ + 174, + 814, + 825, + 869 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/680240d6400757ba8a40ea928e6c20c6e26ac1d8fd8de2eeb0707082c24d3aee.jpg", + "image_caption": [ + "Figure 3: Microarchitectural design space exploration. " + ], + "image_footnote": [], + "bbox": [ + 183, + 108, + 807, + 313 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.1 CNN MICROARCHITECTURE METAPARAMETERS", + "text_level": 1, + "bbox": [ + 174, + 371, + 549, + 383 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In SqueezeNet, each Fire module has three dimensional hyperparameters that we defined in Section 3.2: $s _ { 1 x 1 }$ , $e _ { 1 x 1 }$ , and $e _ { 3 x 3 }$ . SqueezeNet has 8 Fire modules with a total of 24 dimensional hyperparameters. To do broad sweeps of the design space of SqueezeNet-like architectures, we define the following set of higher level metaparameters which control the dimensions of all Fire modules in a CNN. We define $b a s e _ { e }$ as the number of expand filters in the first Fire module in a CNN. After every freq Fire modules, we increase the number of expand filters by $i n c r _ { e }$ . In other words, for Fire module $i$ , the number of expand filters is $\\begin{array} { r } { e _ { i } = b a s e _ { e } + ( i n c r _ { e } * \\left\\lfloor \\frac { i } { f r e q } \\right\\rfloor ) } \\end{array}$ . In the expand layer of a Fire module, some filters are 1x1 and some are $3 { \\bf x } 3$ ; we define $e _ { i } = e _ { i , 1 x 1 } + e _ { i , 3 x 3 }$ with $p c t _ { 3 x 3 }$ (in the range [0, 1], shared over all Fire modules) as the percentage of expand filters that are $3 { \\bf x } 3$ . In other words, $e _ { i , 3 x 3 } = e _ { i } * p c t _ { 3 x 3 }$ , and $e _ { i , 1 x 1 } = e _ { i } * ( 1 - p c t _ { 3 x 3 } )$ . Finally, we define the number of filters in the squeeze layer of a Fire module using a metaparameter called the squeeze ratio (SR) (again, in the range $[ 0 , 1 ]$ , shared by all Fire modules): $s _ { i , 1 x 1 } = S R * e _ { i }$ (or equivalently $s _ { i , 1 x 1 } = S R * ( e _ { i , 1 x 1 } + e _ { i , 3 x \\bar { 3 } } ) \\}$ ). SqueezeNet (Table 1) is an example architecture that we generated with the aforementioned set of metaparameters. Specifically, SqueezeNet has the following metaparameters: $b a s e _ { e } = 1 2 8$ , $i n c r _ { e } = 1 2 8$ , $p c t _ { 3 x 3 } = 0 . 5$ , $f r e q = 2$ , and $S R = 0 . 1 2 5$ . ", + "bbox": [ + 173, + 382, + 825, + 599 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 SQUEEZE RATIO ", + "text_level": 1, + "bbox": [ + 174, + 616, + 330, + 628 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Section 3.1, we proposed decreasing the number of parameters by using squeeze layers to decrease the number of input channels seen by $3 \\mathrm { x } 3$ filters. We defined the squeeze ratio $( S R )$ as the ratio between the number of filters in squeeze layers and the number of filters in expand layers. We now design an experiment to investigate the effect of the squeeze ratio on model size and accuracy. ", + "bbox": [ + 174, + 630, + 825, + 684 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In these experiments, we use SqueezeNet (Figure 2) as a starting point. As in SqueezeNet, these experiments use the following metaparameters: $b a s e _ { e } = 1 2 8$ , $i n c r _ { e } = 1 2 8$ , $p c t _ { 3 x 3 } = 0 . 5$ , and $f r e q = 2$ . We train multiple models, where each model has a different squeeze ratio $( \\mathrm { S R } ) ^ { 7 }$ in the range [0.125, 1.0]. In Figure 3(a), we show the results of this experiment, where each point on the graph is an independent model that was trained from scratch. SqueezeNet is the $\\mathrm { S R } { = } 0 . 1 2 5$ point in this figure.8 From this figure, we learn that increasing SR beyond 0.125 can further increase ImageNet top-5 accuracy from $8 0 . 3 \\%$ (i.e. AlexNet-level) with a 4.8MB model to $8 6 . 0 \\%$ with a 19MB model. Accuracy plateaus at $8 6 . 0 \\%$ with $\\mathrm { S R } { = } 0 . 7 5$ (a 19MB model), and setting $\\mathrm { S R } { = } 1 . 0$ further increases model size without improving accuracy. ", + "bbox": [ + 173, + 691, + 825, + 816 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 TRADING OFF 1X1 AND 3X3 FILTERS ", + "text_level": 1, + "bbox": [ + 176, + 833, + 468, + 847 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Section 3.1, we proposed decreasing the number of parameters in a CNN by replacing some $3 { \\tt X } 3$ filters with 1x1 filters. An open question is, how important is spatial resolution in CNNs? The ", + "bbox": [ + 178, + 847, + 823, + 875 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "VGG (Simonyan & Zisserman, 2014) architectures have 3x3 spatial resolution in most layers’ filters; GoogLeNet (Szegedy et al., 2014) and Network-in-Network (NiN) (Lin et al., 2013) have 1x1 filters in some layers. In GoogLeNet and NiN, the authors simply propose a specific quantity of 1x1 and $3 { \\tt X } 3$ filters without further analysis.9 Here, we attempt to shed light on how the proportion of 1x1 and $3 { \\tt X } 3$ filters affects model size and accuracy. ", + "bbox": [ + 174, + 103, + 823, + 172 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We use the following metaparameters in this experiment: $b a s e _ { e } = i n c r _ { e } = 1 2 8$ , $f r e q = 2$ , $S R =$ 0.500, and we vary $p c t _ { 3 x 3 }$ from $1 \\%$ to $9 9 \\%$ . In other words, each Fire module’s expand layer has a predefined number of filters partitioned between 1x1 and $3 { \\tt X } 3$ , and here we turn the knob on these filters from “mostly 1x1” to “mostly $3 \\mathrm { x } 3 ^ { \\circ }$ . As in the previous experiment, these models have 8 Fire modules, following the same organization of layers as in Figure 2. We show the results of this experiment in Figure 3(b). Note that the 13MB models in Figure 3(a) and Figure 3(b) are the same architecture: $S R = 0 . 5 0 0$ and $p c t _ { 3 x 3 } = 5 0 \\%$ . We see in Figure 3(b) that the top-5 accuracy plateaus at $8 5 . 6 \\%$ using $50 \\%$ 3x3 filters, and further increasing the percentage of 3x3 filters leads to a larger model size but provides no improvement in accuracy on ImageNet. ", + "bbox": [ + 174, + 180, + 825, + 305 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CNN MACROARCHITECTURE DESIGN SPACE EXPLORATION", + "text_level": 1, + "bbox": [ + 174, + 330, + 704, + 347 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "So far we have explored the design space at the microarchitecture level, i.e. the contents of individual modules of the CNN. Now, we explore design decisions at the macroarchitecture level concerning the high-level connections among Fire modules. Inspired by ResNet (He et al., 2015b), we explored three different architectures: ", + "bbox": [ + 176, + 352, + 825, + 409 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "• Vanilla SqueezeNet (as per the prior sections). \n• SqueezeNet with simple bypass connections between some Fire modules. (Inspired by (Srivastava et al., 2015; He et al., 2015b).) \n• SqueezeNet with complex bypass connections between the remaining Fire modules. ", + "bbox": [ + 215, + 417, + 825, + 484 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We illustrate these three variants of SqueezeNet in Figure 2. ", + "bbox": [ + 174, + 500, + 566, + 513 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Our simple bypass architecture adds bypass connections around Fire modules 3, 5, 7, and 9, requiring these modules to learn a residual function between input and output. As in ResNet, to implement a bypass connection around Fire3, we set the input to Fire4 equal to (output of ${ \\mathrm { F i r e } } 2 +$ output of Fire3), where the $^ +$ operator is elementwise addition. This changes the regularization applied to the parameters of these Fire modules, and, as per ResNet, can improve the final accuracy and/or ability to train the full model. ", + "bbox": [ + 174, + 521, + 825, + 604 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "One limitation is that, in the straightforward case, the number of input channels and number of output channels has to be the same; as a result, only half of the Fire modules can have simple bypass connections, as shown in the middle diagram of Fig 2. When the “same number of channels” requirement can’t be met, we use a complex bypass connection, as illustrated on the right of Figure 2. While a simple bypass is “just a wire,” we define a complex bypass as a bypass that includes a 1x1 convolution layer with the number of filters set equal to the number of output channels that are needed. Note that complex bypass connections add extra parameters to the model, while simple bypass connections do not. ", + "bbox": [ + 174, + 612, + 825, + 723 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In addition to changing the regularization, it is intuitive to us that adding bypass connections would help to alleviate the representational bottleneck introduced by squeeze layers. In SqueezeNet, the squeeze ratio (SR) is 0.125, meaning that every squeeze layer has 8x fewer output channels than the accompanying expand layer. Due to this severe dimensionality reduction, a limited amount of information can pass through squeeze layers. However, by adding bypass connections to SqueezeNet, we open up avenues for information to flow around the squeeze layers. ", + "bbox": [ + 174, + 729, + 825, + 814 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We trained SqueezeNet with the three macroarchitectures in Figure 2 and compared the accuracy and model size in Table 3. We fixed the microarchitecture to match SqueezeNet as described in Table 1 throughout the macroarchitecture exploration. Complex and simple bypass connections both yielded an accuracy improvement over the vanilla SqueezeNet architecture. Interestingly, the simple bypass enabled a higher accuracy accuracy improvement than complex bypass. Adding the simple bypass connections yielded an increase of 2.9 percentage-points in top-1 accuracy and 2.2 percentage-points in top-5 accuracy without increasing model size. ", + "bbox": [ + 174, + 820, + 823, + 890 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/7ada205f4e2b7dc80fda12598318ace5aa917cc30f95a41ca6126700a8cb414b.jpg", + "table_caption": [ + "Table 3: SqueezeNet accuracy and model size using different macroarchitecture configurations " + ], + "table_footnote": [], + "table_body": "
ArchitectureTop-1 AccuracyTop-5 AccuracyModel Size
Vanilla SqueezeNet57.5%80.3%4.8MB
SqueezeNet+SimpleBypass60.4%82.5%4.8MB
SqueezeNet+ComplexBypass58.8%82.0%7.7MB
", + "bbox": [ + 236, + 127, + 761, + 181 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 204, + 823, + 232 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "7 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 252, + 330, + 267 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we have proposed steps toward a more disciplined approach to the design-space exploration of convolutional neural networks. Toward this goal we have presented SqueezeNet, a CNN architecture that has $5 0 \\times$ fewer parameters than AlexNet and maintains AlexNet-level accuracy on ImageNet. We also compressed SqueezeNet to less than 0.5MB, or $5 1 0 \\times$ smaller than AlexNet without compression. Since we released this paper as a technical report in 2016, Song Han and his collaborators have experimented further with SqueezeNet and model compression. Using a new approach called Dense-Sparse-Dense (DSD) (Han et al., 2016b), Han et al. use model compression during training as a regularizer to further improve accuracy, producing a compressed set of SqueezeNet parameters that is 1.2 percentage-points more accurate on ImageNet-1k, and also producing an uncompressed set of SqueezeNet parameters that is 4.3 percentage-points more accurate, compared to our results in Table 2. ", + "bbox": [ + 174, + 268, + 825, + 421 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We mentioned near the beginning of this paper that small models are more amenable to on-chip implementations on FPGAs. Since we released the SqueezeNet model, Gschwend has developed a variant of SqueezeNet and implemented it on an FPGA (Gschwend, 2016). As we anticipated, Gschwend was able to able to store the parameters of a SqueezeNet-like model entirely within the FPGA and eliminate the need for off-chip memory accesses to load model parameters. ", + "bbox": [ + 174, + 429, + 825, + 498 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In the context of this paper, we focused on ImageNet as a target dataset. However, it has become common practice to apply ImageNet-trained CNN representations to a variety of applications such as fine-grained object recognition (Zhang et al., 2013; Donahue et al., 2013), logo identification in images (Iandola et al., 2015), and generating sentences about images (Fang et al., 2015). ImageNettrained CNNs have also been applied to a number of applications pertaining to autonomous driving, including pedestrian and vehicle detection in images (Iandola et al., 2014; Girshick et al., 2015; Ashraf et al., 2016) and videos (Chen et al., 2015b), as well as segmenting the shape of the road (Badrinarayanan et al., 2015). We think SqueezeNet will be a good candidate CNN architecture for a variety of applications, especially those in which small model size is of importance. ", + "bbox": [ + 173, + 506, + 825, + 631 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "SqueezeNet is one of several new CNNs that we have discovered while broadly exploring the design space of CNN architectures. We hope that SqueezeNet will inspire the reader to consider and explore the broad range of possibilities in the design space of CNN architectures and to perform that exploration in a more systematic manner. ", + "bbox": [ + 174, + 637, + 825, + 693 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 714, + 285, + 729 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Khalid Ashraf, Bichen Wu, Forrest N. Iandola, Matthew W. Moskewicz, and Kurt Keutzer. Shallow networks for high-accuracy road object-detection. arXiv:1606.01561, 2016. ", + "bbox": [ + 174, + 736, + 821, + 765 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Vijay Badrinarayanan, Alex Kendall, and Roberto Cipolla. 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For a given accuracy level, it is typically possi-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 261, + 470, + 273 + ], + "spans": [ + { + "bbox": [ + 141, + 261, + 470, + 273 + ], + "score": 1.0, + "content": "ble to identify multiple CNN architectures that achieve that accuracy level. With", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 272, + 470, + 286 + ], + "spans": [ + { + "bbox": [ + 141, + 272, + 470, + 286 + ], + "score": 1.0, + "content": "equivalent accuracy, smaller CNN architectures offer at least three advantages: (1)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 282, + 470, + 297 + ], + "spans": [ + { + "bbox": [ + 141, + 282, + 470, + 297 + ], + "score": 1.0, + "content": "Smaller CNNs require less communication across servers during distributed train-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 294, + 470, + 307 + ], + "spans": [ + { + "bbox": [ + 141, + 294, + 470, + 307 + ], + "score": 1.0, + "content": "ing. (2) Smaller CNNs require less bandwidth to export a new model from the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 305, + 470, + 318 + ], + "spans": [ + { + "bbox": [ + 141, + 305, + 470, + 318 + ], + "score": 1.0, + "content": "cloud to an autonomous car. (3) Smaller CNNs are more feasible to deploy on FP-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 315, + 470, + 329 + ], + "spans": [ + { + "bbox": [ + 141, + 315, + 470, + 329 + ], + "score": 1.0, + "content": "GAs and other hardware with limited memory. To provide all of these advantages,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 327, + 470, + 339 + ], + "spans": [ + { + "bbox": [ + 141, + 327, + 470, + 339 + ], + "score": 1.0, + "content": "we propose a small CNN architecture called SqueezeNet. SqueezeNet achieves", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 338, + 469, + 351 + ], + "spans": [ + { + "bbox": [ + 141, + 338, + 317, + 351 + ], + "score": 1.0, + "content": "AlexNet-level accuracy on ImageNet with", + "type": "text" + }, + { + "bbox": [ + 318, + 338, + 334, + 348 + ], + "score": 0.42, + "content": "5 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 338, + 469, + 351 + ], + "score": 1.0, + "content": "fewer parameters. 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For a given accuracy level, there typically exist multiple", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "CNN architectures that achieve that accuracy level. Given equivalent accuracy, a CNN architecture", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 462, + 293, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 293, + 475 + ], + "score": 1.0, + "content": "with fewer parameters has several advantages:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 106, + 429, + 505, + 475 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 483, + 504, + 527 + ], + "lines": [ + { + "bbox": [ + 131, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 131, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "• More efficient distributed training. 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However, over-the-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 141, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "air updates of today’s typical CNN/DNN models can require large data transfers. With", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 142, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 142, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "AlexNet, this would require 240MB of communication from the server to the car. 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For inference, a sufficiently small model", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 142, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 142, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "could be stored directly on the FPGA instead of being bottlenecked by memory band-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 142, + 651, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 142, + 651, + 505, + 664 + ], + "score": 1.0, + "content": "width (Qiu et al., 2016), while video frames stream through the FPGA in real time. 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In addition, we present our attempt at a more disciplined approach to searching the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 278, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 278, + 140 + ], + "score": 1.0, + "content": "design space for novel CNN architectures.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "The rest of the paper is organized as follows. In Section 2 we review the related work. Then, in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "Sections 3 and 4 we describe and evaluate the SqueezeNet architecture. After that, we turn our", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "attention to understanding how CNN architectural design choices impact model size and accuracy.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "We gain this understanding by exploring the design space of SqueezeNet-like architectures. 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The remaining", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 484, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 484, + 254 + ], + "score": 1.0, + "content": "sections are aimed at advanced researchers who intend to design their own CNN architectures.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 272, + 211, + 284 + ], + "lines": [ + { + "bbox": [ + 104, + 270, + 213, + 287 + ], + "spans": [ + { + "bbox": [ + 104, + 270, + 213, + 287 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 288, + 230, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 231, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 231, + 300 + ], + "score": 1.0, + "content": "2.1 MODEL COMPRESSION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 299, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "The overarching goal of our work is to identify a model that has very few parameters while preserv-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "ing accuracy. To address this problem, a sensible approach is to take an existing CNN model and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "compress it in a lossy fashion. In fact, a research community has emerged around the topic of model", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 332, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 504, + 344 + ], + "score": 1.0, + "content": "compression, and several approaches have been reported. A fairly straightforward approach by Den-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "ton et al. is to apply singular value decomposition (SVD) to a pretrained CNN model (Denton et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "2014). Han et al. developed Network Pruning, which begins with a pretrained model, then replaces", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "score": 1.0, + "content": "parameters that are below a certain threshold with zeros to form a sparse matrix, and finally performs", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "a few iterations of training on the sparse CNN (Han et al., 2015b). Recently, Han et al. extended their", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "score": 1.0, + "content": "work by combining Network Pruning with quantization (to 8 bits or less) and huffman encoding to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "create an approach called Deep Compression (Han et al., 2015a), and further designed a hardware", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 407, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 422 + ], + "score": 1.0, + "content": "accelerator called EIE (Han et al., 2016a) that operates directly on the compressed model, achieving", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 419, + 271, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 271, + 432 + ], + "score": 1.0, + "content": "substantial speedups and energy savings.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 108, + 447, + 253, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 254, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 254, + 459 + ], + "score": 1.0, + "content": "2.2 CNN MICROARCHITECTURE", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "Convolutions have been used in artificial neural networks for at least 25 years; LeCun et al. helped", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "to popularize CNNs for digit recognition applications in the late 1980s (LeCun et al., 1989). In", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 478, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 478, + 505, + 493 + ], + "score": 1.0, + "content": "neural networks, convolution filters are typically 3D, with height, width, and channels as the key", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "dimensions. When applied to images, CNN filters typically have 3 channels in their first layer (i.e.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 256, + 514 + ], + "score": 1.0, + "content": "RGB), and in each subsequent layer", + "type": "text" + }, + { + "bbox": [ + 256, + 502, + 268, + 512 + ], + "score": 0.88, + "content": "L _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 501, + 465, + 514 + ], + "score": 1.0, + "content": "the filters have the same number of channels as", + "type": "text" + }, + { + "bbox": [ + 466, + 502, + 487, + 513 + ], + "score": 0.91, + "content": "L _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "has", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "filters. The early work by LeCun et al. (LeCun et al., 1989) uses 5x5xChannels2 filters, and the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "recent VGG (Simonyan & Zisserman, 2014) architectures extensively use 3x3 filters. Models such", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "as Network-in-Network (Lin et al., 2013) and the GoogLeNet family of architectures (Szegedy et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 544, + 459, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 459, + 558 + ], + "score": 1.0, + "content": "2014; Ioffe & Szegedy, 2015; Szegedy et al., 2015; 2016) use 1x1 filters in some layers.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 562, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "score": 1.0, + "content": "With the trend of designing very deep CNNs, it becomes cumbersome to manually select filter di-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "mensions for each layer. To address this, various higher level building blocks, or modules, comprised", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "of multiple convolution layers with a specific fixed organization have been proposed. For example,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "the GoogLeNet papers propose Inception modules, which are comprised of a number of different di-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "mensionalities of filters, usually including 1x1 and 3x3, plus sometimes 5x5 (Szegedy et al., 2014)", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "score": 1.0, + "content": "and sometimes 1x3 and 3x1 (Szegedy et al., 2015). Many such modules are then combined, perhaps", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 629, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 640 + ], + "score": 1.0, + "content": "with additional ad-hoc layers, to form a complete network. We use the term CNN microarchitecture", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 639, + 428, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 428, + 650 + ], + "score": 1.0, + "content": "to refer to the particular organization and dimensions of the individual modules.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5 + }, + { + "type": "title", + "bbox": [ + 107, + 667, + 256, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 258, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 258, + 679 + ], + "score": 1.0, + "content": "2.3 CNN MACROARCHITECTURE", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 710 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "While the CNN microarchitecture refers to individual layers and modules, we define the CNN", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "macroarchitecture as the system-level organization of multiple modules into an end-to-end CNN", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 698, + 159, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 159, + 712 + ], + "score": 1.0, + "content": "architecture.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 721, + 361, + 731 + ], + "lines": [ + { + "bbox": [ + 118, + 719, + 363, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 363, + 734 + ], + "score": 1.0, + "content": "2From now on, we will simply abbreviate HxWxChannels to HxW.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "As you can see, there are several advantages of smaller CNN architectures. With this in mind, we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "focus directly on the problem of identifying a CNN architecture with fewer parameters but equivalent", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "accuracy compared to a well-known model. We have discovered such an architecture, which we call", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "SqueezeNet. In addition, we present our attempt at a more disciplined approach to searching the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 278, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 278, + 140 + ], + "score": 1.0, + "content": "design space for novel CNN architectures.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 83, + 505, + 140 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "The rest of the paper is organized as follows. In Section 2 we review the related work. Then, in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "Sections 3 and 4 we describe and evaluate the SqueezeNet architecture. After that, we turn our", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "attention to understanding how CNN architectural design choices impact model size and accuracy.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "We gain this understanding by exploring the design space of SqueezeNet-like architectures. In", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "Section 5, we do design space exploration on the CNN microarchitecture, which we define as the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "organization and dimensionality of individual layers and modules. 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The remaining", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 484, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 484, + 254 + ], + "score": 1.0, + "content": "sections are aimed at advanced researchers who intend to design their own CNN architectures.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 144, + 506, + 254 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 272, + 211, + 284 + ], + "lines": [ + { + "bbox": [ + 104, + 270, + 213, + 287 + ], + "spans": [ + { + "bbox": [ + 104, + 270, + 213, + 287 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 288, + 230, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 231, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 231, + 300 + ], + "score": 1.0, + "content": "2.1 MODEL COMPRESSION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 299, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "The overarching goal of our work is to identify a model that has very few parameters while preserv-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "ing accuracy. To address this problem, a sensible approach is to take an existing CNN model and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "compress it in a lossy fashion. In fact, a research community has emerged around the topic of model", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 332, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 504, + 344 + ], + "score": 1.0, + "content": "compression, and several approaches have been reported. A fairly straightforward approach by Den-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "ton et al. is to apply singular value decomposition (SVD) to a pretrained CNN model (Denton et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "2014). Han et al. developed Network Pruning, which begins with a pretrained model, then replaces", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "score": 1.0, + "content": "parameters that are below a certain threshold with zeros to form a sparse matrix, and finally performs", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "a few iterations of training on the sparse CNN (Han et al., 2015b). Recently, Han et al. extended their", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "score": 1.0, + "content": "work by combining Network Pruning with quantization (to 8 bits or less) and huffman encoding to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "create an approach called Deep Compression (Han et al., 2015a), and further designed a hardware", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 407, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 422 + ], + "score": 1.0, + "content": "accelerator called EIE (Han et al., 2016a) that operates directly on the compressed model, achieving", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 419, + 271, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 271, + 432 + ], + "score": 1.0, + "content": "substantial speedups and energy savings.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 299, + 506, + 432 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 447, + 253, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 254, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 254, + 459 + ], + "score": 1.0, + "content": "2.2 CNN MICROARCHITECTURE", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "Convolutions have been used in artificial neural networks for at least 25 years; LeCun et al. helped", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "to popularize CNNs for digit recognition applications in the late 1980s (LeCun et al., 1989). In", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 478, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 478, + 505, + 493 + ], + "score": 1.0, + "content": "neural networks, convolution filters are typically 3D, with height, width, and channels as the key", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "dimensions. When applied to images, CNN filters typically have 3 channels in their first layer (i.e.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 256, + 514 + ], + "score": 1.0, + "content": "RGB), and in each subsequent layer", + "type": "text" + }, + { + "bbox": [ + 256, + 502, + 268, + 512 + ], + "score": 0.88, + "content": "L _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 501, + 465, + 514 + ], + "score": 1.0, + "content": "the filters have the same number of channels as", + "type": "text" + }, + { + "bbox": [ + 466, + 502, + 487, + 513 + ], + "score": 0.91, + "content": "L _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "has", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "filters. The early work by LeCun et al. (LeCun et al., 1989) uses 5x5xChannels2 filters, and the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "recent VGG (Simonyan & Zisserman, 2014) architectures extensively use 3x3 filters. Models such", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "as Network-in-Network (Lin et al., 2013) and the GoogLeNet family of architectures (Szegedy et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 544, + 459, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 459, + 558 + ], + "score": 1.0, + "content": "2014; Ioffe & Szegedy, 2015; Szegedy et al., 2015; 2016) use 1x1 filters in some layers.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 457, + 506, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 562, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "score": 1.0, + "content": "With the trend of designing very deep CNNs, it becomes cumbersome to manually select filter di-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "mensions for each layer. To address this, various higher level building blocks, or modules, comprised", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "of multiple convolution layers with a specific fixed organization have been proposed. For example,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "the GoogLeNet papers propose Inception modules, which are comprised of a number of different di-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "mensionalities of filters, usually including 1x1 and 3x3, plus sometimes 5x5 (Szegedy et al., 2014)", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "score": 1.0, + "content": "and sometimes 1x3 and 3x1 (Szegedy et al., 2015). Many such modules are then combined, perhaps", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 629, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 640 + ], + "score": 1.0, + "content": "with additional ad-hoc layers, to form a complete network. We use the term CNN microarchitecture", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 639, + 428, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 428, + 650 + ], + "score": 1.0, + "content": "to refer to the particular organization and dimensions of the individual modules.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 561, + 505, + 650 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 667, + 256, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 258, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 258, + 679 + ], + "score": 1.0, + "content": "2.3 CNN MACROARCHITECTURE", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 710 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "While the CNN microarchitecture refers to individual layers and modules, we define the CNN", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "macroarchitecture as the system-level organization of multiple modules into an end-to-end CNN", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 698, + 159, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 159, + 712 + ], + "score": 1.0, + "content": "architecture.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49, + "bbox_fs": [ + 106, + 677, + 505, + 712 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "Perhaps the mostly widely studied CNN macroarchitecture topic in the recent literature is the impact", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "of depth (i.e. number of layers) in networks. Simoyan and Zisserman proposed the VGG (Simonyan", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "& Zisserman, 2014) family of CNNs with 12 to 19 layers and reported that deeper networks produce", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "higher accuracy on the ImageNet-1k dataset (Deng et al., 2009). K. He et al. proposed deeper CNNs", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 438, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 438, + 140 + ], + "score": 1.0, + "content": "with up to 30 layers that deliver even higher ImageNet accuracy (He et al., 2015a).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "The choice of connections across multiple layers or modules is an emerging area of CNN macroar-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "chitectural research. Residual Networks (ResNet) (He et al., 2015b) and Highway Networks (Sri-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "vastava et al., 2015) each propose the use of connections that skip over multiple layers, for example", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "additively connecting the activations from layer 3 to the activations from layer 6. We refer to these", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "connections as bypass connections. The authors of ResNet provide an A/B comparison of a 34-layer", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "CNN with and without bypass connections; adding bypass connections delivers a 2 percentage-point", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 281, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 281, + 223 + ], + "score": 1.0, + "content": "improvement on Top-5 ImageNet accuracy.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 107, + 235, + 346, + 245 + ], + "lines": [ + { + "bbox": [ + 106, + 234, + 347, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 347, + 247 + ], + "score": 1.0, + "content": "2.4 NEURAL NETWORK DESIGN SPACE EXPLORATION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "Neural networks (including deep and convolutional NNs) have a large design space, with numerous", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 255, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 269 + ], + "score": 1.0, + "content": "options for microarchitectures, macroarchitectures, solvers, and other hyperparameters. It seems", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 280 + ], + "score": 1.0, + "content": "natural that the community would want to gain intuition about how these factors impact a NN’s", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 278, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "accuracy (i.e. the shape of the design space). Much of the work on design space exploration (DSE)", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 288, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 303 + ], + "score": 1.0, + "content": "of NNs has focused on developing automated approaches for finding NN architectures that deliver", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "higher accuracy. These automated DSE approaches include bayesian optimization (Snoek et al.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "score": 1.0, + "content": "2012), simulated annealing (Ludermir et al., 2006), randomized search (Bergstra & Bengio, 2012),", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 336 + ], + "score": 1.0, + "content": "and genetic algorithms (Stanley & Miikkulainen, 2002). To their credit, each of these papers pro-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "score": 1.0, + "content": "vides a case in which the proposed DSE approach produces a NN architecture that achieves higher", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "accuracy compared to a representative baseline. However, these papers make no attempt to provide", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "score": 1.0, + "content": "intuition about the shape of the NN design space. Later in this paper, we eschew automated ap-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 366, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 378 + ], + "score": 1.0, + "content": "proaches – instead, we refactor CNNs in such a way that we can do principled A/B comparisons to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 376, + 427, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 427, + 390 + ], + "score": 1.0, + "content": "investigate how CNN architectural decisions influence model size and accuracy.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 108, + 393, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "In the following sections, we first propose and evaluate the SqueezeNet architecture with and with-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "out model compression. Then, we explore the impact of design choices in microarchitecture and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 416, + 343, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 343, + 428 + ], + "score": 1.0, + "content": "macroarchitecture for SqueezeNet-like CNN architectures.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 444, + 454, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 456, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 456, + 457 + ], + "score": 1.0, + "content": "3 SQUEEZENET: PRESERVING ACCURACY WITH FEW PARAMETERS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "In this section, we begin by outlining our design strategies for CNN architectures with few param-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "eters. Then, we introduce the Fire module, our new building block out of which to build CNN", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "architectures. Finally, we use our design strategies to construct SqueezeNet, which is comprised", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 492, + 205, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 205, + 504 + ], + "score": 1.0, + "content": "mainly of Fire modules.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 109, + 518, + 297, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 298, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 298, + 529 + ], + "score": 1.0, + "content": "3.1 ARCHITECTURAL DESIGN STRATEGIES", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 108, + 528, + 504, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "Our overarching objective in this paper is to identify CNN architectures that have few parameters", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 539, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "while maintaining competitive accuracy. To achieve this, we employ three main strategies when", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 550, + 227, + 562 + ], + "spans": [ + { + "bbox": [ + 107, + 550, + 227, + 562 + ], + "score": 1.0, + "content": "designing CNN architectures:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 108, + 566, + 504, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "Strategy 1. Replace 3x3 filters with 1x1 filters. Given a budget of a certain number of convolution", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "filters, we will choose to make the majority of these filters 1x1, since a 1x1 filter has 9X fewer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 588, + 220, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 179, + 600 + ], + "score": 1.0, + "content": "parameters than a", + "type": "text" + }, + { + "bbox": [ + 179, + 588, + 196, + 599 + ], + "score": 0.29, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 588, + 220, + 600 + ], + "score": 1.0, + "content": "filter.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 612, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "Strategy 2. Decrease the number of input channels to 3x3 filters. Consider a convolution layer", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "that is comprised entirely of 3x3 filters. The total quantity of parameters in this layer is (number of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 257, + 648 + ], + "score": 1.0, + "content": "input channels) * (number of filters)", + "type": "text" + }, + { + "bbox": [ + 257, + 635, + 288, + 646 + ], + "score": 0.84, + "content": "^ { * } \\left( 3 ^ { * } 3 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 633, + 506, + 648 + ], + "score": 1.0, + "content": ". So, to maintain a small total number of parameters", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "score": 1.0, + "content": "in a CNN, it is important not only to decrease the number of 3x3 filters (see Strategy 1 above), but", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 656, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 669 + ], + "score": 1.0, + "content": "also to decrease the number of input channels to the 3x3 filters. We decrease the number of input", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 667, + 437, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 154, + 680 + ], + "score": 1.0, + "content": "channels to", + "type": "text" + }, + { + "bbox": [ + 154, + 667, + 171, + 677 + ], + "score": 0.34, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 667, + 437, + 680 + ], + "score": 1.0, + "content": "filters using squeeze layers, which we describe in the next section.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 107, + 688, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "Strategy 3. Downsample late in the network so that convolution layers have large activation", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "maps. In a convolutional network, each convolution layer produces an output activation map with", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "a spatial resolution that is at least 1x1 and often much larger than 1x1. The height and width of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 402, + 734 + ], + "score": 1.0, + "content": "these activation maps are controlled by: (1) the size of the input data (e.g.", + "type": "text" + }, + { + "bbox": [ + 402, + 721, + 439, + 731 + ], + "score": 0.52, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "images) and (2)", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "Perhaps the mostly widely studied CNN macroarchitecture topic in the recent literature is the impact", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "of depth (i.e. number of layers) in networks. Simoyan and Zisserman proposed the VGG (Simonyan", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "& Zisserman, 2014) family of CNNs with 12 to 19 layers and reported that deeper networks produce", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "higher accuracy on the ImageNet-1k dataset (Deng et al., 2009). K. He et al. proposed deeper CNNs", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 438, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 438, + 140 + ], + "score": 1.0, + "content": "with up to 30 layers that deliver even higher ImageNet accuracy (He et al., 2015a).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 82, + 506, + 140 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "The choice of connections across multiple layers or modules is an emerging area of CNN macroar-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "chitectural research. Residual Networks (ResNet) (He et al., 2015b) and Highway Networks (Sri-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "vastava et al., 2015) each propose the use of connections that skip over multiple layers, for example", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "additively connecting the activations from layer 3 to the activations from layer 6. We refer to these", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "connections as bypass connections. The authors of ResNet provide an A/B comparison of a 34-layer", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "CNN with and without bypass connections; adding bypass connections delivers a 2 percentage-point", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 281, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 281, + 223 + ], + "score": 1.0, + "content": "improvement on Top-5 ImageNet accuracy.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 142, + 506, + 223 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 235, + 346, + 245 + ], + "lines": [ + { + "bbox": [ + 106, + 234, + 347, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 347, + 247 + ], + "score": 1.0, + "content": "2.4 NEURAL NETWORK DESIGN SPACE EXPLORATION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "Neural networks (including deep and convolutional NNs) have a large design space, with numerous", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 255, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 269 + ], + "score": 1.0, + "content": "options for microarchitectures, macroarchitectures, solvers, and other hyperparameters. It seems", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 280 + ], + "score": 1.0, + "content": "natural that the community would want to gain intuition about how these factors impact a NN’s", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 278, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "accuracy (i.e. the shape of the design space). Much of the work on design space exploration (DSE)", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 288, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 303 + ], + "score": 1.0, + "content": "of NNs has focused on developing automated approaches for finding NN architectures that deliver", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "higher accuracy. These automated DSE approaches include bayesian optimization (Snoek et al.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "score": 1.0, + "content": "2012), simulated annealing (Ludermir et al., 2006), randomized search (Bergstra & Bengio, 2012),", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 336 + ], + "score": 1.0, + "content": "and genetic algorithms (Stanley & Miikkulainen, 2002). To their credit, each of these papers pro-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "score": 1.0, + "content": "vides a case in which the proposed DSE approach produces a NN architecture that achieves higher", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "accuracy compared to a representative baseline. However, these papers make no attempt to provide", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "score": 1.0, + "content": "intuition about the shape of the NN design space. Later in this paper, we eschew automated ap-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 366, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 378 + ], + "score": 1.0, + "content": "proaches – instead, we refactor CNNs in such a way that we can do principled A/B comparisons to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 376, + 427, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 427, + 390 + ], + "score": 1.0, + "content": "investigate how CNN architectural decisions influence model size and accuracy.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 245, + 506, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 393, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "In the following sections, we first propose and evaluate the SqueezeNet architecture with and with-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "out model compression. Then, we explore the impact of design choices in microarchitecture and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 416, + 343, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 343, + 428 + ], + "score": 1.0, + "content": "macroarchitecture for SqueezeNet-like CNN architectures.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 393, + 505, + 428 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 444, + 454, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 456, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 456, + 457 + ], + "score": 1.0, + "content": "3 SQUEEZENET: PRESERVING ACCURACY WITH FEW PARAMETERS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "In this section, we begin by outlining our design strategies for CNN architectures with few param-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "eters. Then, we introduce the Fire module, our new building block out of which to build CNN", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "architectures. Finally, we use our design strategies to construct SqueezeNet, which is comprised", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 492, + 205, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 205, + 504 + ], + "score": 1.0, + "content": "mainly of Fire modules.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 457, + 506, + 504 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 518, + 297, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 298, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 298, + 529 + ], + "score": 1.0, + "content": "3.1 ARCHITECTURAL DESIGN STRATEGIES", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 108, + 528, + 504, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "Our overarching objective in this paper is to identify CNN architectures that have few parameters", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 539, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "while maintaining competitive accuracy. To achieve this, we employ three main strategies when", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 550, + 227, + 562 + ], + "spans": [ + { + "bbox": [ + 107, + 550, + 227, + 562 + ], + "score": 1.0, + "content": "designing CNN architectures:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 527, + 505, + 562 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 566, + 504, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "Strategy 1. Replace 3x3 filters with 1x1 filters. Given a budget of a certain number of convolution", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "filters, we will choose to make the majority of these filters 1x1, since a 1x1 filter has 9X fewer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 588, + 220, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 179, + 600 + ], + "score": 1.0, + "content": "parameters than a", + "type": "text" + }, + { + "bbox": [ + 179, + 588, + 196, + 599 + ], + "score": 0.29, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 588, + 220, + 600 + ], + "score": 1.0, + "content": "filter.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 566, + 506, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 612, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "Strategy 2. Decrease the number of input channels to 3x3 filters. Consider a convolution layer", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "that is comprised entirely of 3x3 filters. The total quantity of parameters in this layer is (number of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 257, + 648 + ], + "score": 1.0, + "content": "input channels) * (number of filters)", + "type": "text" + }, + { + "bbox": [ + 257, + 635, + 288, + 646 + ], + "score": 0.84, + "content": "^ { * } \\left( 3 ^ { * } 3 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 633, + 506, + 648 + ], + "score": 1.0, + "content": ". So, to maintain a small total number of parameters", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "score": 1.0, + "content": "in a CNN, it is important not only to decrease the number of 3x3 filters (see Strategy 1 above), but", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 656, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 669 + ], + "score": 1.0, + "content": "also to decrease the number of input channels to the 3x3 filters. 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In this", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 146, + 277 + ], + "score": 1.0, + "content": "example,", + "type": "text" + }, + { + "bbox": [ + 147, + 266, + 189, + 277 + ], + "score": 0.89, + "content": "s _ { 1 x 1 } ~ = ~ 3", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 264, + 193, + 277 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 194, + 266, + 236, + 277 + ], + "score": 0.9, + "content": "e _ { 1 x 1 } = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 264, + 259, + 277 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 259, + 266, + 301, + 277 + ], + "score": 0.91, + "content": "e _ { 3 x 3 } ~ = ~ 4", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 264, + 506, + 277 + ], + "score": 1.0, + "content": ". We illustrate the convolution filters but not the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 276, + 155, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 155, + 288 + ], + "score": 1.0, + "content": "activations.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 323 + ], + "score": 1.0, + "content": "the choice of layers in which to downsample in the CNN architecture. Most commonly, downsam-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 363, + 334 + ], + "score": 1.0, + "content": "pling is engineered into CNN architectures by setting the (stride", + "type": "text" + }, + { + "bbox": [ + 363, + 322, + 381, + 332 + ], + "score": 0.84, + "content": "> 1", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 321, + 506, + 334 + ], + "score": 1.0, + "content": ") in some of the convolution or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "pooling layers (e.g. (Szegedy et al., 2014; Simonyan & Zisserman, 2014; Krizhevsky et al., 2012)).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "If early3 layers in the network have large strides, then most layers will have small activation maps.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "Conversely, if most layers in the network have a stride of 1, and the strides greater than 1 are con-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "score": 1.0, + "content": "centrated toward the end4 of the network, then many layers in the network will have large activation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "maps. Our intuition is that large activation maps (due to delayed downsampling) can lead to higher", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 387, + 504, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 504, + 399 + ], + "score": 1.0, + "content": "classification accuracy, with all else held equal. Indeed, K. He and H. Sun applied delayed down-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "sampling to four different CNN architectures, and in each case delayed downsampling led to higher", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 409, + 275, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 275, + 421 + ], + "score": 1.0, + "content": "classification accuracy (He & Sun, 2015).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "Strategies 1 and 2 are about judiciously decreasing the quantity of parameters in a CNN while", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 437, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 449 + ], + "score": 1.0, + "content": "attempting to preserve accuracy. Strategy 3 is about maximizing accuracy on a limited budget of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "parameters. Next, we describe the Fire module, which is our building block for CNN architectures", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 459, + 351, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 351, + 471 + ], + "score": 1.0, + "content": "that enables us to successfully employ Strategies 1, 2, and 3.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 488, + 214, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 216, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 216, + 501 + ], + "score": 1.0, + "content": "3.2 THE FIRE MODULE", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "We define the Fire module as follows. A Fire module is comprised of: a squeeze convolution layer", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 511, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 522 + ], + "score": 1.0, + "content": "(which has only 1x1 filters), feeding into an expand layer that has a mix of 1x1 and 3x3 convolution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "filters; we illustrate this in Figure 1. The liberal use of 1x1 filters in Fire modules is an application", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 533, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 544 + ], + "score": 1.0, + "content": "of Strategy 1 from Section 3.1. 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We illustrate in Figure 2 that SqueezeNet", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 637, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 653 + ], + "score": 1.0, + "content": "begins with a standalone convolution layer (conv1), followed by 8 Fire modules (fire2-9), ending", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "with a final conv layer (conv10). We gradually increase the number of filters per fire module from", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "the beginning to the end of the network. 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In this", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 146, + 277 + ], + "score": 1.0, + "content": "example,", + "type": "text" + }, + { + "bbox": [ + 147, + 266, + 189, + 277 + ], + "score": 0.89, + "content": "s _ { 1 x 1 } ~ = ~ 3", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 264, + 193, + 277 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 194, + 266, + 236, + 277 + ], + "score": 0.9, + "content": "e _ { 1 x 1 } = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 264, + 259, + 277 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 259, + 266, + 301, + 277 + ], + "score": 0.91, + "content": "e _ { 3 x 3 } ~ = ~ 4", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 264, + 506, + 277 + ], + "score": 1.0, + "content": ". We illustrate the convolution filters but not the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 276, + 155, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 155, + 288 + ], + "score": 1.0, + "content": "activations.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 323 + ], + "score": 1.0, + "content": "the choice of layers in which to downsample in the CNN architecture. Most commonly, downsam-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 363, + 334 + ], + "score": 1.0, + "content": "pling is engineered into CNN architectures by setting the (stride", + "type": "text" + }, + { + "bbox": [ + 363, + 322, + 381, + 332 + ], + "score": 0.84, + "content": "> 1", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 321, + 506, + 334 + ], + "score": 1.0, + "content": ") in some of the convolution or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "pooling layers (e.g. (Szegedy et al., 2014; Simonyan & Zisserman, 2014; Krizhevsky et al., 2012)).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "If early3 layers in the network have large strides, then most layers will have small activation maps.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "Conversely, if most layers in the network have a stride of 1, and the strides greater than 1 are con-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "score": 1.0, + "content": "centrated toward the end4 of the network, then many layers in the network will have large activation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "maps. Our intuition is that large activation maps (due to delayed downsampling) can lead to higher", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 387, + 504, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 504, + 399 + ], + "score": 1.0, + "content": "classification accuracy, with all else held equal. Indeed, K. He and H. Sun applied delayed down-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "sampling to four different CNN architectures, and in each case delayed downsampling led to higher", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 409, + 275, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 275, + 421 + ], + "score": 1.0, + "content": "classification accuracy (He & Sun, 2015).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 309, + 506, + 421 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "Strategies 1 and 2 are about judiciously decreasing the quantity of parameters in a CNN while", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 437, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 449 + ], + "score": 1.0, + "content": "attempting to preserve accuracy. Strategy 3 is about maximizing accuracy on a limited budget of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "parameters. Next, we describe the Fire module, which is our building block for CNN architectures", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 459, + 351, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 351, + 471 + ], + "score": 1.0, + "content": "that enables us to successfully employ Strategies 1, 2, and 3.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 425, + 506, + 471 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 488, + 214, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 216, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 216, + 501 + ], + "score": 1.0, + "content": "3.2 THE FIRE MODULE", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "We define the Fire module as follows. A Fire module is comprised of: a squeeze convolution layer", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 511, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 522 + ], + "score": 1.0, + "content": "(which has only 1x1 filters), feeding into an expand layer that has a mix of 1x1 and 3x3 convolution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "filters; we illustrate this in Figure 1. The liberal use of 1x1 filters in Fire modules is an application", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 533, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 544 + ], + "score": 1.0, + "content": "of Strategy 1 from Section 3.1. We expose three tunable dimensions (hyperparameters) in a Fire", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 143, + 557 + ], + "score": 1.0, + "content": "module:", + "type": "text" + }, + { + "bbox": [ + 144, + 545, + 162, + 555 + ], + "score": 0.81, + "content": "s _ { 1 x 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 543, + 166, + 557 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 167, + 545, + 185, + 555 + ], + "score": 0.77, + "content": "e _ { 1 x 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 543, + 207, + 557 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 207, + 545, + 226, + 555 + ], + "score": 0.88, + "content": "e _ { 3 x 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 543, + 306, + 557 + ], + "score": 1.0, + "content": ". 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When we use Fire modules we set", + "type": "text" + }, + { + "bbox": [ + 335, + 567, + 354, + 576 + ], + "score": 0.88, + "content": "s _ { 1 x 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 564, + 419, + 578 + ], + "score": 1.0, + "content": "to be less than", + "type": "text" + }, + { + "bbox": [ + 419, + 566, + 472, + 577 + ], + "score": 0.87, + "content": "( e _ { 1 x 1 } + e _ { 3 x 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "), so the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 366, + 589 + ], + "score": 1.0, + "content": "squeeze layer helps to limit the number of input channels to the", + "type": "text" + }, + { + "bbox": [ + 366, + 576, + 383, + 586 + ], + "score": 0.3, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "filters, as per Strategy 2 from", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 587, + 156, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 156, + 598 + ], + "score": 1.0, + "content": "Section 3.1.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 499, + 506, + 598 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 617, + 280, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 616, + 281, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 281, + 629 + ], + "score": 1.0, + "content": "3.3 THE SQUEEZENET ARCHITECTURE", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "We now describe the SqueezeNet CNN architecture. We illustrate in Figure 2 that SqueezeNet", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 637, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 653 + ], + "score": 1.0, + "content": "begins with a standalone convolution layer (conv1), followed by 8 Fire modules (fire2-9), ending", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "with a final conv layer (conv10). We gradually increase the number of filters per fire module from", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "the beginning to the end of the network. SqueezeNet performs max-pooling with a stride of 2 after", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "layers conv1, fire4, fire8, and conv10; these relatively late placements of pooling are per Strategy 3", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 683, + 401, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 401, + 695 + ], + "score": 1.0, + "content": "from Section 3.1. 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Left: SqueezeNet (Section 3.3);", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 347, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 362 + ], + "score": 1.0, + "content": "Middle: SqueezeNet with simple bypass (Section 6); Right: SqueezeNet with complex bypass (Sec-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 357, + 139, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 139, + 373 + ], + "score": 1.0, + "content": "tion 6).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 269, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 271, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 271, + 406 + ], + "score": 1.0, + "content": "3.3.1 OTHER SQUEEZENET DETAILS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 108, + 406, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "For brevity, we have omitted number of details and design choices about SqueezeNet from Table 1", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "and Figure 2. We provide these design choices in the following. The intuition behind these choices", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 428, + 267, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 267, + 442 + ], + "score": 1.0, + "content": "may be found in the papers cited below.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 132, + 451, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 132, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 132, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "• So that the output activations from 1x1 and 3x3 filters have the same height and width, we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 462, + 491, + 475 + ], + "spans": [ + { + "bbox": [ + 141, + 462, + 368, + 475 + ], + "score": 1.0, + "content": "add a 1-pixel border of zero-padding in the input data to", + "type": "text" + }, + { + "bbox": [ + 368, + 463, + 385, + 473 + ], + "score": 0.7, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 462, + 491, + 475 + ], + "score": 1.0, + "content": "filters of expand modules.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 131, + 476, + 490, + 490 + ], + "spans": [ + { + "bbox": [ + 131, + 476, + 490, + 490 + ], + "score": 1.0, + "content": "• ReLU (Nair & Hinton, 2010) is applied to activations from squeeze and expand layers.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 131, + 491, + 486, + 504 + ], + "spans": [ + { + "bbox": [ + 131, + 491, + 333, + 504 + ], + "score": 1.0, + "content": "• Dropout (Srivastava et al., 2014) with a ratio of", + "type": "text" + }, + { + "bbox": [ + 333, + 491, + 353, + 501 + ], + "score": 0.85, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 491, + 486, + 504 + ], + "score": 1.0, + "content": "is applied after the fire9 module.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 133, + 504, + 504, + 518 + ], + "spans": [ + { + "bbox": [ + 133, + 504, + 504, + 518 + ], + "score": 1.0, + "content": "• Note the lack of fully-connected layers in SqueezeNet; this design choice was inspired by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 516, + 298, + 528 + ], + "spans": [ + { + "bbox": [ + 141, + 516, + 298, + 528 + ], + "score": 1.0, + "content": "the NiN (Lin et al., 2013) architecture.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 140, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 140, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "When training SqueezeNet, we begin with a learning rate of 0.04, and we lin-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 540, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 141, + 540, + 505, + 553 + ], + "score": 1.0, + "content": "early decrease the learning rate throughout training, as described in (Mishkin et al.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 141, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "2016). For details on the training protocol (e.g. batch size, learning rate, parame-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 563, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 142, + 563, + 505, + 574 + ], + "score": 1.0, + "content": "ter initialization), please refer to our Caffe-compatible configuration files located here:", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 574, + 315, + 586 + ], + "spans": [ + { + "bbox": [ + 142, + 574, + 315, + 586 + ], + "score": 1.0, + "content": "https://github.com/DeepScale/SqueezeNet.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 139, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 139, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "The Caffe framework does not natively support a convolution layer that contains multiple", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 141, + 598, + 269, + 611 + ], + "score": 1.0, + "content": "filter resolutions (e.g. 1x1 and", + "type": "text" + }, + { + "bbox": [ + 270, + 599, + 286, + 609 + ], + "score": 0.34, + "content": "3 { \\bf x } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 598, + 506, + 611 + ], + "score": 1.0, + "content": ") (Jia et al., 2014). To get around this, we implement", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 141, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "our expand layer with two separate convolution layers: a layer with 1x1 filters, and a layer", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 142, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "with 3x3 filters. Then, we concatenate the outputs of these layers together in the channel", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 141, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "dimension. 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However, in addition to Caffe, several other CNN frameworks have emerged, including", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "MXNet (Chen et al., 2015a), Chainer (Tokui et al., 2015), Keras (Chollet, 2016), and Torch (Col-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "lobert et al., 2011). Each of these has its own native format for representing a CNN architec-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "ture. 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Left: SqueezeNet (Section 3.3);", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 347, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 362 + ], + "score": 1.0, + "content": "Middle: SqueezeNet with simple bypass (Section 6); Right: SqueezeNet with complex bypass (Sec-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 357, + 139, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 139, + 373 + ], + "score": 1.0, + "content": "tion 6).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 269, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 271, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 271, + 406 + ], + "score": 1.0, + "content": "3.3.1 OTHER SQUEEZENET DETAILS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 108, + 406, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "For brevity, we have omitted number of details and design choices about SqueezeNet from Table 1", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "and Figure 2. We provide these design choices in the following. The intuition behind these choices", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 428, + 267, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 267, + 442 + ], + "score": 1.0, + "content": "may be found in the papers cited below.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 406, + 505, + 442 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 451, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 132, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 132, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "• So that the output activations from 1x1 and 3x3 filters have the same height and width, we", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 462, + 491, + 475 + ], + "spans": [ + { + "bbox": [ + 141, + 462, + 368, + 475 + ], + "score": 1.0, + "content": "add a 1-pixel border of zero-padding in the input data to", + "type": "text" + }, + { + "bbox": [ + 368, + 463, + 385, + 473 + ], + "score": 0.7, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 462, + 491, + 475 + ], + "score": 1.0, + "content": "filters of expand modules.", + "type": "text" + } + ], + "index": 11, + "is_list_end_line": true + }, + { + "bbox": [ + 131, + 476, + 490, + 490 + ], + "spans": [ + { + "bbox": [ + 131, + 476, + 490, + 490 + ], + "score": 1.0, + "content": "• ReLU (Nair & Hinton, 2010) is applied to activations from squeeze and expand layers.", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 131, + 491, + 486, + 504 + ], + "spans": [ + { + "bbox": [ + 131, + 491, + 333, + 504 + ], + "score": 1.0, + "content": "• Dropout (Srivastava et al., 2014) with a ratio of", + "type": "text" + }, + { + "bbox": [ + 333, + 491, + 353, + 501 + ], + "score": 0.85, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 491, + 486, + 504 + ], + "score": 1.0, + "content": "is applied after the fire9 module.", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 133, + 504, + 504, + 518 + ], + "spans": [ + { + "bbox": [ + 133, + 504, + 504, + 518 + ], + "score": 1.0, + "content": "• Note the lack of fully-connected layers in SqueezeNet; this design choice was inspired by", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 516, + 298, + 528 + ], + "spans": [ + { + "bbox": [ + 141, + 516, + 298, + 528 + ], + "score": 1.0, + "content": "the NiN (Lin et al., 2013) architecture.", + "type": "text" + } + ], + "index": 15, + "is_list_end_line": true + }, + { + "bbox": [ + 140, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 140, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "When training SqueezeNet, we begin with a learning rate of 0.04, and we lin-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 540, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 141, + 540, + 505, + 553 + ], + "score": 1.0, + "content": "early decrease the learning rate throughout training, as described in (Mishkin et al.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 141, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "2016). For details on the training protocol (e.g. batch size, learning rate, parame-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 563, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 142, + 563, + 505, + 574 + ], + "score": 1.0, + "content": "ter initialization), please refer to our Caffe-compatible configuration files located here:", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 574, + 315, + 586 + ], + "spans": [ + { + "bbox": [ + 142, + 574, + 315, + 586 + ], + "score": 1.0, + "content": "https://github.com/DeepScale/SqueezeNet.", + "type": "text" + } + ], + "index": 20, + "is_list_end_line": true + }, + { + "bbox": [ + 139, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 139, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "The Caffe framework does not natively support a convolution layer that contains multiple", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 141, + 598, + 269, + 611 + ], + "score": 1.0, + "content": "filter resolutions (e.g. 1x1 and", + "type": "text" + }, + { + "bbox": [ + 270, + 599, + 286, + 609 + ], + "score": 0.34, + "content": "3 { \\bf x } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 598, + 506, + 611 + ], + "score": 1.0, + "content": ") (Jia et al., 2014). To get around this, we implement", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 141, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "our expand layer with two separate convolution layers: a layer with 1x1 filters, and a layer", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 142, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "with 3x3 filters. Then, we concatenate the outputs of these layers together in the channel", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 141, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "dimension. This is numerically equivalent to implementing one layer that contains both", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 642, + 222, + 654 + ], + "spans": [ + { + "bbox": [ + 142, + 642, + 222, + 654 + ], + "score": 1.0, + "content": "1x1 and 3x3 filters.", + "type": "text" + } + ], + "index": 26, + "is_list_end_line": true + } + ], + "index": 18, + "bbox_fs": [ + 131, + 451, + 506, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "We released the SqueezeNet configuration files in the format defined by the Caffe CNN frame-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 675, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 506, + 691 + ], + "score": 1.0, + "content": "work. 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In each of the CNN model compression papers", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "score": 1.0, + "content": "reviewed in Section 2.1, the goal was to compress an AlexNet (Krizhevsky et al., 2012) model", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "that was trained to classify images using the ImageNet (Deng et al., 2009) (ILSVRC 2012) dataset.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 237, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 252 + ], + "score": 1.0, + "content": "Therefore, we use AlexNet5 and the associated model compression results as a basis for comparison", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 249, + 225, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 225, + 261 + ], + "score": 1.0, + "content": "when evaluating SqueezeNet.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "table", + "bbox": [ + 108, + 304, + 503, + 536 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 280, + 503, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "score": 1.0, + "content": "Table 1: SqueezeNet architectural dimensions. 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conv1111x111x967×7/2 (x96)1100%(7×7)6bit14,20814,208
maxpool155x55x963x3/2。
fire255x55x1282166464100%100%33%6bit11,9205,746
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Deep Compression achieves a", + "type": "text" + }, + { + "bbox": [ + 286, + 596, + 302, + 605 + ], + "score": 0.61, + "content": "3 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "reduction in model size while still maintaining the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "baseline accuracy level (Han et al., 2015a). 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In each of the CNN model compression papers", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "score": 1.0, + "content": "reviewed in Section 2.1, the goal was to compress an AlexNet (Krizhevsky et al., 2012) model", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "that was trained to classify images using the ImageNet (Deng et al., 2009) (ILSVRC 2012) dataset.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 237, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 252 + ], + "score": 1.0, + "content": "Therefore, we use AlexNet5 and the associated model compression results as a basis for comparison", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 249, + 225, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 225, + 261 + ], + "score": 1.0, + "content": "when evaluating SqueezeNet.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 204, + 505, + 261 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 304, + 503, + 536 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 280, + 503, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "score": 1.0, + "content": "Table 1: SqueezeNet architectural dimensions. 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layername/typeoutput sizefilter size/stride(if not a firelayer)depthS1x1(#1x1squeeze)e1x1(#1x1expand)e3x3(#3x3expand)S1x1sparsitye1x1sparsitye3x3sparsity# bits#parameterbefore pruning#parameterafter pruning
input image224x224x3--
conv1111x111x967×7/2 (x96)1100%(7×7)6bit14,20814,208
maxpool155x55x963x3/2。
fire255x55x1282166464100%100%33%6bit11,9205,746
fire355x55x1282166464100%100%33%6bit12,4326,258
fire455x55×256232128128100%100%33%6bit45,34420,646
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fire727×27x38424819219250%100%33%6bit111,02446,236
fire827×27×512264256256100%50%33%6bit188,99277,581
maxpool813x12x5123x3/20
fire913x13x51226425625650%100%30%6bit197,18477,581
conv1013x13x10001x1/1 (x1000)120%(3x3)6bit513,000103,400
avgpool101x1x100013x13/1。
1 Jactivations parameters compression info1,248,424(total)421,098(total)
", + "type": "table", + "image_path": "9b5479672bc826dd062e7b49fab2cd62ed9127784630f2286d4f07b2cf61c8aa.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 108, + 304, + 503, + 381.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 108, + 381.3333333333333, + 503, + 458.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 108, + 458.66666666666663, + 503, + 536.0 + ], + "spans": [], + "index": 16 + } + ] + } + ], + "index": 13.75 + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "In Table 2, we review SqueezeNet in the context of recent model compression results. The SVD-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 561, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 415, + 576 + ], + "score": 1.0, + "content": "based approach is able to compress a pretrained AlexNet model by a factor of", + "type": "text" + }, + { + "bbox": [ + 416, + 563, + 427, + 573 + ], + "score": 0.54, + "content": "{ 5 } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 561, + 506, + 576 + ], + "score": 1.0, + "content": ", while diminishing", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 178, + 585 + ], + "score": 1.0, + "content": "top-1 accuracy to", + "type": "text" + }, + { + "bbox": [ + 179, + 573, + 206, + 584 + ], + "score": 0.87, + "content": "5 6 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 573, + 413, + 585 + ], + "score": 1.0, + "content": "(Denton et al., 2014). Network Pruning achieves a", + "type": "text" + }, + { + "bbox": [ + 414, + 574, + 425, + 584 + ], + "score": 0.7, + "content": "9 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "reduction in model", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 264, + 597 + ], + "score": 1.0, + "content": "size while maintaining the baseline of", + "type": "text" + }, + { + "bbox": [ + 264, + 585, + 291, + 595 + ], + "score": 0.88, + "content": "5 7 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 585, + 335, + 597 + ], + "score": 1.0, + "content": "top-1 and", + "type": "text" + }, + { + "bbox": [ + 335, + 585, + 362, + 595 + ], + "score": 0.87, + "content": "8 0 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "top-5 accuracy on ImageNet (Han", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 286, + 608 + ], + "score": 1.0, + "content": "et al., 2015b). Deep Compression achieves a", + "type": "text" + }, + { + "bbox": [ + 286, + 596, + 302, + 605 + ], + "score": 0.61, + "content": "3 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "reduction in model size while still maintaining the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "baseline accuracy level (Han et al., 2015a). Now, with SqueezeNet, we achieve a 50X reduction in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 617, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 630 + ], + "score": 1.0, + "content": "model size compared to AlexNet, while meeting or exceeding the top-1 and top-5 accuracy of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 629, + 384, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 384, + 640 + ], + "score": 1.0, + "content": "AlexNet. We summarize all of the aforementioned results in Table 2.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 552, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 645, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "It appears that we have surpassed the state-of-the-art results from the model compression commu-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 656, + 504, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 482, + 668 + ], + "score": 1.0, + "content": "nity: even when using uncompressed 32-bit values to represent the model, SqueezeNet has a", + "type": "text" + }, + { + "bbox": [ + 482, + 656, + 504, + 667 + ], + "score": 0.85, + "content": "1 . 4 \\times", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "smaller model size than the best efforts from the model compression community while maintain-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "ing or exceeding the baseline accuracy. Until now, an open question has been: are small models", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "amenable to compression, or do small models “need” all of the representational power afforded", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "by dense floating-point values? To find out, we applied Deep Compression (Han et al., 2015a)", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 645, + 506, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 109, + 111, + 502, + 228 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 89, + 504, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 89, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 505, + 102 + ], + "score": 1.0, + "content": "Table 2: Comparing SqueezeNet to model compression approaches. By model size, we mean the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 100, + 411, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 411, + 111 + ], + "score": 1.0, + "content": "number of bytes required to store all of the parameters in the trained model.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 109, + 111, + 502, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 111, + 502, + 228 + ], + "spans": [ + { + "bbox": [ + 109, + 111, + 502, + 228 + ], + "score": 0.985, + "html": "
CNN architectureCompression ApproachDataTypeOriginal→Compressed ModelSizeReduction inModel Sizevs.AlexNetTop-1ImageNetAccuracyTop-5ImageNetAccuracy
AlexNetNone (baseline)32 bit240MB1x57.2%80.3%
AlexNetSVD (Denton et al.,2014)32 bit240MB→48MB5x56.0%79.4%
AlexNetNetwork Pruning (Hanet al.,2015b)32 bit240MB→27MB9x57.2%80.3%
AlexNetDeepCompression (Hanet al.,2015a)5-8bit240MB→6.9MB35x57.2%80.3%
SqueezeNet (ours)None32 bit4.8MB50x57.5%80.3%
SqueezeNet (ours)Deep Compression8bit4.8MB→0.66MB363x57.5%80.3%
SqueezeNet (ours)Deep Compression6bit4.8MB→0.47MB510x57.5%80.3%
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This yields a", + "type": "text" + }, + { + "bbox": [ + 408, + 255, + 446, + 266 + ], + "score": 0.41, + "content": "0 . 6 6 ~ \\mathrm { M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 254, + 478, + 268 + ], + "score": 1.0, + "content": "model", + "type": "text" + }, + { + "bbox": [ + 479, + 255, + 504, + 267 + ], + "score": 0.83, + "content": "( 3 6 3 \\times", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "score": 1.0, + "content": "smaller than 32-bit AlexNet) with equivalent accuracy to AlexNet. Further, applying Deep Compres-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 277, + 504, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 236, + 289 + ], + "score": 1.0, + "content": "sion with 6-bit quantization and", + "type": "text" + }, + { + "bbox": [ + 237, + 277, + 257, + 288 + ], + "score": 0.87, + "content": "33 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 277, + 478, + 289 + ], + "score": 1.0, + "content": "sparsity on SqueezeNet, we produce a 0.47MB model", + "type": "text" + }, + { + "bbox": [ + 479, + 277, + 504, + 288 + ], + "score": 0.84, + "content": "( 5 1 0 \\times", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "score": 1.0, + "content": "smaller than 32-bit AlexNet) with equivalent accuracy. Our small model is indeed amenable to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 299, + 164, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 164, + 312 + ], + "score": 1.0, + "content": "compression.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "score": 1.0, + "content": "In addition, these results demonstrate that Deep Compression (Han et al., 2015a) not only works", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "well on CNN architectures with many parameters (e.g. AlexNet and VGG), but it is also able to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "compress the already compact, fully convolutional SqueezeNet architecture. Deep Compression", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 219, + 362 + ], + "score": 1.0, + "content": "compressed SqueezeNet by", + "type": "text" + }, + { + "bbox": [ + 219, + 349, + 239, + 360 + ], + "score": 0.88, + "content": "1 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "while preserving the baseline accuracy. In summary: by combin-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 359, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 104, + 359, + 505, + 374 + ], + "score": 1.0, + "content": "ing CNN architectural innovation (SqueezeNet) with state-of-the-art compression techniques (Deep", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 225, + 384 + ], + "score": 1.0, + "content": "Compression), we achieved a", + "type": "text" + }, + { + "bbox": [ + 226, + 371, + 249, + 381 + ], + "score": 0.9, + "content": "5 1 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "reduction in model size with no decrease in accuracy compared", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 382, + 168, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 168, + 394 + ], + "score": 1.0, + "content": "to the baseline.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "Finally, note that Deep Compression (Han et al., 2015b) uses a codebook as part of its scheme for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "score": 1.0, + "content": "quantizing CNN parameters to 6- or 8-bits of precision. Therefore, on most commodity processors,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 102, + 417, + 507, + 438 + ], + "spans": [ + { + "bbox": [ + 102, + 417, + 268, + 438 + ], + "score": 1.0, + "content": "it is not trivial to achieve a speedup of", + "type": "text" + }, + { + "bbox": [ + 268, + 420, + 306, + 434 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 3 2 } { 8 } = 4 x } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 417, + 415, + 438 + ], + "score": 1.0, + "content": "with 8-bit quantization or", + "type": "text" + }, + { + "bbox": [ + 415, + 420, + 461, + 434 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 3 2 } { 6 } = 5 . \\dot { 3 } x } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 417, + 507, + 438 + ], + "score": 1.0, + "content": "with 6-bit", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "score": 1.0, + "content": "quantization using the scheme developed in Deep Compression. However, Han et al. developed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "custom hardware – Efficient Inference Engine (EIE) – that can compute codebook-quantized CNNs", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "more efficiently (Han et al., 2016a). In addition, in the months since we released SqueezeNet,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 463, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 104, + 463, + 505, + 478 + ], + "score": 1.0, + "content": "P. Gysel developed a strategy called Ristretto for linearly quantizing SqueezeNet to 8 bits (Gysel,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "2016). Specifically, Ristretto does computation in 8 bits, and it stores parameters and activations in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 485, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 104, + 485, + 506, + 499 + ], + "score": 1.0, + "content": "8-bit data types. Using the Ristretto strategy for 8-bit computation in SqueezeNet inference, Gysel", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "observed less than 1 percentage-point of drop in accuracy when using 8-bit instead of 32-bit data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 509, + 134, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 134, + 523 + ], + "score": 1.0, + "content": "types.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 108, + 543, + 428, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 429, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 429, + 558 + ], + "score": 1.0, + "content": "5 CNN MICROARCHITECTURE DESIGN SPACE EXPLORATION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 562, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "score": 1.0, + "content": "So far, we have proposed architectural design strategies for small models, followed these principles", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "to create SqueezeNet, and discovered that SqueezeNet is 50x smaller than AlexNet with equivalent", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 583, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 598 + ], + "score": 1.0, + "content": "accuracy. 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By model size, we mean the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 100, + 411, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 411, + 111 + ], + "score": 1.0, + "content": "number of bytes required to store all of the parameters in the trained model.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 109, + 111, + 502, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 111, + 502, + 228 + ], + "spans": [ + { + "bbox": [ + 109, + 111, + 502, + 228 + ], + "score": 0.985, + "html": "
CNN architectureCompression ApproachDataTypeOriginal→Compressed ModelSizeReduction inModel Sizevs.AlexNetTop-1ImageNetAccuracyTop-5ImageNetAccuracy
AlexNetNone (baseline)32 bit240MB1x57.2%80.3%
AlexNetSVD (Denton et al.,2014)32 bit240MB→48MB5x56.0%79.4%
AlexNetNetwork Pruning (Hanet al.,2015b)32 bit240MB→27MB9x57.2%80.3%
AlexNetDeepCompression (Hanet al.,2015a)5-8bit240MB→6.9MB35x57.2%80.3%
SqueezeNet (ours)None32 bit4.8MB50x57.5%80.3%
SqueezeNet (ours)Deep Compression8bit4.8MB→0.66MB363x57.5%80.3%
SqueezeNet (ours)Deep Compression6bit4.8MB→0.47MB510x57.5%80.3%
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Note that our goal here is not to maximize accuracy in every experiment, but rather", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 677, + 441, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 441, + 691 + ], + "score": 1.0, + "content": "to understand the impact of CNN architectural choices on model size and accuracy.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 645, + 506, + 691 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 86, + 494, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 86, + 494, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 86, + 494, + 248 + ], + "spans": [ + { + "bbox": [ + 112, + 86, + 494, + 248 + ], + "score": 0.963, + "type": "image", + "image_path": "680240d6400757ba8a40ea928e6c20c6e26ac1d8fd8de2eeb0707082c24d3aee.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 86, + 494, + 140.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 140.0, + 494, + 194.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 194.0, + 494, + 248.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 196, + 262, + 414, + 273 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 194, + 260, + 415, + 276 + ], + "spans": [ + { + "bbox": [ + 194, + 260, + 415, + 276 + ], + "score": 1.0, + "content": "Figure 3: Microarchitectural design space exploration.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "title", + "bbox": [ + 107, + 294, + 336, + 304 + ], + "lines": [ + { + "bbox": [ + 106, + 293, + 338, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 338, + 306 + ], + "score": 1.0, + "content": "5.1 CNN MICROARCHITECTURE METAPARAMETERS", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 303, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "In SqueezeNet, each Fire module has three dimensional hyperparameters that we defined in Sec-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 312, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 146, + 328 + ], + "score": 1.0, + "content": "tion 3.2:", + "type": "text" + }, + { + "bbox": [ + 147, + 315, + 165, + 325 + ], + "score": 0.82, + "content": "s _ { 1 x 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 312, + 171, + 328 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 171, + 316, + 190, + 325 + ], + "score": 0.81, + "content": "e _ { 1 x 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 312, + 213, + 328 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 214, + 316, + 232, + 325 + ], + "score": 0.88, + "content": "e _ { 3 x 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 312, + 506, + 328 + ], + "score": 1.0, + "content": ". SqueezeNet has 8 Fire modules with a total of 24 dimensional", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 325, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 337 + ], + "score": 1.0, + "content": "hyperparameters. To do broad sweeps of the design space of SqueezeNet-like architectures, we", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "define the following set of higher level metaparameters which control the dimensions of all Fire", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 232, + 360 + ], + "score": 1.0, + "content": "modules in a CNN. 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In other", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 370, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 203, + 387 + ], + "score": 1.0, + "content": "words, for Fire module", + "type": "text" + }, + { + "bbox": [ + 204, + 373, + 208, + 383 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 370, + 341, + 387 + ], + "score": 1.0, + "content": ", the number of expand filters is", + "type": "text" + }, + { + "bbox": [ + 342, + 370, + 472, + 388 + ], + "score": 0.91, + "content": "\\begin{array} { r } { e _ { i } = b a s e _ { e } + ( i n c r _ { e } * \\left\\lfloor \\frac { i } { f r e q } \\right\\rfloor ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 371, + 505, + 388 + ], + "score": 1.0, + "content": ". In the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 384, + 504, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 364, + 402 + ], + "score": 1.0, + "content": "expand layer of a Fire module, some filters are 1x1 and some are", + "type": "text" + }, + { + "bbox": [ + 365, + 387, + 380, + 397 + ], + "score": 0.35, + "content": "3 { \\bf x } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 384, + 425, + 402 + ], + "score": 1.0, + "content": "; we define", + "type": "text" + }, + { + "bbox": [ + 425, + 389, + 504, + 399 + ], + "score": 0.89, + "content": "e _ { i } = e _ { i , 1 x 1 } + e _ { i , 3 x 3 }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 126, + 410 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 398, + 154, + 409 + ], + "score": 0.9, + "content": "p c t _ { 3 x 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "(in the range [0, 1], shared over all Fire modules) as the percentage of expand filters that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 407, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 121, + 423 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 122, + 409, + 138, + 419 + ], + "score": 0.36, + "content": "3 { \\bf x } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 407, + 209, + 423 + ], + "score": 1.0, + "content": ". 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Finally, we define", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "the number of filters in the squeeze layer of a Fire module using a metaparameter called the squeeze", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 228, + 444 + ], + "score": 1.0, + "content": "ratio (SR) (again, in the range", + "type": "text" + }, + { + "bbox": [ + 229, + 430, + 249, + 442 + ], + "score": 0.44, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 429, + 369, + 444 + ], + "score": 1.0, + "content": ", shared by all Fire modules):", + "type": "text" + }, + { + "bbox": [ + 369, + 431, + 438, + 443 + ], + "score": 0.92, + "content": "s _ { i , 1 x 1 } = S R * e _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 429, + 505, + 444 + ], + "score": 1.0, + "content": "(or equivalently", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 241, + 454 + ], + "score": 0.91, + "content": "s _ { i , 1 x 1 } = S R * ( e _ { i , 1 x 1 } + e _ { i , 3 x \\bar { 3 } } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 441, + 505, + 455 + ], + "score": 1.0, + "content": "). 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We now", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 529, + 484, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 484, + 545 + ], + "score": 1.0, + "content": "design an experiment to investigate the effect of the squeeze ratio on model size and accuracy.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 548, + 505, + 647 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "In these experiments, we use SqueezeNet (Figure 2) as a starting point. 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Accuracy plateaus at", + "type": "text" + }, + { + "bbox": [ + 259, + 625, + 286, + 635 + ], + "score": 0.87, + "content": "8 6 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 624, + 309, + 637 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 309, + 625, + 346, + 636 + ], + "score": 0.81, + "content": "\\mathrm { S R } { = } 0 . 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 624, + 472, + 637 + ], + "score": 1.0, + "content": "(a 19MB model), and setting", + "type": "text" + }, + { + "bbox": [ + 472, + 625, + 504, + 636 + ], + "score": 0.82, + "content": "\\mathrm { S R } { = } 1 . 0", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 634, + 336, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 336, + 649 + ], + "score": 1.0, + "content": "further increases model size without improving accuracy.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 660, + 287, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 289, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 289, + 673 + ], + "score": 1.0, + "content": "5.3 TRADING OFF 1X1 AND 3X3 FILTERS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 109, + 671, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 669, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 683 + ], + "score": 1.0, + "content": "In Section 3.1, we proposed decreasing the number of parameters in a CNN by replacing some", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 123, + 692 + ], + "score": 0.32, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "filters with 1x1 filters. An open question is, how important is spatial resolution in CNNs? The", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 700, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 698, + 386, + 713 + ], + "spans": [ + { + "bbox": [ + 119, + 698, + 386, + 713 + ], + "score": 1.0, + "content": "7Note that, for a given model, all Fire layers share the same squeeze ratio.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "8Note that we named it SqueezeNet because it has a low squeeze ratio (SR). 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In other", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 370, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 203, + 387 + ], + "score": 1.0, + "content": "words, for Fire module", + "type": "text" + }, + { + "bbox": [ + 204, + 373, + 208, + 383 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 370, + 341, + 387 + ], + "score": 1.0, + "content": ", the number of expand filters is", + "type": "text" + }, + { + "bbox": [ + 342, + 370, + 472, + 388 + ], + "score": 0.91, + "content": "\\begin{array} { r } { e _ { i } = b a s e _ { e } + ( i n c r _ { e } * \\left\\lfloor \\frac { i } { f r e q } \\right\\rfloor ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 371, + 505, + 388 + ], + "score": 1.0, + "content": ". In the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 384, + 504, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 364, + 402 + ], + "score": 1.0, + "content": "expand layer of a Fire module, some filters are 1x1 and some are", + "type": "text" + }, + { + "bbox": [ + 365, + 387, + 380, + 397 + ], + "score": 0.35, + "content": "3 { \\bf x } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 384, + 425, + 402 + ], + "score": 1.0, + "content": "; we define", + "type": "text" + }, + { + "bbox": [ + 425, + 389, + 504, + 399 + ], + "score": 0.89, + "content": "e _ { i } = e _ { i , 1 x 1 } + e _ { i , 3 x 3 }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 126, + 410 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 398, + 154, + 409 + ], + "score": 0.9, + "content": "p c t _ { 3 x 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "(in the range [0, 1], shared over all Fire modules) as the percentage of expand filters that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 407, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 121, + 423 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 122, + 409, + 138, + 419 + ], + "score": 0.36, + "content": "3 { \\bf x } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 407, + 209, + 423 + ], + "score": 1.0, + "content": ". 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We defined the squeeze ratio", + "type": "text" + }, + { + "bbox": [ + 436, + 510, + 455, + 520 + ], + "score": 0.45, + "content": "( S R )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "as the ratio", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "between the number of filters in squeeze layers and the number of filters in expand layers. We now", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 529, + 484, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 484, + 545 + ], + "score": 1.0, + "content": "design an experiment to investigate the effect of the squeeze ratio on model size and accuracy.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 498, + 505, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 548, + 505, + 647 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "In these experiments, we use SqueezeNet (Figure 2) as a starting point. As in SqueezeNet, these", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 304, + 572 + ], + "score": 1.0, + "content": "experiments use the following metaparameters:", + "type": "text" + }, + { + "bbox": [ + 304, + 559, + 360, + 570 + ], + "score": 0.8, + "content": "b a s e _ { e } = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 558, + 366, + 572 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 366, + 559, + 421, + 570 + ], + "score": 0.76, + "content": "i n c r _ { e } = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 558, + 426, + 572 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 426, + 559, + 483, + 570 + ], + "score": 0.79, + "content": "p c t _ { 3 x 3 } = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 558, + 506, + 572 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 107, + 570, + 148, + 581 + ], + "score": 0.89, + "content": "f r e q = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 570, + 454, + 581 + ], + "score": 1.0, + "content": ". We train multiple models, where each model has a different squeeze ratio", + "type": "text" + }, + { + "bbox": [ + 454, + 569, + 478, + 581 + ], + "score": 0.73, + "content": "( \\mathrm { S R } ) ^ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 570, + 505, + 581 + ], + "score": 1.0, + "content": "in the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 581, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 593 + ], + "score": 1.0, + "content": "range [0.125, 1.0]. In Figure 3(a), we show the results of this experiment, where each point on the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 438, + 604 + ], + "score": 1.0, + "content": "graph is an independent model that was trained from scratch. SqueezeNet is the", + "type": "text" + }, + { + "bbox": [ + 439, + 592, + 480, + 602 + ], + "score": 0.82, + "content": "\\mathrm { S R } { = } 0 . 1 2 5", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "point", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 600, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 104, + 600, + 506, + 616 + ], + "score": 1.0, + "content": "in this figure.8 From this figure, we learn that increasing SR beyond 0.125 can further increase", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 235, + 626 + ], + "score": 1.0, + "content": "ImageNet top-5 accuracy from", + "type": "text" + }, + { + "bbox": [ + 235, + 614, + 262, + 624 + ], + "score": 0.87, + "content": "8 0 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 613, + 447, + 626 + ], + "score": 1.0, + "content": "(i.e. AlexNet-level) with a 4.8MB model to", + "type": "text" + }, + { + "bbox": [ + 448, + 614, + 475, + 624 + ], + "score": 0.87, + "content": "8 6 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "with a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 624, + 504, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 258, + 637 + ], + "score": 1.0, + "content": "19MB model. Accuracy plateaus at", + "type": "text" + }, + { + "bbox": [ + 259, + 625, + 286, + 635 + ], + "score": 0.87, + "content": "8 6 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 624, + 309, + 637 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 309, + 625, + 346, + 636 + ], + "score": 0.81, + "content": "\\mathrm { S R } { = } 0 . 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 624, + 472, + 637 + ], + "score": 1.0, + "content": "(a 19MB model), and setting", + "type": "text" + }, + { + "bbox": [ + 472, + 625, + 504, + 636 + ], + "score": 0.82, + "content": "\\mathrm { S R } { = } 1 . 0", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 634, + 336, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 336, + 649 + ], + "score": 1.0, + "content": "further increases model size without improving accuracy.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 547, + 506, + 649 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 660, + 287, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 289, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 289, + 673 + ], + "score": 1.0, + "content": "5.3 TRADING OFF 1X1 AND 3X3 FILTERS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 109, + 671, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 669, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 683 + ], + "score": 1.0, + "content": "In Section 3.1, we proposed decreasing the number of parameters in a CNN by replacing some", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 123, + 692 + ], + "score": 0.32, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "filters with 1x1 filters. An open question is, how important is spatial resolution in CNNs? The", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 106, + 669, + 505, + 694 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "VGG (Simonyan & Zisserman, 2014) architectures have 3x3 spatial resolution in most layers’ filters;", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "GoogLeNet (Szegedy et al., 2014) and Network-in-Network (NiN) (Lin et al., 2013) have 1x1 filters", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "in some layers. In GoogLeNet and NiN, the authors simply propose a specific quantity of 1x1 and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 123, + 126 + ], + "score": 0.53, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "filters without further analysis.9 Here, we attempt to shed light on how the proportion of 1x1", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 297, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 123, + 140 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 127, + 140, + 137 + ], + "score": 0.34, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 126, + 297, + 140 + ], + "score": 1.0, + "content": "filters affects model size and accuracy.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 339, + 156 + ], + "score": 1.0, + "content": "We use the following metaparameters in this experiment:", + "type": "text" + }, + { + "bbox": [ + 339, + 144, + 429, + 154 + ], + "score": 0.89, + "content": "b a s e _ { e } = i n c r _ { e } = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 143, + 433, + 156 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 433, + 144, + 473, + 155 + ], + "score": 0.79, + "content": "f r e q = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 143, + 477, + 156 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 477, + 143, + 505, + 154 + ], + "score": 0.82, + "content": "S R =", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 185, + 167 + ], + "score": 1.0, + "content": "0.500, and we vary", + "type": "text" + }, + { + "bbox": [ + 186, + 155, + 213, + 165 + ], + "score": 0.91, + "content": "p c t _ { 3 x 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 154, + 236, + 167 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 236, + 154, + 250, + 165 + ], + "score": 0.86, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 154, + 262, + 167 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 262, + 154, + 282, + 165 + ], + "score": 0.88, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 154, + 506, + 167 + ], + "score": 1.0, + "content": ". In other words, each Fire module’s expand layer has a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 339, + 178 + ], + "score": 1.0, + "content": "predefined number of filters partitioned between 1x1 and", + "type": "text" + }, + { + "bbox": [ + 339, + 165, + 356, + 176 + ], + "score": 0.45, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 164, + 506, + 178 + ], + "score": 1.0, + "content": ", and here we turn the knob on these", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 260, + 188 + ], + "score": 1.0, + "content": "filters from “mostly 1x1” to “mostly", + "type": "text" + }, + { + "bbox": [ + 260, + 176, + 281, + 187 + ], + "score": 0.45, + "content": "3 \\mathrm { x } 3 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 176, + 505, + 188 + ], + "score": 1.0, + "content": ". As in the previous experiment, these models have 8", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "Fire modules, following the same organization of layers as in Figure 2. We show the results of this", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "score": 1.0, + "content": "experiment in Figure 3(b). Note that the 13MB models in Figure 3(a) and Figure 3(b) are the same", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 504, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 159, + 222 + ], + "score": 1.0, + "content": "architecture:", + "type": "text" + }, + { + "bbox": [ + 159, + 209, + 211, + 219 + ], + "score": 0.89, + "content": "S R = 0 . 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 209, + 228, + 222 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 228, + 209, + 287, + 220 + ], + "score": 0.92, + "content": "p c t _ { 3 x 3 } = 5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 209, + 504, + 222 + ], + "score": 1.0, + "content": ". We see in Figure 3(b) that the top-5 accuracy plateaus", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 218, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 116, + 234 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 117, + 220, + 144, + 231 + ], + "score": 0.87, + "content": "8 5 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 218, + 168, + 234 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 169, + 220, + 188, + 231 + ], + "score": 0.75, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 218, + 505, + 234 + ], + "score": 1.0, + "content": "3x3 filters, and further increasing the percentage of 3x3 filters leads to a larger", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 376, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 376, + 244 + ], + "score": 1.0, + "content": "model size but provides no improvement in accuracy on ImageNet.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 107, + 262, + 431, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 262, + 433, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 433, + 277 + ], + "score": 1.0, + "content": "6 CNN MACROARCHITECTURE DESIGN SPACE EXPLORATION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 108, + 279, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "So far we have explored the design space at the microarchitecture level, i.e. the contents of individual", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "modules of the CNN. Now, we explore design decisions at the macroarchitecture level concerning", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "the high-level connections among Fire modules. Inspired by ResNet (He et al., 2015b), we explored", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 312, + 221, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 221, + 324 + ], + "score": 1.0, + "content": "three different architectures:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 132, + 331, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 132, + 331, + 329, + 343 + ], + "spans": [ + { + "bbox": [ + 132, + 331, + 329, + 343 + ], + "score": 1.0, + "content": "• Vanilla SqueezeNet (as per the prior sections).", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 131, + 345, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 131, + 345, + 505, + 359 + ], + "score": 1.0, + "content": "• SqueezeNet with simple bypass connections between some Fire modules. (Inspired by (Sri-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 357, + 299, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 299, + 370 + ], + "score": 1.0, + "content": "vastava et al., 2015; He et al., 2015b).)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 132, + 371, + 480, + 385 + ], + "spans": [ + { + "bbox": [ + 132, + 371, + 480, + 385 + ], + "score": 1.0, + "content": "• SqueezeNet with complex bypass connections between the remaining Fire modules.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 347, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 348, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 348, + 409 + ], + "score": 1.0, + "content": "We illustrate these three variants of SqueezeNet in Figure 2.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 427 + ], + "score": 1.0, + "content": "Our simple bypass architecture adds bypass connections around Fire modules 3, 5, 7, and 9, requiring", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "these modules to learn a residual function between input and output. As in ResNet, to implement", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 432, + 448 + ], + "score": 1.0, + "content": "a bypass connection around Fire3, we set the input to Fire4 equal to (output of", + "type": "text" + }, + { + "bbox": [ + 432, + 435, + 465, + 446 + ], + "score": 0.26, + "content": "{ \\mathrm { F i r e } } 2 +", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "output of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 176, + 459 + ], + "score": 1.0, + "content": "Fire3), where the", + "type": "text" + }, + { + "bbox": [ + 177, + 447, + 185, + 456 + ], + "score": 0.77, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "operator is elementwise addition. This changes the regularization applied to the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "parameters of these Fire modules, and, as per ResNet, can improve the final accuracy and/or ability", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 468, + 199, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 199, + 480 + ], + "score": 1.0, + "content": "to train the full model.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 485, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 497 + ], + "score": 1.0, + "content": "One limitation is that, in the straightforward case, the number of input channels and number of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "output channels has to be the same; as a result, only half of the Fire modules can have simple", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "bypass connections, as shown in the middle diagram of Fig 2. When the “same number of channels”", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 518, + 504, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 504, + 530 + ], + "score": 1.0, + "content": "requirement can’t be met, we use a complex bypass connection, as illustrated on the right of Figure 2.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "While a simple bypass is “just a wire,” we define a complex bypass as a bypass that includes a 1x1", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "convolution layer with the number of filters set equal to the number of output channels that are", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "needed. Note that complex bypass connections add extra parameters to the model, while simple", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 562, + 216, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 216, + 574 + ], + "score": 1.0, + "content": "bypass connections do not.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "In addition to changing the regularization, it is intuitive to us that adding bypass connections would", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "help to alleviate the representational bottleneck introduced by squeeze layers. In SqueezeNet, the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "squeeze ratio (SR) is 0.125, meaning that every squeeze layer has 8x fewer output channels than the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "accompanying expand layer. Due to this severe dimensionality reduction, a limited amount of in-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "formation can pass through squeeze layers. However, by adding bypass connections to SqueezeNet,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 633, + 392, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 392, + 647 + ], + "score": 1.0, + "content": "we open up avenues for information to flow around the squeeze layers.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 504, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 664 + ], + "score": 1.0, + "content": "We trained SqueezeNet with the three macroarchitectures in Figure 2 and compared the accuracy", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 660, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 674 + ], + "score": 1.0, + "content": "and model size in Table 3. We fixed the microarchitecture to match SqueezeNet as described in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 672, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 506, + 685 + ], + "score": 1.0, + "content": "Table 1 throughout the macroarchitecture exploration. Complex and simple bypass connections", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 682, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 696 + ], + "score": 1.0, + "content": "both yielded an accuracy improvement over the vanilla SqueezeNet architecture. Interestingly, the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 694, + 506, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 506, + 708 + ], + "score": 1.0, + "content": "simple bypass enabled a higher accuracy accuracy improvement than complex bypass. Adding the", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 116, + 721, + 464, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 466, + 732 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 466, + 732 + ], + "score": 1.0, + "content": "9To be clear, each filter is 1x1xChannels or 3x3xChannels, which we abbreviate to 1x1 and 3x3.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "VGG (Simonyan & Zisserman, 2014) architectures have 3x3 spatial resolution in most layers’ filters;", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "GoogLeNet (Szegedy et al., 2014) and Network-in-Network (NiN) (Lin et al., 2013) have 1x1 filters", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "in some layers. In GoogLeNet and NiN, the authors simply propose a specific quantity of 1x1 and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 123, + 126 + ], + "score": 0.53, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "filters without further analysis.9 Here, we attempt to shed light on how the proportion of 1x1", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 297, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 123, + 140 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 127, + 140, + 137 + ], + "score": 0.34, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 126, + 297, + 140 + ], + "score": 1.0, + "content": "filters affects model size and accuracy.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 82, + 506, + 140 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 339, + 156 + ], + "score": 1.0, + "content": "We use the following metaparameters in this experiment:", + "type": "text" + }, + { + "bbox": [ + 339, + 144, + 429, + 154 + ], + "score": 0.89, + "content": "b a s e _ { e } = i n c r _ { e } = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 143, + 433, + 156 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 433, + 144, + 473, + 155 + ], + "score": 0.79, + "content": "f r e q = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 143, + 477, + 156 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 477, + 143, + 505, + 154 + ], + "score": 0.82, + "content": "S R =", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 185, + 167 + ], + "score": 1.0, + "content": "0.500, and we vary", + "type": "text" + }, + { + "bbox": [ + 186, + 155, + 213, + 165 + ], + "score": 0.91, + "content": "p c t _ { 3 x 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 154, + 236, + 167 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 236, + 154, + 250, + 165 + ], + "score": 0.86, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 154, + 262, + 167 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 262, + 154, + 282, + 165 + ], + "score": 0.88, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 154, + 506, + 167 + ], + "score": 1.0, + "content": ". In other words, each Fire module’s expand layer has a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 339, + 178 + ], + "score": 1.0, + "content": "predefined number of filters partitioned between 1x1 and", + "type": "text" + }, + { + "bbox": [ + 339, + 165, + 356, + 176 + ], + "score": 0.45, + "content": "3 { \\tt X } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 164, + 506, + 178 + ], + "score": 1.0, + "content": ", and here we turn the knob on these", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 260, + 188 + ], + "score": 1.0, + "content": "filters from “mostly 1x1” to “mostly", + "type": "text" + }, + { + "bbox": [ + 260, + 176, + 281, + 187 + ], + "score": 0.45, + "content": "3 \\mathrm { x } 3 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 176, + 505, + 188 + ], + "score": 1.0, + "content": ". As in the previous experiment, these models have 8", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "Fire modules, following the same organization of layers as in Figure 2. We show the results of this", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "score": 1.0, + "content": "experiment in Figure 3(b). Note that the 13MB models in Figure 3(a) and Figure 3(b) are the same", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 504, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 159, + 222 + ], + "score": 1.0, + "content": "architecture:", + "type": "text" + }, + { + "bbox": [ + 159, + 209, + 211, + 219 + ], + "score": 0.89, + "content": "S R = 0 . 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 209, + 228, + 222 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 228, + 209, + 287, + 220 + ], + "score": 0.92, + "content": "p c t _ { 3 x 3 } = 5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 209, + 504, + 222 + ], + "score": 1.0, + "content": ". We see in Figure 3(b) that the top-5 accuracy plateaus", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 218, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 116, + 234 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 117, + 220, + 144, + 231 + ], + "score": 0.87, + "content": "8 5 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 218, + 168, + 234 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 169, + 220, + 188, + 231 + ], + "score": 0.75, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 218, + 505, + 234 + ], + "score": 1.0, + "content": "3x3 filters, and further increasing the percentage of 3x3 filters leads to a larger", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 376, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 376, + 244 + ], + "score": 1.0, + "content": "model size but provides no improvement in accuracy on ImageNet.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 143, + 506, + 244 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 262, + 431, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 262, + 433, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 433, + 277 + ], + "score": 1.0, + "content": "6 CNN MACROARCHITECTURE DESIGN SPACE EXPLORATION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 108, + 279, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "So far we have explored the design space at the microarchitecture level, i.e. the contents of individual", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "modules of the CNN. Now, we explore design decisions at the macroarchitecture level concerning", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "the high-level connections among Fire modules. Inspired by ResNet (He et al., 2015b), we explored", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 312, + 221, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 221, + 324 + ], + "score": 1.0, + "content": "three different architectures:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 279, + 505, + 324 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 331, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 132, + 331, + 329, + 343 + ], + "spans": [ + { + "bbox": [ + 132, + 331, + 329, + 343 + ], + "score": 1.0, + "content": "• Vanilla SqueezeNet (as per the prior sections).", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 131, + 345, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 131, + 345, + 505, + 359 + ], + "score": 1.0, + "content": "• SqueezeNet with simple bypass connections between some Fire modules. (Inspired by (Sri-", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 357, + 299, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 299, + 370 + ], + "score": 1.0, + "content": "vastava et al., 2015; He et al., 2015b).)", + "type": "text" + } + ], + "index": 21, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 371, + 480, + 385 + ], + "spans": [ + { + "bbox": [ + 132, + 371, + 480, + 385 + ], + "score": 1.0, + "content": "• SqueezeNet with complex bypass connections between the remaining Fire modules.", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 20.5, + "bbox_fs": [ + 131, + 331, + 505, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 347, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 348, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 348, + 409 + ], + "score": 1.0, + "content": "We illustrate these three variants of SqueezeNet in Figure 2.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 394, + 348, + 409 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 427 + ], + "score": 1.0, + "content": "Our simple bypass architecture adds bypass connections around Fire modules 3, 5, 7, and 9, requiring", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "these modules to learn a residual function between input and output. As in ResNet, to implement", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 432, + 448 + ], + "score": 1.0, + "content": "a bypass connection around Fire3, we set the input to Fire4 equal to (output of", + "type": "text" + }, + { + "bbox": [ + 432, + 435, + 465, + 446 + ], + "score": 0.26, + "content": "{ \\mathrm { F i r e } } 2 +", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "output of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 176, + 459 + ], + "score": 1.0, + "content": "Fire3), where the", + "type": "text" + }, + { + "bbox": [ + 177, + 447, + 185, + 456 + ], + "score": 0.77, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "operator is elementwise addition. This changes the regularization applied to the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "parameters of these Fire modules, and, as per ResNet, can improve the final accuracy and/or ability", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 468, + 199, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 199, + 480 + ], + "score": 1.0, + "content": "to train the full model.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 411, + 506, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 485, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 497 + ], + "score": 1.0, + "content": "One limitation is that, in the straightforward case, the number of input channels and number of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "output channels has to be the same; as a result, only half of the Fire modules can have simple", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "bypass connections, as shown in the middle diagram of Fig 2. When the “same number of channels”", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 518, + 504, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 504, + 530 + ], + "score": 1.0, + "content": "requirement can’t be met, we use a complex bypass connection, as illustrated on the right of Figure 2.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "While a simple bypass is “just a wire,” we define a complex bypass as a bypass that includes a 1x1", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "convolution layer with the number of filters set equal to the number of output channels that are", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "needed. Note that complex bypass connections add extra parameters to the model, while simple", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 562, + 216, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 216, + 574 + ], + "score": 1.0, + "content": "bypass connections do not.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 485, + 506, + 574 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "In addition to changing the regularization, it is intuitive to us that adding bypass connections would", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "help to alleviate the representational bottleneck introduced by squeeze layers. 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However, it has become", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "common practice to apply ImageNet-trained CNN representations to a variety of applications such", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "as fine-grained object recognition (Zhang et al., 2013; Donahue et al., 2013), logo identification in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 432, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 447 + ], + "score": 1.0, + "content": "images (Iandola et al., 2015), and generating sentences about images (Fang et al., 2015). ImageNet-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 444, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 458 + ], + "score": 1.0, + "content": "trained CNNs have also been applied to a number of applications pertaining to autonomous driv-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "ing, including pedestrian and vehicle detection in images (Iandola et al., 2014; Girshick et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 465, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 480 + ], + "score": 1.0, + "content": "2015; Ashraf et al., 2016) and videos (Chen et al., 2015b), as well as segmenting the shape of the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 476, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 104, + 476, + 506, + 491 + ], + "score": 1.0, + "content": "road (Badrinarayanan et al., 2015). 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layername/typeoutput sizefilter size/stride(if not a firelayer)depthS1x1(#1x1squeeze)e1x1(#1x1expand)e3x3(#3x3expand)S1x1sparsitye1x1sparsitye3x3sparsity# bits#parameterbefore pruning#parameterafter pruning
input image224x224x3--
conv1111x111x967×7/2 (x96)1100%(7×7)6bit14,20814,208
maxpool155x55x963x3/2。
fire255x55x1282166464100%100%33%6bit11,9205,746
fire355x55x1282166464100%100%33%6bit12,4326,258
fire455x55×256232128128100%100%33%6bit45,34420,646
maxpool427x27x2563x3/20
fire527×27×256232128128100%100%33%6bit49,44024,742
fire627x27x384248192192100%50%33%6bit104,88044,700
fire727×27x38424819219250%100%33%6bit111,02446,236
fire827×27×512264256256100%50%33%6bit188,99277,581
maxpool813x12x5123x3/20
fire913x13x51226425625650%100%30%6bit197,18477,581
conv1013x13x10001x1/1 (x1000)120%(3x3)6bit513,000103,400
avgpool101x1x100013x13/1。
1 Jactivations parameters compression info1,248,424(total)421,098(total)
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CNN architectureCompression ApproachDataTypeOriginal→Compressed ModelSizeReduction inModel Sizevs.AlexNetTop-1ImageNetAccuracyTop-5ImageNetAccuracy
AlexNetNone (baseline)32 bit240MB1x57.2%80.3%
AlexNetSVD (Denton et al.,2014)32 bit240MB→48MB5x56.0%79.4%
AlexNetNetwork Pruning (Hanet al.,2015b)32 bit240MB→27MB9x57.2%80.3%
AlexNetDeepCompression (Hanet al.,2015a)5-8bit240MB→6.9MB35x57.2%80.3%
SqueezeNet (ours)None32 bit4.8MB50x57.5%80.3%
SqueezeNet (ours)Deep Compression8bit4.8MB→0.66MB363x57.5%80.3%
SqueezeNet (ours)Deep Compression6bit4.8MB→0.47MB510x57.5%80.3%
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--- /dev/null +++ b/parse/train/r1efr3C9Ym/images/89f8aa1252eaf18bd0408dbfb937d5324ab23247418c862e99a4d92c1223d6e2.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cf1f58c8372d16c2a6e9aa61f13f05fcc594b07b1558dc47f040d6a725eed193 +size 39514 diff --git a/parse/train/r1xwKoR9Y7/r1xwKoR9Y7.md b/parse/train/r1xwKoR9Y7/r1xwKoR9Y7.md new file mode 100644 index 0000000000000000000000000000000000000000..f8b2d26303812f0cd3b2c8cfb1a0cfa003a8452c --- /dev/null +++ b/parse/train/r1xwKoR9Y7/r1xwKoR9Y7.md @@ -0,0 +1,242 @@ +# GAMEPAD: A LEARNING ENVIRONMENT FOR THEOREM PROVING + +Daniel Huang∗† dehuang@berkeley.edu + +Prafulla Dhariwal∗‡ prafulla@openai.com + +Dawn Song† + +dawnsong@cs.berkeley.edu + +Ilya Sutskever‡ ilyasu@openai.com + +# ABSTRACT + +In this paper, we introduce a system called GamePad that can be used to explore the application of machine learning methods to theorem proving in the Coq proof assistant. Interactive theorem provers such as Coq enable users to construct machine-checkable proofs in a step-by-step manner. Hence, they provide an opportunity to explore theorem proving with human supervision. We use GamePad to synthesize proofs for a simple algebraic rewrite problem and train baseline models for a formalization of the Feit-Thompson theorem. We address position evaluation (i.e., predict the number of proof steps left) and tactic prediction (i.e., predict the next proof step) tasks, which arise naturally in tactic-based theorem proving. + +# 1 INTRODUCTION + +Theorem proving is a challenging AI task that involves symbolic reasoning (e.g., SMT solvers (De Moura & Bjørner, 2008)) and intuition guided search. Recent work (Irving et al., 2016; Loos et al., 2017; Kaliszyk et al., 2017) has shown the promise of applying deep learning techniques in this domain, primarily on tasks useful for automated theorem provers (e.g., premise selection) which operate with little to no human supervision. In this work, we aim to move closer to learning on proofs constructed with human supervision. + +We look at theorem proving in the realm of formal proofs. A formal proof is systematically derived in a formal system, which makes it possible to algorithmically (i.e., with a computer) check these proofs for correctness. Thus, formal proofs provide perfect learning signal—theorem statements and proofs are unambiguous. Human mathematicians usually do not write proofs in this style, and instead, construct and communicate proofs in natural language. Although the form and level of detail involved in each kind of proof differ, the logical content is similar in both contexts. + +Our work focuses on interactive theorem provers (ITPs), which are software tools that enable human users to construct formal proofs. ITPs have at least two features that make them compelling environments for exploring the application of learning techniques to theorem proving. First and foremost, ITPs provide full-fledged programmable environments. Consequently, any machine learning infrastructure built for an ITP can be reused across any problem domain crafted to study an aspect of learning and theorem proving. Second, the proofs are constructed by humans, and thus, have the constraint that they must be relatively human-understandable. Hence, ITPs provide access to large amounts of supervised data (i.e., expert-constructed proofs of theorems that are mathematically interesting). For example, ITPs have been used to build and check the proofs of large mathematical theorems such as the Feit-Thompson theorem (Gonthier et al., 2013) and provide provable guarantees on complex pieces of software such as the CompCert C compiler (Leroy et al., 2012). + +We introduce a system called GamePad1 that exposes parts of the Coq ITP to enable machine learning tasks and explore a few use cases. We focus on the Coq proof assistant for two reasons. First, + +![](images/1fec650b161e90747296d39eababa690f2aa0ff7a0fc43bc4747eed66f79eeba.jpg) +Figure 1: A proof script in Coq (left) and the resulting proof states, proof steps, and the complete proof tree (right). A proof state consists of a context (pink rectangles) and a goal (white rectangles). The initial proof state has as its goal the statement we are trying to prove and an empty context. The arrows indicate what tactic the prover used. The final states of the proof are indicated by the red circles and can be transitioned to only when the goal in the previous state is trivially true. + +Coq is a mature system with an active developer community that has been used to formalize nontrivial theorems, including Feit-Thompson and CompCert. Second, Coq supports the extraction of verified software. Consequently, one can prove that a program is correct and then run the verified program. The ease of extraction makes Coq a popular choice for program verification. + +Our contributions are the following. First, we introduce GamePad, which provides a structured Python representation of Coq proofs (Section 3), including all the proof states encountered in a proof, the steps taken, and expression abstract syntax trees (ASTs). The tool also enables lightweight interaction with Coq so that it can be used to dynamically build proofs (e.g., used as an environment for reinforcement learning). Tasks that can leverage this structured representation (Section 4) include position evaluation (i.e., predict the number of proof steps left) and tactic prediction (i.e., predict the next proof step to take). We also discuss how we can use the structured representation to embed proof states into $\bar { \mathbb { R } } ^ { D }$ (Section 5). Second, we demonstrate the synthesis of Coq proof scripts that makes use of a tactic prediction model for a hand-crafted algebraic rewriting problem (Section 6). Third and finally, we apply baseline models for position evaluation and tactic prediction to the FeitThompson formalization using data extracted by GamePad (Section 7). + +The code for GamePad, as well as the associated data sets, models and results, are open source on GitHub at https://github.com/ml4tp/gamepad. + +# 2 BACKGROUND + +To provide context for the rest of this paper, we begin by illustrating the formal proof process and how it can be modeled as a game. We then walk through a simple example of a proof in Coq and end by summarizing the constructs that we will encounter in the rest of the paper. + +Theorem proving process When humans write a pencil-paper proof, we typically maintain a mental and/or written scratchpad that keeps track of (1) what we need to show and (2) the facts we currently have at our disposal. As the proof progresses, the items in this scratchpad changes. For instance, we might transform what we need to show (e.g., it suffices to show another statement) and/or discover new facts (e.g., we derive an additional fact as a consequence of known facts). When we move from the pencil-paper setting to an ITP such as Coq, we will still have such a scratchpad, but use the ITP’s term language to express mathematical statements and the rules of the ITP’s formal system to construct the proof instead. + +Formal theorem proving as a game A game serves as useful mental model of the proving process. The state of the game is a proof state, which looks like the scratchpad and is comprised of (1) a goal (expressed in the term language) stating what we need to prove and (2) a context containing the assumptions (also expressed in the term language). The starting state has as its goal the statement of a theorem and an empty context, while a final state has as its goal a statement that is trivially true (i.e., definitionally equal) given the context. The aim of a prover is to transform the starting state into a collection of final states using only logically valid transitions, which can affect both the context and the goal. It is possible to obtain multiple final states because some transitions may split a goal into multiple subgoals. For a given (sub)goal, a successful proof corresponds to a proof tree, where the root is the start state and all the leaves are final states. + +An example Coq proof Consider showing that adding 0 to any natural number $n$ is $n$ itself. A paper-pencil proof might proceed by induction on $n$ , where we check that the result holds on the base case (i.e., $n = 0$ ) and the inductive case (i.e., $n = n + 1 )$ . We can carry out a similar process in Coq, using a sequence of commands called tactics in Coq’s tactic language Ltac that indicate what proofs steps to take. For instance, we can use the code induction n; simpl. to start a proof by induction (on n) in Coq. The expressions induction and simpl are tactics (of arity 1 and 0 respectively), the semicolon ; sequences two tactics, and the period . signals the end of one proof step. The effect of the entire command (up to the period) is to run induction and then simplify all resulting proof states and is shown in Figure 1 via the green arrows, which connect proof states 1 with 2 and 1 with 3. After we perform the induction, we see that proof state 2 contains the base case and proof state 3 contains the inductive case. The inductive case has an inductive hypothesis in the context. Proofs in Coq are finished when the goal is trivially true given the context (e.g., $0 ~ = ~ 0$ given an empty context). The collection of proof states and tactic invocations forms a proof tree. + +Formal definitions Coq provides three equi-expressive term languages that encode logical formulas such as forall n: nat, ${ \mathrm { ~ ~ n ~ } } + 0 = { \mathrm { ~ ~ n ~ } }$ . They differ in the amount of annotation they support. Coq’s user-facing term language Gallina supports type-inference and other forms of notational convenience (e.g., syntactic sugar and extensible parsing). Coq’s mid-level term language removes all forms of notational convenience. Finally, Coq’s kernel-level term language instantiates all types and is the level at which proofs are checked. Thus, the mid-level and kernel-level languages are similar, with the exception that the mid-level term language has a notion of an implicit argument. + +Implicit arguments occur in application constructs—M $M _ { 1 } \dots M _ { n }$ (apply function $M$ to arguments $M _ { 1 } \ldots M _ { n } )$ at the kernel-level versus $M M _ { 1 } ^ { \iota _ { 1 } } \dots M _ { n } ^ { \iota _ { n } }$ at the mid-level, where $\iota _ { i }$ marks implicit versus non-implicit arguments. Implicit arguments take care of the book-keeping aspects of a formal proof such as the types of all terms involved; a human prover would usually omit them in a paperpencil proof. Hence, learning with mid-level terms without implicit arguments is closer to humanlevel proving, whereas learning with kernel-level terms is closer to machine-level proving. + +Coq represents a proof state $P S$ as a tuple $\langle \Gamma , M , n \rangle$ of a local context $\Gamma$ , a goal term $M$ , and a unique proof state identifier $n$ . A context $\Gamma$ provides a list of identifiers and their types that expresses all the current assumptions. For instance, the proof state with identifier 3 has $\Gamma = \mathrm { ~ n ~ }$ : nat, IHn : $\mathrm { ~ ~ n ~ } = \mathrm { ~ ~ n ~ } + \mathrm { ~ ~ 0 ~ }$ . A tactic invocation creates edges between the appropriate proof states, which forms a proof tree. + +Proof states are interpreted in the presence of global state. The global state contains the interpretation of constructs such as constants and constructors that have global scope. It also contains the interpretation of constructs that have stateful effects such as existentials. In particular, it is possible to posit the existence of an object in one proof state and then proceed to perform case analysis resulting in multiple proof states that share the same existential. The GamePad tool (Section 3) exposes the constructs described above (with the exception of Gallina terms) as Python data structures, which we can use for building models. + +# 3 GAMEPAD TOOL + +We briefly describe the GamePad tool with an emphasis on the data that it exposes and any other design decisions that affect the modeling process. + +Obtaining proof traces We obtain the proof trace by implementing a patch to Coq (version 8.6.1) that instruments the Coq Ltac interpreter to log the intermediate proof states. The patch also supports SSreflect (Gonthier & Stephane Le, 2009), a popular Coq plugin for writing proofs about mathemat- ´ ics. Importantly, the patch does not touch the Coq proof checker, and hence, does not affect the critical part of the system that verifies the correctness of proofs. + +This implementation choice affords us flexibility in deciding what proof steps to consider atomic. As a reminder, the sequenced tactic induction n; simpl. (Section 2) can be considered as a single proof step (read up to the delimiting .) as in the example, but it is comprised of two primitive tactics—induction and simpl. As the granularity of a proof step has consequences for setting up a tactic prediction task (Section 4), we made the choice to obtain the finer-grained proof states by modifying Coq directly while maintaining the mapping to the human-level proof script. Thus, the tool records proof states at the granularity of primitive tactics (i.e., breaks up ;) as well as at the granularity of the proof script (i.e., .) + +Representing Coq proofs We expose the proof trace obtained from Coq as Python data structures, including Coq’s mid-level and kernel-level term languages, proof states, proof steps, and proof trees so that they can be manipulated with arbitrary Python code. Consequently, we can build models using this structure. For efficiency reasons, we represent Coq terms in a shared form. Representing terms in a shared form is a necessary technique to scale compilers and ITPs to real-world programs and proofs. In our setting, it is also essential to scaling training/inference to real-world data sets (see Section 7). + +Light-weight interaction The API provides a thin wrapper around coqtop, the Coq repl, which can be used to interactively construct proof scripts from within Python (see Section 6). This component mediates all interaction with Coq, including proof state parsing into GamePad’s representation of Coq proofs and sending tactic actions. + +# 4 TASKS + +In the theorem proving process, we encounter two important tasks: position evaluation and tactic prediction. A position evaluator $\mathcal { P }$ tells how easy it is to prove a given proof state, while a tactic predictor $\tau$ tells how to take an action that can make our proof state easier to prove. As a reminder, proof states are interpreted in the presence of global state, which we have left implicit here. Note that in general, $\mathcal { P }$ depends on $\tau$ ; a good tactic predictor would find it easier to prove a given proof state. Also, an action taken by $\tau$ can lead to multiple child proof states, and thus $\mathcal { P }$ must consider the provability of all child proof states. + +In this work, we aim to learn parametric functions $\mathcal { P } ^ { \mathcal { T } _ { h } }$ and $\mathcal { T } _ { h }$ using supervised learning on a data set of human proofs, where $\mathcal { T } _ { h }$ is the human tactic predictor implicit in the data set. In the future, one can use $\mathcal { P }$ and $\tau$ within an end-to-end prover, and learn them directly using reinforcement learning. This requires the ability to manipulate proof states by sending actions, which is possible using the light-weight interaction provided by GamePad. + +Position evaluation The goal of the position evaluation task is to predict the approximate number of steps required to finish a proof given an input proof state. We define this as the function $\mathcal { P } ^ { \mathcal { T } }$ : $P S \stackrel { - } { } \{ 1 , \stackrel { - } { \cdot } \cdot \cdot , K \}$ for a given tactic predictor $\tau$ , where we bin the sizes of the proof trees into $K$ classes to make learning easy. Given a data set of steps taken by a human prover, we aim to learn $\mathcal { P } ^ { \mathcal { T } _ { h } }$ by supervised learning on $N$ training tuples $\{ ( s _ { n } , d _ { n } ) \} _ { 1 \leq n \leq N }$ , where each $s _ { n }$ is a proof state and each $d _ { n }$ is the binned size of the proof tree below the corresponding $s _ { n }$ . + +The position evaluator defined above provides a proxy for how difficult it is to complete the proof from the current state—a lower number indicates that the goal is easy to deduce given the context. A model for position evaluation can be used to create a sequence of tasks for curriculum learning or by human provers to measure the progress they are making in a proof. We also note that position evaluation contains aspects of the premise selection problem (Irving et al., 2016) in that it should assign a high number to proof states which do not yet contain the requisite hypotheses to prove the current goal. + +Tactic prediction The goal of the tactic prediction task is to predict the next tactic to apply given an input proof state. We define this as the function $\mathcal { T } : P S T$ , where $T$ represents the set of possible tactic actions. Given a data set of steps taken by a human prover, we aim to learn the human tactic predictor $\mathcal { T } _ { h }$ given $N$ training tuples $\{ ( { \bar { s } } _ { n } , t _ { n } ) \} _ { 1 \leq n \leq N }$ , where $t _ { n }$ is the tactic the human prover used in state $s _ { n }$ . + +Tactic prediction is more localized to a single proof state compared to position evaluation, although there are two additional challenges. First, we must choose the granularity of a proof step, which ranges from considering only atomic tactics to human-level proof steps (i.e., compound tactics). Currently, our tool provides access to atomic tactics and as well as the capability to treat sequences of atomic tactics as a single proof step. + +Second, tactic prediction may additionally require the synthesis of an argument. Some arguments to tactics such as induction or rewrite require the user to select an identifier in the local or global context to apply—what to do induction on and what to equality to rewrite by respectively. This can be considered a premise selection problem. Other arguments to tactics include synthesizing entire Coq terms. For example, the tactic have: x : $\qquad = ~ \mathrm { ~ M ~ }$ declares a local lemma that asserts that M is true and to introduce it into the context as $_ \textrm { x }$ after it has been proven. Our tool provides the tactics, the arguments, and extracts local and global identifiers referenced in the arguments for convenience. This decomposes the problem of synthesizing tactic arguments into (1) predicting the identifiers involved and (2) constructing a term given a set of identifiers. The problem of synthesizing a term is difficult and a topic of research in itself—we do not address it in this paper. + +# 5 REPRESENTING PROOF STATES + +As we have just seen, both position evaluation and tactic prediction require a representation of proof states. Hence, we now discuss the representation of proof states in a form amenable for learning. + +One manner in which the structured representation of proofs states provided by GamePad can be leveraged is to apply recurrent neural networks (RNNs) in a similar manner to how they are applied to parse trees in natural language processing to obtain an embedding vector. We can embed terms using their structure with a recursive embedding function $\mathcal { E } : \mathrm { T e r m } \mathbb { R } ^ { D }$ . For example, the embedding of an application term $M _ { 0 } M _ { 1 } \ldots M _ { r }$ is obtained recursively as + +$$ +\mathcal { E } ( M _ { 0 } \ldots \mathit { M _ { r } } ) = \mathbf { R } \mathbf { N } \mathbf { N } ( \mathcal { E ^ { \prime } } [ \mathbb { A } \mathrm { p p } ] , \mathcal { E } ( M _ { 0 } ) , \ldots , \mathcal { E } ( M _ { r } ) ) +$$ + +where $\mathcal { E } ^ { \prime } [ [ \cdot ] ]$ is a learnable embedding table indexed by the kind of the AST node (e.g., App for J Kapplication node). The leaves of the AST consist of constants, inductive types, constructors, existentials, and variables. Each constant, inductive type, constructor, and existential is given an entry in a learnable embedding table, which encodes the global state. + +Towards interpreter-inspired embeddings One way to add inductive bias to the embedding is to use the known reduction semantics of terms. For instance, a programming language interpreter uses an environment, a lookup table mapping variables to their values, so that the meaning of a variable is the meaning that the environment assigns it. For example, the meaning of the program expression $_ \textrm { x }$ under an empty environment is a run-time error whereas the meaning of the same expression under the environment $\{ \mathrm { x } \mapsto \underline { { 4 2 } } \}$ is 42. We can apply this idea to obtain a new embedding function $\mathcal { E } : \mathrm { T e r m } \times \mathrm { E n v } \mathbb { R } ^ { D }$ that additionally takes an environment $\rho$ of type Env which is a lookup table mapping variables to embedding vectors. Whenever we encounter a binding form such as a dependent product $\Pi x : M _ { 1 } . M _ { 2 }$ (similar to an anonymous function $\lambda x : M _ { 1 } . M _ { 2 } )$ , we bind a new meaning for $x$ within its scope (i.e., the term $M _ { 2 }$ ). To do so, we sample a random vector $v \sim \mathcal { N } ^ { D }$ according to a standard (multivariate) normal distribution2 $\mathcal { N } ^ { D }$ and extend the local environment $\rho$ with the mapping $x \mapsto v$ , written $\rho [ x \mapsto v ]$ , so that mentions of the variable $x$ in $M _ { 2 }$ can be looked up. Then the embedding of a binding form, such as a dependent product term (Prod), is obtained recursively as + +$$ +\mathcal { E } ( \Pi x : M _ { 1 } . M _ { 2 } , \rho ) = \mathrm { R N N } ( \mathcal { E } ^ { \prime } [ \mathrm { P r } \mathrm { c o d } ] , \mathcal { E } ( M _ { 1 } , \rho ) , \mathcal { E } ( M _ { 2 } , \rho [ x \mapsto v ] ) ) \quad \mathrm { w h e r e ~ } v \sim \mathcal { N } ^ { D } . +$$ + +The embedding for the variable case (which occur at the leaves of the AST) is $\mathcal { E } ( x , \rho ) = \rho ( x )$ which corresponds to a variable lookup from the environment. + +We resample the corresponding $v$ from $\mathcal { N } ^ { D }$ every forward pass in training when we embed the term $\Pi x : M _ { 1 } . M _ { 2 }$ . By doing so, we encode the semantics that $x$ is just a placeholder and we should get the same result if we had used a different vector $v$ to embed it, while also preserving the environment lookup semantics that the embedding $v$ for $x$ is constant within the scope of $x$ in a single pass. Note that the embedding is invariant by construction to variable renaming. Wang et al. (2017) propose another structured approach based on a De Bruijn term representation that is invariant under variable renaming that would be interesting to compare against. It would also be interesting to extend the entire embedding to more closely follow the structure of an interpreter so that it better reflects the semantics as opposed to the syntax, although we leave these extensions to future work. + +Embedding proof states After obtaining the embedding for each type3 in the proof state and the embedding for the goal, we use another RNN over the embeddings. Note that the proof state context must be traversed in-order because the types of items later in the context may depend on the types of items that appear earlier in the context. As expected, the result of embedding a proof state is a vector in $\mathbb { R } ^ { D }$ . + +# 6 END-TO-END PROOF GENERATION FOR ALGEBRAIC REWRITES + +In this section, we walk through a basic setup that uses GamePad to learn a simple algebraic rewriter. First, we use a deterministic procedure to synthesize Coq proofs for our domain and use GamePad to extract the resulting proof trees. Second, we train a tactic predictor and then deploy it to synthesize end-to-end proofs using GamePad’s interactive mechanisms. This setup applies to any other domain of interest, provided we have a method of generating Coq proof scripts for that domain. + +# 6.1 SIMPLE ALGEBRAIC REWRITE PROBLEM + +We consider a problem that involves showing that two algebraic expressions are equivalent to one another. More concretely, we consider statements of the form: + +$$ +\forall b \in G , X = b , +$$ + +where $X$ is an arbitrary expression from the grammar $X \ { \mathrel { \mathop : } } { = } \ b \ | \ e \ | \ m \ | \ X \oplus X$ composed of elements with left-identity $e$ , right-identity $m$ , and binary operator $\oplus$ . We have two simplification rules: $\forall b \in G , b \oplus m = b$ (right identity) and $\forall b \in G , e \oplus b = b$ (left identity). + +Although the problem is simple, it involves bits of non-trivial reasoning. For instance, consider showing the equivalence $\forall b \in G , b \oplus ( e \oplus m ) = b$ . Here, we can choose to eliminate the $e$ (left identity) or the $m$ (right identity). Notably, choosing to eliminate $m$ does not progress the proof because we cannot simplify the proof state $b \oplus e = b$ . Note that the proof is not stuck because we can expand $b \oplus e$ back to $b \oplus ( e \oplus m )$ and then get rid of $e$ the second time around. Thus, a prover has at least two choices in solving such problems: (1) maintain a global perspective to choose which parts of the goal to rewrite or (2) learn to expand terms. For this problem, we write a deterministic procedure that generates proofs of the first form that selects a position and an identity law, and attempt to learn this algorithm. We do not generate proofs that require backtracking (e.g., due to a greedy rewrite) although it would be an interesting direction of future work. + +# 6.2 END-TO-END PROOF SYNTHESIS + +Tactic prediction The tactic prediction model embeds a proof state into $\mathbb { R } ^ { D }$ and uses a fullyconnected layer for prediction. We can model the proofs for this problem as a tactic prediction problem where the predicted category is a pair of the position in the AST and the identity to apply. We convert the position in the AST into a number using a preorder traversal of the Coq AST. For example, the second $\oplus$ in the expression $b \oplus ( e \oplus m )$ has position 2. We can encode each identity as a single number. We obtain the prediction class as the pair of both numbers. + +We implement end-to-end proof synthesis by combining a tactic prediction model that is trained offline with GamePad’s lightweight interaction module. The interaction module takes care of reading in the current Coq proof state and sending tactic calls to advance the proof. We use the trained model to do inference on the current Coq proof state to obtain a position to rewrite and the identity law to apply. We then translate this to a tactic and take the corresponding action in Coq. For the current problem, we do not consider backtracking. + +Results For this problem, we generate 400 unique theorems of the form $\forall b \in G , X = b$ and their proofs where $X$ is a randomly generated algebraic expression of length 10 that evaluates to $b$ . We construct $X$ by recursively expanding the left and right expressions of $\oplus$ subject to the constraint that only one side, chosen at random, reduces to $b$ and the other side reduces to the appropriate identity (left or right). We stop when the length of the expression is 10. We then extract the proof states using GamePad and train the tactic prediction model. + +To test the model, we generate a distinct set of 50 randomly generated algebraic expressions of length 10 and test how many proofs our model can complete using a greedy approach. That is, at each proof state, we use the trained model to perform inference and choose the (rewrite position, identity law) tuple with the highest probability as our action. We find that such an approach can generate 14 complete proofs, where we score a proof as a failure if any proof step fails. For expressions of length 10, there are 9 proof steps. We can relax this setting so that when any single proof step fails, we use the deterministic procedure to supply a proof step. This approach completes all 50 proofs with an average failure rate of 1 proof step per a proof. + +We have observed cases where a good position is selected but the wrong identity law is paired with it. As we predict the position and rewrite jointly, this behavior is somewhat surprising. We also observe that the accuracy on the same test set for the tactic prediction task is $9 5 \%$ . Thus, while the imitation learning model has good generalization in the traditional sense, more work needs to be done to leverage the policy when synthesizing complete proofs. + +# 7 REAL-WORLD DATA SETS + +We can also apply GamePad to data extracted from real-world formalizations. In this section, we apply baseline position evaluation and tactic prediction models to data extracted from the FeitThompson formalization. We encounter difficulties not present in the simple algebraic rewrite domain here, including scaling and a more difficult tactic prediction problem. + +The Feit-Thompson data set The Feit-Thompson theorem states that every finite group of oddorder is solvable, and is a deep result in group theory. The formalization has the interesting property that the researchers attempted to follow the book proofs as closely as possible. The extracted data set consists of 1602 lemmas and expands into 83478 proof states. For our tasks, we split the lemmas in the data set into a training, validation and test set in the ratio $8 : 1 : 1$ , and ensure that the number of proof states in the splits are in a similar ratio. Note that we split by lemmas and not by proof states because predictions for proof states within the proof of the same lemma are not independent and can lead to a more optimistic evaluation of the generalization capability of the models (particularly for the case of position evaluation). + +As a reminder, each proof state consists of a context and an associated goal. Each context contains on average 60 terms or identifiers, and on average 4000 nodes at the kernel level (and 1000 at the mid level without implicit arguments). The most common nodes include constants, applications, and variables, and as such, we focus on those during the design of our embedding. A formalization such as CompCert would contain a different distribution of AST nodes. In particular, as it concerns program verification, there would be more AST nodes involving fixed-points4, case analysis, and inductive type constructors. + +The most prevalent tactics include rewrite and have. As a reminder, a rewrite tactic requires the user to indicate which equality in the current local or global context to apply and is akin to premise selection. The have tactic introduces an intermediate lemma in the proof and is the hardest to learn as the user usually uses some mathematical insight about the problem before conjecturing a statement. We currently do not synthesize arguments for have tactics, although our tool extracts such data. We believe a generative model for synthesizing them would be a great direction for future work. + +Table 1: Training time speedups for the GRU model with state size of 128 on the position evaluation task obtained compared to an un-optimized baseline. ∗ indicates CPU and † indicates GPU. + +
ModelBase*Embedding Sharing*Dynamic Batching*Both*Both†
Speedup (approx.)110×10×130×190×
+ +Table 2: Test accuracies for position evaluation (Pos) and tactic prediction (Tac). $\dagger$ indicates kernellevel. $^ \ddag$ indicates mid-level without implicit arguments. For tactic argument prediction, we report validation recall for models with a minimum precision of $1 0 \%$ + +
ModelPostPostTactTactTact t arguments
Constant53.6653.6644.7544.751
SVM57.3757.5248.9449.45=
GRU65.3065.7458.2357.7025.98
TreeLSTM68.4466.3060.6360.5523.91
+ +Scaling to large proof trees In practice, ASTs can be on the order of thousands of nodes. We can apply two optimizations to scale to larger trees. The first optimization involves embedding sharing, where we memoize the embedding and the associated computation graph for any expression or subtree when it appears another time in the same proof state. For instance, if the context has terms $M _ { 1 } M _ { 2 }$ and $M _ { 1 }$ , the embedding for $M _ { 1 }$ is computed once. A single forward and backward pass is thus performed on this computation graph, with the gradients automatically accumulating from all places where the embedding was used. It thus helps save both memory and computation time of our models. The second optimization involves dynamic batching (Looks et al., 2017; Polosukhin & Zavershynskyi, 2018), which enables us to efficiently batch the computation of ops that perform the same operation albeit on different inputs, and thus make better use of the parallel acceleration provided by multi-core CPU’s and GPU’s. 5 + +Table 1 shows the approximate speedups obtained from applying embedding sharing and dynamic batching to representing proof states. Note that the GPU speedup will increase with larger models. + +# 7.1 POSITION EVALUATION AND TACTIC PREDICTION + +We use the interpreter-inspired embeddings to embed proof states into $\mathbb { R } ^ { D }$ , and then aim to train models for position evaluation and tactic prediction tasks. For the position evaluation task, we binned the target predictions into $K = 3$ classes—close ( $\mathit { \Theta } _ { \prec } 5$ steps), medium (between $6 - 1 9$ steps, inclusive), and far $> 2 0$ steps), and perform a simple three way classification. + +The tactic prediction is more complicated. First, we group tactics into equivalence classes if they perform the same mathematical operation. For example, we group the tactics reflexivity and done together because they are applied at the end of proofs to show that a statement is trivially true. Second, we now also have tactic arguments. We train two models, one that predicts only the tactic and another that additionally predicts the arguments. The first is a 23 way classification problem. For the second, the arguments could be (1) a term from the local context, (2) a term from the global context (e.g., a lemma), or (3) a term created by the human user. As arguments in the third category can be any Coq term, the space of arguments is potentially infinite. As a reminder, the data set makes extensive use of have tactics—in essence, this would require the model to conjecture a statement. For our baseline models, we focus on arguments in the first category and predict the presence or absence of each term in the context at any position in the arguments. For each term, we use the final hidden state and the embedding of each term followed by a linear layer to produce a two way prediction. Note that the distribution of labels is skewed towards the absent category. Thus, we are more interested in the precision-recall curve. We weigh the cross-entropy loss higher for presence class, and also try to balance the distribution of labels by randomly sampling only a subset of the negative class at training time. + +Results We first start by training a simple SVM (Cortes & Vapnik, 1995) on a set of heuristic features like context size, goal size, number of hypothesis in context and the smallest edit distance between a hypothesis and the context. The SVM performs better than the constant baseline of guessing the most common class. We then train RNN models to utilise our embedding strategy. We train GRU (Cho et al., 2014) and TreeLSTM (Tai et al., 2015) models using mini-batches of 32 proof states, set the RNN state size to 128, and use the Adam optimizer Kingma & Ba (2015) with a learning rate of 0.001. We use input (Srivastava et al., 2014) and weight (Merity et al., 2017) dropout with a rate of 0.1 for the TreeLSTM models (higher rates led to much slower learning). All neural net models were trained using PyTorch (Paszke et al., 2017). Table 2 show the results for the tasks. We were able to improve upon the SVM baseline, which indicates that it is possible to learn useful representations using the human supervision data and utilizing our proof state embeddings. We then experiment with removing the bookkeeping aspects of the prover by switching from kernel level to mid level proof states without implicit arguments. We obtain similar accuracies, indicating that most of that data is redundant. + +# 8 RELATED WORK + +The level of abstraction and representation of proofs that learning is applied to are salient points of comparison between work on learning and theorem proving. These choices inform the setup and challenges associated with the learning problem. + +As in our work, there are systems that experiment with learning in ITPs. Duncan (2002) explores how to learn (user-defined) tactics from a corpus of proofs in Isabelle so that they can be applied to future proofs. ML4PG (Komendantskaya et al., 2012) interfaces to ITPs at the level of the user interface for entering proof scripts. Thus, ML4PG is applicable to multiple ITPs although it obtains less granular proof states. Holphrasm (Whalen, 2016) uses string encodings of proof states and focuses on tactic argument synthesis (there is essentially only one tactic in the underlying ITP MetaMath (Megill, 2007)). HolStep (Kaliszyk et al., 2017) addresses premise selection using stringlevel encodings of proof states. Wang et al. (2017) extend the HolStep work to show the advantage of using a DeBruijn representation of proof terms as opposed to string-level encodings for premise selection. The structured representation provided by GamePad would support experimenting with such extensions. Gauthier et al. (2017) explores learning tactic-level proof search in Isabelle using hand-crafted features on string encodings of proof states. It would be interesting to experiment with their algorithm to our algebraic rewrite problem. Nagashima & He (2018) looks at explainable tactic prediction in Isabelle. + +Other approaches focus on automated theorem provers, which are designed to prove theorems with no human interaction. Irving et al. (2016) describes the premise selection problem and trains neural network models on proof traces obtained from applying E (Schulz, 2002) to the Mizar Mathematical Corpus. Loos et al. (2017), in addition to addressing premise selection, also address a clause selection task by applying neural network models to this problem in E. Kaliszyk & Urban (2014) demonstrate that similar learning based methods can prove $3 9 \%$ of the lemmas in the Flyspeck project (Kaliszyk & Urban, 2014). + +Another take on learning and theorem proving is to replace an entire theorem proving (sub)routine with a learned algorithm instead of using learning for heuristics. For instance, end-to-end differentiable proving (Rocktaschel & Riedel, 2017) replaces traditional ¨ unification with a trained neural network and demonstrates the efficacy of this approach for knowledge base completion. Neurosat (Selsam et al., 2018) applies a neural network model to predict satisfaction problems and shows how to recover a satisfying assignment. + +# 9 CONCLUSION + +In this work, we look at theorem proving problem through the lens of a system that enables learning with proofs constructed with human supervision. We highlight three key aspects of the problem at this level. The first concerns obtaining inputs to a learning algorithm that approximate the level of abstraction faced by a human prover. For this, we use an ITP, as it retains aspects of human supervision. GamePad preserves the structure of the proofs (e.g., annotations regarding implicit arguments) so they can be used for building models. The second involves building models that employ the game-like structure of ITP proofs. Here, we experiment with tactic prediction for toy and real world data sets. Finally, as a consequence of theorem proving at a higher-level (compared to SMT solvers), we will need to be careful to distinguish the syntax from the semantics of terms. Our current approach is to provide structured representations of terms so that more semantic structure can be exploited. While our results are preliminary, our hope is that GamePad provides an accessible starting point to explore the application of machine learning in the context of interactive theorem proving. + +# 10 FUTURE WORK + +We end by suggesting a few avenues for extending our work. The first concerns the design of new benchmarks for human-level proofs. In this paper, we designed a relatively simple algebraic rewrite problem to test the system end-to-end. Designing more difficult problems that still admit tractable learning (such as solving infinite sums or integrals) would be a great direction for future work. Note that you can define a new domain inside Coq and use GamePad to build provers that learn from example proofs. A second direction concerns building models that conjecture and explore the space of true statements, in addition to proving statements. This is particularly important in synthesizing arguments to tactics like have. Currently, we only predict the tactic identifiers and do not synthesize the entire term. Building a generative model would be a great next step. Lastly, it would be interesting to see if using end-to-end training with reinforcement learning and utilizing Monte-Carlo tree search to efficiently explore the search space can be effectively applied to humanlevel proofs.6 + +# 11 ACKNOWLEDGEMENTS + +We would like to thank Diederik Kingma, Tim Salimans, and Geoffrey Irving for reviewing initial drafts of the work. We thank Daniel Selsam for suggesting that we consider removing implicit arguments, and Jonathan Cai for discussions about the toy problem. Daniel Huang was supported by DARPA FA8750-17-2-0091. + +# REFERENCES + +Kyunghyun Cho, Bart van Merrienboer, C¸ alar G ¨ ulc¸ehre, Dzmitry Bahdanau, Fethi Bougares, Holger ¨ Schwenk, and Yoshua Bengio. Learning Phrase Representations using RNN Encoder–Decoder for Statistical Machine Translation. 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In International Conference on Learning Representations (ICLR), 2017. + +Cezary Kaliszyk, Josef Urban, Henryk Michalewski, and Mirek Olk. Reinforcement Learning of Theorem Proving. arXiv preprint arXiv:1805.07563, 2018. + +Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. Proceedings of the International Conference on Learning Representations 2015, 2015. + +Ekaterina Komendantskaya, Jonathan Heras, and Gudmund Grov. Machine Learning in Proof Gen- ´ eral: Interfacing Interfaces. arXiv preprint arXiv:1212.3618, 2012. + +Xavier Leroy et al. The compcert verified compiler. Documentation and users manual. INRIA Paris-Rocquencourt, 2012. + +Moshe Looks, Marcello Herreshoff, DeLesley Hutchins, and Peter Norvig. Deep learning with dynamic computation graphs. Proceedings of the International Conference on Learning Representations 2017, 2017. + +Sarah M. Loos, Geoffrey Irving, Christian Szegedy, and Cezary Kaliszyk. Deep network guided proof search. In LPAR, volume 46 of EPiC Series in Computing, pp. 85–105. EasyChair, 2017. + +Norman D. Megill. Metamath: A Computer Language for Pure Mathematics. Lulu Press, Morrisville, North Carolina, 2007. http://us.metamath.org/downloads/metamath.pdf. + +Stephen Merity, Nitish Shirish Keskar, and Richard Socher. Regularizing and Optimizing LSTM Language Models. arXiv preprint arXiv:1708.02182, 2017. + +Yutaka Nagashima and Yilun He. Pamper: Proof method recommendation system for isabelle/hol. arXiv preprint arXiv:1806.07239, 2018. + +Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in PyTorch. In NIPS-W, 2017. + +Illia Polosukhin and Maksym Zavershynskyi. Pytorch fold. https://github.com/nearai/ pytorch-tools, 2018. + +Tim Rocktaschel and Sebastian Riedel. End-to-end differentiable proving. In ¨ Advances in Neural Information Processing Systems, pp. 3791–3803, 2017. + +Stephan Schulz. E–a brainiac theorem prover. AI Communications, 15(2, 3):111–126, 2002. + +Daniel Selsam, Matthew Lamm, Benedikt Bunz, Percy Liang, Leonardo de Moura, and David L. ¨ Dill. Learning a SAT Solver from Single-Bit Supervision. arXiv preprint arXiv:1802.03685, 2018. + +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. + +Kai Sheng Tai, Richard Socher, and Christopher D. Manning. Improved semantic representations from tree-structured long short-term memory networks. In ACL (1), pp. 1556–1566. The Association for Computer Linguistics, 2015. + +Mingzhe Wang, Yihe Tang, Jian Wang, and Jia Deng. Premise Selection for Theorem Proving by Deep Graph Embedding. In Advances in Neural Information Processing Systems, pp. 2783–2793, 2017. + +Daniel Whalen. Holophrasm: a neural Automated Theorem Prover for higher-order logic. arXiv preprint arXiv:1608.02644, 2016. \ No newline at end of file diff --git a/parse/train/r1xwKoR9Y7/r1xwKoR9Y7_content_list.json b/parse/train/r1xwKoR9Y7/r1xwKoR9Y7_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..fb4ca3919381eaa9319b380a3ec60ccabc30b909 --- /dev/null +++ b/parse/train/r1xwKoR9Y7/r1xwKoR9Y7_content_list.json @@ -0,0 +1,1348 @@ +[ + { + "type": "text", + "text": "GAMEPAD: A LEARNING ENVIRONMENT FOR THEOREM PROVING ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Daniel Huang∗† dehuang@berkeley.edu ", + "bbox": [ + 305, + 162, + 503, + 191 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Prafulla Dhariwal∗‡ prafulla@openai.com ", + "bbox": [ + 527, + 162, + 715, + 191 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Dawn Song† ", + "bbox": [ + 356, + 205, + 442, + 219 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "dawnsong@cs.berkeley.edu ", + "bbox": [ + 281, + 222, + 516, + 234 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ilya Sutskever‡ ilyasu@openai.com ", + "bbox": [ + 534, + 205, + 700, + 234 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 452, + 271, + 544, + 286 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we introduce a system called GamePad that can be used to explore the application of machine learning methods to theorem proving in the Coq proof assistant. Interactive theorem provers such as Coq enable users to construct machine-checkable proofs in a step-by-step manner. Hence, they provide an opportunity to explore theorem proving with human supervision. We use GamePad to synthesize proofs for a simple algebraic rewrite problem and train baseline models for a formalization of the Feit-Thompson theorem. We address position evaluation (i.e., predict the number of proof steps left) and tactic prediction (i.e., predict the next proof step) tasks, which arise naturally in tactic-based theorem proving. ", + "bbox": [ + 232, + 303, + 764, + 429 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 457, + 336, + 473 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Theorem proving is a challenging AI task that involves symbolic reasoning (e.g., SMT solvers (De Moura & Bjørner, 2008)) and intuition guided search. Recent work (Irving et al., 2016; Loos et al., 2017; Kaliszyk et al., 2017) has shown the promise of applying deep learning techniques in this domain, primarily on tasks useful for automated theorem provers (e.g., premise selection) which operate with little to no human supervision. In this work, we aim to move closer to learning on proofs constructed with human supervision. ", + "bbox": [ + 174, + 489, + 825, + 571 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We look at theorem proving in the realm of formal proofs. A formal proof is systematically derived in a formal system, which makes it possible to algorithmically (i.e., with a computer) check these proofs for correctness. Thus, formal proofs provide perfect learning signal—theorem statements and proofs are unambiguous. Human mathematicians usually do not write proofs in this style, and instead, construct and communicate proofs in natural language. Although the form and level of detail involved in each kind of proof differ, the logical content is similar in both contexts. ", + "bbox": [ + 174, + 579, + 825, + 662 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Our work focuses on interactive theorem provers (ITPs), which are software tools that enable human users to construct formal proofs. ITPs have at least two features that make them compelling environments for exploring the application of learning techniques to theorem proving. First and foremost, ITPs provide full-fledged programmable environments. Consequently, any machine learning infrastructure built for an ITP can be reused across any problem domain crafted to study an aspect of learning and theorem proving. Second, the proofs are constructed by humans, and thus, have the constraint that they must be relatively human-understandable. Hence, ITPs provide access to large amounts of supervised data (i.e., expert-constructed proofs of theorems that are mathematically interesting). For example, ITPs have been used to build and check the proofs of large mathematical theorems such as the Feit-Thompson theorem (Gonthier et al., 2013) and provide provable guarantees on complex pieces of software such as the CompCert C compiler (Leroy et al., 2012). ", + "bbox": [ + 174, + 670, + 825, + 823 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We introduce a system called GamePad1 that exposes parts of the Coq ITP to enable machine learning tasks and explore a few use cases. We focus on the Coq proof assistant for two reasons. First, ", + "bbox": [ + 176, + 829, + 821, + 858 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/1fec650b161e90747296d39eababa690f2aa0ff7a0fc43bc4747eed66f79eeba.jpg", + "image_caption": [ + "Figure 1: A proof script in Coq (left) and the resulting proof states, proof steps, and the complete proof tree (right). A proof state consists of a context (pink rectangles) and a goal (white rectangles). The initial proof state has as its goal the statement we are trying to prove and an empty context. The arrows indicate what tactic the prover used. The final states of the proof are indicated by the red circles and can be transitioned to only when the goal in the previous state is trivially true. " + ], + "image_footnote": [], + "bbox": [ + 173, + 109, + 828, + 247 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Coq is a mature system with an active developer community that has been used to formalize nontrivial theorems, including Feit-Thompson and CompCert. Second, Coq supports the extraction of verified software. Consequently, one can prove that a program is correct and then run the verified program. The ease of extraction makes Coq a popular choice for program verification. ", + "bbox": [ + 174, + 364, + 825, + 420 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our contributions are the following. First, we introduce GamePad, which provides a structured Python representation of Coq proofs (Section 3), including all the proof states encountered in a proof, the steps taken, and expression abstract syntax trees (ASTs). The tool also enables lightweight interaction with Coq so that it can be used to dynamically build proofs (e.g., used as an environment for reinforcement learning). Tasks that can leverage this structured representation (Section 4) include position evaluation (i.e., predict the number of proof steps left) and tactic prediction (i.e., predict the next proof step to take). We also discuss how we can use the structured representation to embed proof states into $\\bar { \\mathbb { R } } ^ { D }$ (Section 5). Second, we demonstrate the synthesis of Coq proof scripts that makes use of a tactic prediction model for a hand-crafted algebraic rewriting problem (Section 6). Third and finally, we apply baseline models for position evaluation and tactic prediction to the FeitThompson formalization using data extracted by GamePad (Section 7). ", + "bbox": [ + 174, + 428, + 825, + 580 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The code for GamePad, as well as the associated data sets, models and results, are open source on GitHub at https://github.com/ml4tp/gamepad. ", + "bbox": [ + 176, + 587, + 823, + 616 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 636, + 326, + 652 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To provide context for the rest of this paper, we begin by illustrating the formal proof process and how it can be modeled as a game. We then walk through a simple example of a proof in Coq and end by summarizing the constructs that we will encounter in the rest of the paper. ", + "bbox": [ + 174, + 667, + 825, + 710 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Theorem proving process When humans write a pencil-paper proof, we typically maintain a mental and/or written scratchpad that keeps track of (1) what we need to show and (2) the facts we currently have at our disposal. As the proof progresses, the items in this scratchpad changes. For instance, we might transform what we need to show (e.g., it suffices to show another statement) and/or discover new facts (e.g., we derive an additional fact as a consequence of known facts). When we move from the pencil-paper setting to an ITP such as Coq, we will still have such a scratchpad, but use the ITP’s term language to express mathematical statements and the rules of the ITP’s formal system to construct the proof instead. ", + "bbox": [ + 174, + 726, + 825, + 838 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Formal theorem proving as a game A game serves as useful mental model of the proving process. The state of the game is a proof state, which looks like the scratchpad and is comprised of (1) a goal (expressed in the term language) stating what we need to prove and (2) a context containing the assumptions (also expressed in the term language). The starting state has as its goal the statement of a theorem and an empty context, while a final state has as its goal a statement that is trivially true (i.e., definitionally equal) given the context. The aim of a prover is to transform the starting state into a collection of final states using only logically valid transitions, which can affect both the context and the goal. It is possible to obtain multiple final states because some transitions may split a goal into multiple subgoals. For a given (sub)goal, a successful proof corresponds to a proof tree, where the root is the start state and all the leaves are final states. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "An example Coq proof Consider showing that adding 0 to any natural number $n$ is $n$ itself. A paper-pencil proof might proceed by induction on $n$ , where we check that the result holds on the base case (i.e., $n = 0$ ) and the inductive case (i.e., $n = n + 1 )$ . We can carry out a similar process in Coq, using a sequence of commands called tactics in Coq’s tactic language Ltac that indicate what proofs steps to take. For instance, we can use the code induction n; simpl. to start a proof by induction (on n) in Coq. The expressions induction and simpl are tactics (of arity 1 and 0 respectively), the semicolon ; sequences two tactics, and the period . signals the end of one proof step. The effect of the entire command (up to the period) is to run induction and then simplify all resulting proof states and is shown in Figure 1 via the green arrows, which connect proof states 1 with 2 and 1 with 3. After we perform the induction, we see that proof state 2 contains the base case and proof state 3 contains the inductive case. The inductive case has an inductive hypothesis in the context. Proofs in Coq are finished when the goal is trivially true given the context (e.g., $0 ~ = ~ 0$ given an empty context). The collection of proof states and tactic invocations forms a proof tree. ", + "bbox": [ + 173, + 189, + 825, + 369 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Formal definitions Coq provides three equi-expressive term languages that encode logical formulas such as forall n: nat, ${ \\mathrm { ~ ~ n ~ } } + 0 = { \\mathrm { ~ ~ n ~ } }$ . They differ in the amount of annotation they support. Coq’s user-facing term language Gallina supports type-inference and other forms of notational convenience (e.g., syntactic sugar and extensible parsing). Coq’s mid-level term language removes all forms of notational convenience. Finally, Coq’s kernel-level term language instantiates all types and is the level at which proofs are checked. Thus, the mid-level and kernel-level languages are similar, with the exception that the mid-level term language has a notion of an implicit argument. ", + "bbox": [ + 174, + 387, + 825, + 484 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Implicit arguments occur in application constructs—M $M _ { 1 } \\dots M _ { n }$ (apply function $M$ to arguments $M _ { 1 } \\ldots M _ { n } )$ at the kernel-level versus $M M _ { 1 } ^ { \\iota _ { 1 } } \\dots M _ { n } ^ { \\iota _ { n } }$ at the mid-level, where $\\iota _ { i }$ marks implicit versus non-implicit arguments. Implicit arguments take care of the book-keeping aspects of a formal proof such as the types of all terms involved; a human prover would usually omit them in a paperpencil proof. Hence, learning with mid-level terms without implicit arguments is closer to humanlevel proving, whereas learning with kernel-level terms is closer to machine-level proving. ", + "bbox": [ + 174, + 491, + 823, + 574 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Coq represents a proof state $P S$ as a tuple $\\langle \\Gamma , M , n \\rangle$ of a local context $\\Gamma$ , a goal term $M$ , and a unique proof state identifier $n$ . A context $\\Gamma$ provides a list of identifiers and their types that expresses all the current assumptions. For instance, the proof state with identifier 3 has $\\Gamma = \\mathrm { ~ n ~ }$ : nat, IHn : $\\mathrm { ~ ~ n ~ } = \\mathrm { ~ ~ n ~ } + \\mathrm { ~ ~ 0 ~ }$ . A tactic invocation creates edges between the appropriate proof states, which forms a proof tree. ", + "bbox": [ + 174, + 582, + 825, + 651 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Proof states are interpreted in the presence of global state. The global state contains the interpretation of constructs such as constants and constructors that have global scope. It also contains the interpretation of constructs that have stateful effects such as existentials. In particular, it is possible to posit the existence of an object in one proof state and then proceed to perform case analysis resulting in multiple proof states that share the same existential. The GamePad tool (Section 3) exposes the constructs described above (with the exception of Gallina terms) as Python data structures, which we can use for building models. ", + "bbox": [ + 174, + 657, + 825, + 756 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 GAMEPAD TOOL ", + "text_level": 1, + "bbox": [ + 176, + 776, + 344, + 792 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We briefly describe the GamePad tool with an emphasis on the data that it exposes and any other design decisions that affect the modeling process. ", + "bbox": [ + 174, + 809, + 821, + 838 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Obtaining proof traces We obtain the proof trace by implementing a patch to Coq (version 8.6.1) that instruments the Coq Ltac interpreter to log the intermediate proof states. The patch also supports SSreflect (Gonthier & Stephane Le, 2009), a popular Coq plugin for writing proofs about mathemat- ´ ics. Importantly, the patch does not touch the Coq proof checker, and hence, does not affect the critical part of the system that verifies the correctness of proofs. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This implementation choice affords us flexibility in deciding what proof steps to consider atomic. As a reminder, the sequenced tactic induction n; simpl. (Section 2) can be considered as a single proof step (read up to the delimiting .) as in the example, but it is comprised of two primitive tactics—induction and simpl. As the granularity of a proof step has consequences for setting up a tactic prediction task (Section 4), we made the choice to obtain the finer-grained proof states by modifying Coq directly while maintaining the mapping to the human-level proof script. Thus, the tool records proof states at the granularity of primitive tactics (i.e., breaks up ;) as well as at the granularity of the proof script (i.e., .) ", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Representing Coq proofs We expose the proof trace obtained from Coq as Python data structures, including Coq’s mid-level and kernel-level term languages, proof states, proof steps, and proof trees so that they can be manipulated with arbitrary Python code. Consequently, we can build models using this structure. For efficiency reasons, we represent Coq terms in a shared form. Representing terms in a shared form is a necessary technique to scale compilers and ITPs to real-world programs and proofs. In our setting, it is also essential to scaling training/inference to real-world data sets (see Section 7). ", + "bbox": [ + 174, + 232, + 825, + 329 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Light-weight interaction The API provides a thin wrapper around coqtop, the Coq repl, which can be used to interactively construct proof scripts from within Python (see Section 6). This component mediates all interaction with Coq, including proof state parsing into GamePad’s representation of Coq proofs and sending tactic actions. ", + "bbox": [ + 174, + 347, + 825, + 402 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 TASKS ", + "text_level": 1, + "bbox": [ + 176, + 424, + 261, + 440 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the theorem proving process, we encounter two important tasks: position evaluation and tactic prediction. A position evaluator $\\mathcal { P }$ tells how easy it is to prove a given proof state, while a tactic predictor $\\tau$ tells how to take an action that can make our proof state easier to prove. As a reminder, proof states are interpreted in the presence of global state, which we have left implicit here. Note that in general, $\\mathcal { P }$ depends on $\\tau$ ; a good tactic predictor would find it easier to prove a given proof state. Also, an action taken by $\\tau$ can lead to multiple child proof states, and thus $\\mathcal { P }$ must consider the provability of all child proof states. ", + "bbox": [ + 174, + 457, + 825, + 555 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this work, we aim to learn parametric functions $\\mathcal { P } ^ { \\mathcal { T } _ { h } }$ and $\\mathcal { T } _ { h }$ using supervised learning on a data set of human proofs, where $\\mathcal { T } _ { h }$ is the human tactic predictor implicit in the data set. In the future, one can use $\\mathcal { P }$ and $\\tau$ within an end-to-end prover, and learn them directly using reinforcement learning. This requires the ability to manipulate proof states by sending actions, which is possible using the light-weight interaction provided by GamePad. ", + "bbox": [ + 174, + 560, + 825, + 632 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Position evaluation The goal of the position evaluation task is to predict the approximate number of steps required to finish a proof given an input proof state. We define this as the function $\\mathcal { P } ^ { \\mathcal { T } }$ : $P S \\stackrel { - } { } \\{ 1 , \\stackrel { - } { \\cdot } \\cdot \\cdot , K \\}$ for a given tactic predictor $\\tau$ , where we bin the sizes of the proof trees into $K$ classes to make learning easy. Given a data set of steps taken by a human prover, we aim to learn $\\mathcal { P } ^ { \\mathcal { T } _ { h } }$ by supervised learning on $N$ training tuples $\\{ ( s _ { n } , d _ { n } ) \\} _ { 1 \\leq n \\leq N }$ , where each $s _ { n }$ is a proof state and each $d _ { n }$ is the binned size of the proof tree below the corresponding $s _ { n }$ . ", + "bbox": [ + 174, + 648, + 825, + 732 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The position evaluator defined above provides a proxy for how difficult it is to complete the proof from the current state—a lower number indicates that the goal is easy to deduce given the context. A model for position evaluation can be used to create a sequence of tasks for curriculum learning or by human provers to measure the progress they are making in a proof. We also note that position evaluation contains aspects of the premise selection problem (Irving et al., 2016) in that it should assign a high number to proof states which do not yet contain the requisite hypotheses to prove the current goal. ", + "bbox": [ + 174, + 739, + 825, + 837 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Tactic prediction The goal of the tactic prediction task is to predict the next tactic to apply given an input proof state. We define this as the function $\\mathcal { T } : P S T$ , where $T$ represents the set of possible tactic actions. Given a data set of steps taken by a human prover, we aim to learn the human tactic predictor $\\mathcal { T } _ { h }$ given $N$ training tuples $\\{ ( { \\bar { s } } _ { n } , t _ { n } ) \\} _ { 1 \\leq n \\leq N }$ , where $t _ { n }$ is the tactic the human prover used in state $s _ { n }$ . ", + "bbox": [ + 176, + 854, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Tactic prediction is more localized to a single proof state compared to position evaluation, although there are two additional challenges. First, we must choose the granularity of a proof step, which ranges from considering only atomic tactics to human-level proof steps (i.e., compound tactics). Currently, our tool provides access to atomic tactics and as well as the capability to treat sequences of atomic tactics as a single proof step. ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Second, tactic prediction may additionally require the synthesis of an argument. Some arguments to tactics such as induction or rewrite require the user to select an identifier in the local or global context to apply—what to do induction on and what to equality to rewrite by respectively. This can be considered a premise selection problem. Other arguments to tactics include synthesizing entire Coq terms. For example, the tactic have: x : $\\qquad = ~ \\mathrm { ~ M ~ }$ declares a local lemma that asserts that M is true and to introduce it into the context as $_ \\textrm { x }$ after it has been proven. Our tool provides the tactics, the arguments, and extracts local and global identifiers referenced in the arguments for convenience. This decomposes the problem of synthesizing tactic arguments into (1) predicting the identifiers involved and (2) constructing a term given a set of identifiers. The problem of synthesizing a term is difficult and a topic of research in itself—we do not address it in this paper. ", + "bbox": [ + 173, + 180, + 825, + 319 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 REPRESENTING PROOF STATES ", + "text_level": 1, + "bbox": [ + 176, + 339, + 464, + 356 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As we have just seen, both position evaluation and tactic prediction require a representation of proof states. Hence, we now discuss the representation of proof states in a form amenable for learning. ", + "bbox": [ + 174, + 372, + 825, + 400 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "One manner in which the structured representation of proofs states provided by GamePad can be leveraged is to apply recurrent neural networks (RNNs) in a similar manner to how they are applied to parse trees in natural language processing to obtain an embedding vector. We can embed terms using their structure with a recursive embedding function $\\mathcal { E } : \\mathrm { T e r m } \\mathbb { R } ^ { D }$ . For example, the embedding of an application term $M _ { 0 } M _ { 1 } \\ldots M _ { r }$ is obtained recursively as ", + "bbox": [ + 174, + 406, + 825, + 478 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/c022d3a4b407faf827ee813709d84a0f0ed665288326d76bfcd476b29304c1a8.jpg", + "text": "$$\n\\mathcal { E } ( M _ { 0 } \\ldots \\mathit { M _ { r } } ) = \\mathbf { R } \\mathbf { N } \\mathbf { N } ( \\mathcal { E ^ { \\prime } } [ \\mathbb { A } \\mathrm { p p } ] , \\mathcal { E } ( M _ { 0 } ) , \\ldots , \\mathcal { E } ( M _ { r } ) )\n$$", + "text_format": "latex", + "bbox": [ + 318, + 484, + 676, + 502 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mathcal { E } ^ { \\prime } [ [ \\cdot ] ]$ is a learnable embedding table indexed by the kind of the AST node (e.g., App for J Kapplication node). The leaves of the AST consist of constants, inductive types, constructors, existentials, and variables. Each constant, inductive type, constructor, and existential is given an entry in a learnable embedding table, which encodes the global state. ", + "bbox": [ + 174, + 508, + 825, + 565 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Towards interpreter-inspired embeddings One way to add inductive bias to the embedding is to use the known reduction semantics of terms. For instance, a programming language interpreter uses an environment, a lookup table mapping variables to their values, so that the meaning of a variable is the meaning that the environment assigns it. For example, the meaning of the program expression $_ \\textrm { x }$ under an empty environment is a run-time error whereas the meaning of the same expression under the environment $\\{ \\mathrm { x } \\mapsto \\underline { { 4 2 } } \\}$ is 42. We can apply this idea to obtain a new embedding function $\\mathcal { E } : \\mathrm { T e r m } \\times \\mathrm { E n v } \\mathbb { R } ^ { D }$ that additionally takes an environment $\\rho$ of type Env which is a lookup table mapping variables to embedding vectors. Whenever we encounter a binding form such as a dependent product $\\Pi x : M _ { 1 } . M _ { 2 }$ (similar to an anonymous function $\\lambda x : M _ { 1 } . M _ { 2 } )$ , we bind a new meaning for $x$ within its scope (i.e., the term $M _ { 2 }$ ). To do so, we sample a random vector $v \\sim \\mathcal { N } ^ { D }$ according to a standard (multivariate) normal distribution2 $\\mathcal { N } ^ { D }$ and extend the local environment $\\rho$ with the mapping $x \\mapsto v$ , written $\\rho [ x \\mapsto v ]$ , so that mentions of the variable $x$ in $M _ { 2 }$ can be looked up. Then the embedding of a binding form, such as a dependent product term (Prod), is obtained recursively as ", + "bbox": [ + 174, + 580, + 825, + 775 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/41ecd72483433f1646f3a316a196f391c4690e8f94f87afe8c2353afb701e364.jpg", + "text": "$$\n\\mathcal { E } ( \\Pi x : M _ { 1 } . M _ { 2 } , \\rho ) = \\mathrm { R N N } ( \\mathcal { E } ^ { \\prime } [ \\mathrm { P r } \\mathrm { c o d } ] , \\mathcal { E } ( M _ { 1 } , \\rho ) , \\mathcal { E } ( M _ { 2 } , \\rho [ x \\mapsto v ] ) ) \\quad \\mathrm { w h e r e ~ } v \\sim \\mathcal { N } ^ { D } .\n$$", + "text_format": "latex", + "bbox": [ + 207, + 782, + 789, + 800 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The embedding for the variable case (which occur at the leaves of the AST) is $\\mathcal { E } ( x , \\rho ) = \\rho ( x )$ which corresponds to a variable lookup from the environment. ", + "bbox": [ + 176, + 806, + 820, + 835 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We resample the corresponding $v$ from $\\mathcal { N } ^ { D }$ every forward pass in training when we embed the term $\\Pi x : M _ { 1 } . M _ { 2 }$ . By doing so, we encode the semantics that $x$ is just a placeholder and we should get the same result if we had used a different vector $v$ to embed it, while also preserving the environment lookup semantics that the embedding $v$ for $x$ is constant within the scope of $x$ in a single pass. Note that the embedding is invariant by construction to variable renaming. Wang et al. (2017) propose another structured approach based on a De Bruijn term representation that is invariant under variable renaming that would be interesting to compare against. It would also be interesting to extend the entire embedding to more closely follow the structure of an interpreter so that it better reflects the semantics as opposed to the syntax, although we leave these extensions to future work. ", + "bbox": [ + 174, + 842, + 825, + 898 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Embedding proof states After obtaining the embedding for each type3 in the proof state and the embedding for the goal, we use another RNN over the embeddings. Note that the proof state context must be traversed in-order because the types of items later in the context may depend on the types of items that appear earlier in the context. As expected, the result of embedding a proof state is a vector in $\\mathbb { R } ^ { D }$ . ", + "bbox": [ + 174, + 189, + 825, + 260 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6 END-TO-END PROOF GENERATION FOR ALGEBRAIC REWRITES ", + "text_level": 1, + "bbox": [ + 176, + 281, + 730, + 297 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we walk through a basic setup that uses GamePad to learn a simple algebraic rewriter. First, we use a deterministic procedure to synthesize Coq proofs for our domain and use GamePad to extract the resulting proof trees. Second, we train a tactic predictor and then deploy it to synthesize end-to-end proofs using GamePad’s interactive mechanisms. This setup applies to any other domain of interest, provided we have a method of generating Coq proof scripts for that domain. ", + "bbox": [ + 174, + 314, + 825, + 385 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6.1 SIMPLE ALGEBRAIC REWRITE PROBLEM", + "text_level": 1, + "bbox": [ + 174, + 401, + 500, + 416 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We consider a problem that involves showing that two algebraic expressions are equivalent to one another. More concretely, we consider statements of the form: ", + "bbox": [ + 173, + 428, + 823, + 457 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/1d78beec4a0cf37ab5484fbb39cdaa5eec604fa1cc1da24e30030e79fac30b2b.jpg", + "text": "$$\n\\forall b \\in G , X = b ,\n$$", + "text_format": "latex", + "bbox": [ + 442, + 465, + 553, + 482 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $X$ is an arbitrary expression from the grammar $X \\ { \\mathrel { \\mathop : } } { = } \\ b \\ | \\ e \\ | \\ m \\ | \\ X \\oplus X$ composed of elements with left-identity $e$ , right-identity $m$ , and binary operator $\\oplus$ . We have two simplification rules: $\\forall b \\in G , b \\oplus m = b$ (right identity) and $\\forall b \\in G , e \\oplus b = b$ (left identity). ", + "bbox": [ + 174, + 489, + 823, + 532 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Although the problem is simple, it involves bits of non-trivial reasoning. For instance, consider showing the equivalence $\\forall b \\in G , b \\oplus ( e \\oplus m ) = b$ . Here, we can choose to eliminate the $e$ (left identity) or the $m$ (right identity). Notably, choosing to eliminate $m$ does not progress the proof because we cannot simplify the proof state $b \\oplus e = b$ . Note that the proof is not stuck because we can expand $b \\oplus e$ back to $b \\oplus ( e \\oplus m )$ and then get rid of $e$ the second time around. Thus, a prover has at least two choices in solving such problems: (1) maintain a global perspective to choose which parts of the goal to rewrite or (2) learn to expand terms. For this problem, we write a deterministic procedure that generates proofs of the first form that selects a position and an identity law, and attempt to learn this algorithm. We do not generate proofs that require backtracking (e.g., due to a greedy rewrite) although it would be an interesting direction of future work. ", + "bbox": [ + 173, + 539, + 825, + 678 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6.2 END-TO-END PROOF SYNTHESIS ", + "text_level": 1, + "bbox": [ + 176, + 696, + 441, + 710 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Tactic prediction The tactic prediction model embeds a proof state into $\\mathbb { R } ^ { D }$ and uses a fullyconnected layer for prediction. We can model the proofs for this problem as a tactic prediction problem where the predicted category is a pair of the position in the AST and the identity to apply. We convert the position in the AST into a number using a preorder traversal of the Coq AST. For example, the second $\\oplus$ in the expression $b \\oplus ( e \\oplus m )$ has position 2. We can encode each identity as a single number. We obtain the prediction class as the pair of both numbers. ", + "bbox": [ + 174, + 722, + 823, + 806 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We implement end-to-end proof synthesis by combining a tactic prediction model that is trained offline with GamePad’s lightweight interaction module. The interaction module takes care of reading in the current Coq proof state and sending tactic calls to advance the proof. We use the trained model to do inference on the current Coq proof state to obtain a position to rewrite and the identity law to apply. We then translate this to a tactic and take the corresponding action in Coq. For the current problem, we do not consider backtracking. ", + "bbox": [ + 174, + 814, + 825, + 897 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Results For this problem, we generate 400 unique theorems of the form $\\forall b \\in G , X = b$ and their proofs where $X$ is a randomly generated algebraic expression of length 10 that evaluates to $b$ . We construct $X$ by recursively expanding the left and right expressions of $\\oplus$ subject to the constraint that only one side, chosen at random, reduces to $b$ and the other side reduces to the appropriate identity (left or right). We stop when the length of the expression is 10. We then extract the proof states using GamePad and train the tactic prediction model. ", + "bbox": [ + 174, + 103, + 825, + 186 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To test the model, we generate a distinct set of 50 randomly generated algebraic expressions of length 10 and test how many proofs our model can complete using a greedy approach. That is, at each proof state, we use the trained model to perform inference and choose the (rewrite position, identity law) tuple with the highest probability as our action. We find that such an approach can generate 14 complete proofs, where we score a proof as a failure if any proof step fails. For expressions of length 10, there are 9 proof steps. We can relax this setting so that when any single proof step fails, we use the deterministic procedure to supply a proof step. This approach completes all 50 proofs with an average failure rate of 1 proof step per a proof. ", + "bbox": [ + 174, + 194, + 825, + 306 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We have observed cases where a good position is selected but the wrong identity law is paired with it. As we predict the position and rewrite jointly, this behavior is somewhat surprising. We also observe that the accuracy on the same test set for the tactic prediction task is $9 5 \\%$ . Thus, while the imitation learning model has good generalization in the traditional sense, more work needs to be done to leverage the policy when synthesizing complete proofs. ", + "bbox": [ + 174, + 313, + 823, + 382 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "7 REAL-WORLD DATA SETS ", + "text_level": 1, + "bbox": [ + 176, + 410, + 419, + 426 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We can also apply GamePad to data extracted from real-world formalizations. In this section, we apply baseline position evaluation and tactic prediction models to data extracted from the FeitThompson formalization. We encounter difficulties not present in the simple algebraic rewrite domain here, including scaling and a more difficult tactic prediction problem. ", + "bbox": [ + 174, + 448, + 823, + 503 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The Feit-Thompson data set The Feit-Thompson theorem states that every finite group of oddorder is solvable, and is a deep result in group theory. The formalization has the interesting property that the researchers attempted to follow the book proofs as closely as possible. The extracted data set consists of 1602 lemmas and expands into 83478 proof states. For our tasks, we split the lemmas in the data set into a training, validation and test set in the ratio $8 : 1 : 1$ , and ensure that the number of proof states in the splits are in a similar ratio. Note that we split by lemmas and not by proof states because predictions for proof states within the proof of the same lemma are not independent and can lead to a more optimistic evaluation of the generalization capability of the models (particularly for the case of position evaluation). ", + "bbox": [ + 174, + 527, + 825, + 652 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As a reminder, each proof state consists of a context and an associated goal. Each context contains on average 60 terms or identifiers, and on average 4000 nodes at the kernel level (and 1000 at the mid level without implicit arguments). The most common nodes include constants, applications, and variables, and as such, we focus on those during the design of our embedding. A formalization such as CompCert would contain a different distribution of AST nodes. In particular, as it concerns program verification, there would be more AST nodes involving fixed-points4, case analysis, and inductive type constructors. ", + "bbox": [ + 174, + 660, + 825, + 757 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The most prevalent tactics include rewrite and have. As a reminder, a rewrite tactic requires the user to indicate which equality in the current local or global context to apply and is akin to premise selection. The have tactic introduces an intermediate lemma in the proof and is the hardest to learn as the user usually uses some mathematical insight about the problem before conjecturing a statement. We currently do not synthesize arguments for have tactics, although our tool extracts such data. We believe a generative model for synthesizing them would be a great direction for future work. ", + "bbox": [ + 174, + 765, + 825, + 861 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/12d59a7edba237b79cf54954537d743cd7d2e1d7969be4e9d0b38c527e84958c.jpg", + "table_caption": [ + "Table 1: Training time speedups for the GRU model with state size of 128 on the position evaluation task obtained compared to an un-optimized baseline. ∗ indicates CPU and † indicates GPU. " + ], + "table_footnote": [], + "table_body": "
ModelBase*Embedding Sharing*Dynamic Batching*Both*Both†
Speedup (approx.)110×10×130×190×
", + "bbox": [ + 179, + 140, + 812, + 171 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/99adce68b3d8955c819a6394c3d94501d6ca410f2f85519ada62c2de36dd7717.jpg", + "table_caption": [ + "Table 2: Test accuracies for position evaluation (Pos) and tactic prediction (Tac). $\\dagger$ indicates kernellevel. $^ \\ddag$ indicates mid-level without implicit arguments. For tactic argument prediction, we report validation recall for models with a minimum precision of $1 0 \\%$ " + ], + "table_footnote": [], + "table_body": "
ModelPostPostTactTactTact t arguments
Constant53.6653.6644.7544.751
SVM57.3757.5248.9449.45=
GRU65.3065.7458.2357.7025.98
TreeLSTM68.4466.3060.6360.5523.91
", + "bbox": [ + 277, + 242, + 718, + 315 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Scaling to large proof trees In practice, ASTs can be on the order of thousands of nodes. We can apply two optimizations to scale to larger trees. The first optimization involves embedding sharing, where we memoize the embedding and the associated computation graph for any expression or subtree when it appears another time in the same proof state. For instance, if the context has terms $M _ { 1 } M _ { 2 }$ and $M _ { 1 }$ , the embedding for $M _ { 1 }$ is computed once. A single forward and backward pass is thus performed on this computation graph, with the gradients automatically accumulating from all places where the embedding was used. It thus helps save both memory and computation time of our models. The second optimization involves dynamic batching (Looks et al., 2017; Polosukhin & Zavershynskyi, 2018), which enables us to efficiently batch the computation of ops that perform the same operation albeit on different inputs, and thus make better use of the parallel acceleration provided by multi-core CPU’s and GPU’s. 5 ", + "bbox": [ + 173, + 344, + 825, + 497 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 1 shows the approximate speedups obtained from applying embedding sharing and dynamic batching to representing proof states. Note that the GPU speedup will increase with larger models. ", + "bbox": [ + 173, + 505, + 823, + 534 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7.1 POSITION EVALUATION AND TACTIC PREDICTION ", + "text_level": 1, + "bbox": [ + 178, + 554, + 555, + 569 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We use the interpreter-inspired embeddings to embed proof states into $\\mathbb { R } ^ { D }$ , and then aim to train models for position evaluation and tactic prediction tasks. For the position evaluation task, we binned the target predictions into $K = 3$ classes—close ( $\\mathit { \\Theta } _ { \\prec } 5$ steps), medium (between $6 - 1 9$ steps, inclusive), and far $> 2 0$ steps), and perform a simple three way classification. ", + "bbox": [ + 174, + 582, + 825, + 637 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The tactic prediction is more complicated. First, we group tactics into equivalence classes if they perform the same mathematical operation. For example, we group the tactics reflexivity and done together because they are applied at the end of proofs to show that a statement is trivially true. Second, we now also have tactic arguments. We train two models, one that predicts only the tactic and another that additionally predicts the arguments. The first is a 23 way classification problem. For the second, the arguments could be (1) a term from the local context, (2) a term from the global context (e.g., a lemma), or (3) a term created by the human user. As arguments in the third category can be any Coq term, the space of arguments is potentially infinite. As a reminder, the data set makes extensive use of have tactics—in essence, this would require the model to conjecture a statement. For our baseline models, we focus on arguments in the first category and predict the presence or absence of each term in the context at any position in the arguments. For each term, we use the final hidden state and the embedding of each term followed by a linear layer to produce a two way prediction. Note that the distribution of labels is skewed towards the absent category. Thus, we are more interested in the precision-recall curve. We weigh the cross-entropy loss higher for presence class, and also try to balance the distribution of labels by randomly sampling only a subset of the negative class at training time. ", + "bbox": [ + 174, + 645, + 825, + 867 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Results We first start by training a simple SVM (Cortes & Vapnik, 1995) on a set of heuristic features like context size, goal size, number of hypothesis in context and the smallest edit distance between a hypothesis and the context. The SVM performs better than the constant baseline of guessing the most common class. We then train RNN models to utilise our embedding strategy. We train GRU (Cho et al., 2014) and TreeLSTM (Tai et al., 2015) models using mini-batches of 32 proof states, set the RNN state size to 128, and use the Adam optimizer Kingma & Ba (2015) with a learning rate of 0.001. We use input (Srivastava et al., 2014) and weight (Merity et al., 2017) dropout with a rate of 0.1 for the TreeLSTM models (higher rates led to much slower learning). All neural net models were trained using PyTorch (Paszke et al., 2017). Table 2 show the results for the tasks. We were able to improve upon the SVM baseline, which indicates that it is possible to learn useful representations using the human supervision data and utilizing our proof state embeddings. We then experiment with removing the bookkeeping aspects of the prover by switching from kernel level to mid level proof states without implicit arguments. We obtain similar accuracies, indicating that most of that data is redundant. ", + "bbox": [ + 174, + 103, + 825, + 297 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "8 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 318, + 343, + 334 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The level of abstraction and representation of proofs that learning is applied to are salient points of comparison between work on learning and theorem proving. These choices inform the setup and challenges associated with the learning problem. ", + "bbox": [ + 176, + 349, + 825, + 392 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "As in our work, there are systems that experiment with learning in ITPs. Duncan (2002) explores how to learn (user-defined) tactics from a corpus of proofs in Isabelle so that they can be applied to future proofs. ML4PG (Komendantskaya et al., 2012) interfaces to ITPs at the level of the user interface for entering proof scripts. Thus, ML4PG is applicable to multiple ITPs although it obtains less granular proof states. Holphrasm (Whalen, 2016) uses string encodings of proof states and focuses on tactic argument synthesis (there is essentially only one tactic in the underlying ITP MetaMath (Megill, 2007)). HolStep (Kaliszyk et al., 2017) addresses premise selection using stringlevel encodings of proof states. Wang et al. (2017) extend the HolStep work to show the advantage of using a DeBruijn representation of proof terms as opposed to string-level encodings for premise selection. The structured representation provided by GamePad would support experimenting with such extensions. Gauthier et al. (2017) explores learning tactic-level proof search in Isabelle using hand-crafted features on string encodings of proof states. It would be interesting to experiment with their algorithm to our algebraic rewrite problem. Nagashima & He (2018) looks at explainable tactic prediction in Isabelle. ", + "bbox": [ + 174, + 398, + 825, + 593 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Other approaches focus on automated theorem provers, which are designed to prove theorems with no human interaction. Irving et al. (2016) describes the premise selection problem and trains neural network models on proof traces obtained from applying E (Schulz, 2002) to the Mizar Mathematical Corpus. Loos et al. (2017), in addition to addressing premise selection, also address a clause selection task by applying neural network models to this problem in E. Kaliszyk & Urban (2014) demonstrate that similar learning based methods can prove $3 9 \\%$ of the lemmas in the Flyspeck project (Kaliszyk & Urban, 2014). ", + "bbox": [ + 174, + 599, + 825, + 698 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Another take on learning and theorem proving is to replace an entire theorem proving (sub)routine with a learned algorithm instead of using learning for heuristics. For instance, end-to-end differentiable proving (Rocktaschel & Riedel, 2017) replaces traditional ¨ unification with a trained neural network and demonstrates the efficacy of this approach for knowledge base completion. Neurosat (Selsam et al., 2018) applies a neural network model to predict satisfaction problems and shows how to recover a satisfying assignment. ", + "bbox": [ + 174, + 704, + 823, + 787 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "9 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 809, + 318, + 825 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work, we look at theorem proving problem through the lens of a system that enables learning with proofs constructed with human supervision. We highlight three key aspects of the problem at this level. The first concerns obtaining inputs to a learning algorithm that approximate the level of abstraction faced by a human prover. For this, we use an ITP, as it retains aspects of human supervision. GamePad preserves the structure of the proofs (e.g., annotations regarding implicit arguments) so they can be used for building models. The second involves building models that employ the game-like structure of ITP proofs. Here, we experiment with tactic prediction for toy and real world data sets. Finally, as a consequence of theorem proving at a higher-level (compared to SMT solvers), we will need to be careful to distinguish the syntax from the semantics of terms. Our current approach is to provide structured representations of terms so that more semantic structure can be exploited. While our results are preliminary, our hope is that GamePad provides an accessible starting point to explore the application of machine learning in the context of interactive theorem proving. ", + "bbox": [ + 174, + 840, + 823, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "10 FUTURE WORK ", + "text_level": 1, + "bbox": [ + 176, + 223, + 344, + 239 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We end by suggesting a few avenues for extending our work. The first concerns the design of new benchmarks for human-level proofs. In this paper, we designed a relatively simple algebraic rewrite problem to test the system end-to-end. Designing more difficult problems that still admit tractable learning (such as solving infinite sums or integrals) would be a great direction for future work. Note that you can define a new domain inside Coq and use GamePad to build provers that learn from example proofs. A second direction concerns building models that conjecture and explore the space of true statements, in addition to proving statements. This is particularly important in synthesizing arguments to tactics like have. Currently, we only predict the tactic identifiers and do not synthesize the entire term. Building a generative model would be a great next step. Lastly, it would be interesting to see if using end-to-end training with reinforcement learning and utilizing Monte-Carlo tree search to efficiently explore the search space can be effectively applied to humanlevel proofs.6 ", + "bbox": [ + 174, + 256, + 825, + 422 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "11 ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 178, + 445, + 408, + 460 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We would like to thank Diederik Kingma, Tim Salimans, and Geoffrey Irving for reviewing initial drafts of the work. We thank Daniel Selsam for suggesting that we consider removing implicit arguments, and Jonathan Cai for discussions about the toy problem. Daniel Huang was supported by DARPA FA8750-17-2-0091. 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Interactive theorem provers such as Coq enable users to construct", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 273, + 469, + 286 + ], + "spans": [ + { + "bbox": [ + 141, + 273, + 469, + 286 + ], + "score": 1.0, + "content": "machine-checkable proofs in a step-by-step manner. Hence, they provide an op-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 285, + 470, + 296 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 470, + 296 + ], + "score": 1.0, + "content": "portunity to explore theorem proving with human supervision. We use GamePad", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 295, + 469, + 307 + ], + "spans": [ + { + "bbox": [ + 141, + 295, + 469, + 307 + ], + "score": 1.0, + "content": "to synthesize proofs for a simple algebraic rewrite problem and train baseline mod-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 306, + 469, + 318 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 469, + 318 + ], + "score": 1.0, + "content": "els for a formalization of the Feit-Thompson theorem. We address position evalu-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 318, + 469, + 330 + ], + "spans": [ + { + "bbox": [ + 141, + 318, + 469, + 330 + ], + "score": 1.0, + "content": "ation (i.e., predict the number of proof steps left) and tactic prediction (i.e., predict", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 328, + 465, + 342 + ], + "spans": [ + { + "bbox": [ + 141, + 328, + 465, + 342 + ], + "score": 1.0, + "content": "the next proof step) tasks, which arise naturally in tactic-based theorem proving.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 362, + 206, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 208, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 208, + 378 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "score": 1.0, + "content": "Theorem proving is a challenging AI task that involves symbolic reasoning (e.g., SMT", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "solvers (De Moura & Bjørner, 2008)) and intuition guided search. Recent work (Irving et al., 2016;", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "score": 1.0, + "content": "Loos et al., 2017; Kaliszyk et al., 2017) has shown the promise of applying deep learning techniques", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "in this domain, primarily on tasks useful for automated theorem provers (e.g., premise selection)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 445 + ], + "score": 1.0, + "content": "which operate with little to no human supervision. In this work, we aim to move closer to learning", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 443, + 296, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 296, + 456 + ], + "score": 1.0, + "content": "on proofs constructed with human supervision.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "We look at theorem proving in the realm of formal proofs. A formal proof is systematically derived", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "in a formal system, which makes it possible to algorithmically (i.e., with a computer) check these", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "proofs for correctness. Thus, formal proofs provide perfect learning signal—theorem statements", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 493, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 504 + ], + "score": 1.0, + "content": "and proofs are unambiguous. Human mathematicians usually do not write proofs in this style, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "instead, construct and communicate proofs in natural language. Although the form and level of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 514, + 464, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 464, + 526 + ], + "score": 1.0, + "content": "detail involved in each kind of proof differ, the logical content is similar in both contexts.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "Our work focuses on interactive theorem provers (ITPs), which are software tools that enable human", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 542, + 504, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 504, + 554 + ], + "score": 1.0, + "content": "users to construct formal proofs. 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Human mathematicians usually do not write proofs in this style, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "instead, construct and communicate proofs in natural language. Although the form and level of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 514, + 464, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 464, + 526 + ], + "score": 1.0, + "content": "detail involved in each kind of proof differ, the logical content is similar in both contexts.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 459, + 506, + 526 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "Our work focuses on interactive theorem provers (ITPs), which are software tools that enable human", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 542, + 504, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 504, + 554 + ], + "score": 1.0, + "content": "users to construct formal proofs. ITPs have at least two features that make them compelling environ-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 552, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 567 + ], + "score": 1.0, + "content": "ments for exploring the application of learning techniques to theorem proving. First and foremost,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "ITPs provide full-fledged programmable environments. Consequently, any machine learning infras-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "tructure built for an ITP can be reused across any problem domain crafted to study an aspect of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "learning and theorem proving. Second, the proofs are constructed by humans, and thus, have the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "constraint that they must be relatively human-understandable. Hence, ITPs provide access to large", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "amounts of supervised data (i.e., expert-constructed proofs of theorems that are mathematically in-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "teresting). For example, ITPs have been used to build and check the proofs of large mathematical", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "theorems such as the Feit-Thompson theorem (Gonthier et al., 2013) and provide provable guaran-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 641, + 468, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 468, + 654 + ], + "score": 1.0, + "content": "tees on complex pieces of software such as the CompCert C compiler (Leroy et al., 2012).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 531, + 506, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 657, + 503, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "score": 1.0, + "content": "We introduce a system called GamePad1 that exposes parts of the Coq ITP to enable machine learn-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "ing tasks and explore a few use cases. We focus on the Coq proof assistant for two reasons. 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A proof state consists of a context (pink rectangles) and a goal (white rectangles).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 232, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 505, + 247 + ], + "score": 1.0, + "content": "The initial proof state has as its goal the statement we are trying to prove and an empty context. The", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 245, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 257 + ], + "score": 1.0, + "content": "arrows indicate what tactic the prover used. The final states of the proof are indicated by the red", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 255, + 463, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 463, + 268 + ], + "score": 1.0, + "content": "circles and can be transitioned to only when the goal in the previous state is trivially true.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 505, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "Coq is a mature system with an active developer community that has been used to formalize non-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "trivial theorems, including Feit-Thompson and CompCert. Second, Coq supports the extraction of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "verified software. Consequently, one can prove that a program is correct and then run the verified", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 322, + 452, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 452, + 334 + ], + "score": 1.0, + "content": "program. The ease of extraction makes Coq a popular choice for program verification.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "Our contributions are the following. First, we introduce GamePad, which provides a structured", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 350, + 507, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 507, + 363 + ], + "score": 1.0, + "content": "Python representation of Coq proofs (Section 3), including all the proof states encountered in a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "proof, the steps taken, and expression abstract syntax trees (ASTs). The tool also enables lightweight", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "score": 1.0, + "content": "interaction with Coq so that it can be used to dynamically build proofs (e.g., used as an environment", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 382, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 396 + ], + "score": 1.0, + "content": "for reinforcement learning). Tasks that can leverage this structured representation (Section 4) include", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "position evaluation (i.e., predict the number of proof steps left) and tactic prediction (i.e., predict", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "the next proof step to take). We also discuss how we can use the structured representation to embed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 414, + 507, + 429 + ], + "spans": [ + { + "bbox": [ + 104, + 414, + 175, + 429 + ], + "score": 1.0, + "content": "proof states into", + "type": "text" + }, + { + "bbox": [ + 175, + 415, + 191, + 426 + ], + "score": 0.88, + "content": "\\bar { \\mathbb { R } } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 414, + 507, + 429 + ], + "score": 1.0, + "content": "(Section 5). Second, we demonstrate the synthesis of Coq proof scripts that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 425, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 440 + ], + "score": 1.0, + "content": "makes use of a tactic prediction model for a hand-crafted algebraic rewriting problem (Section 6).", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "Third and finally, we apply baseline models for position evaluation and tactic prediction to the Feit-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 448, + 392, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 392, + 461 + ], + "score": 1.0, + "content": "Thompson formalization using data extracted by GamePad (Section 7).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 108, + 465, + 504, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 479 + ], + "score": 1.0, + "content": "The code for GamePad, as well as the associated data sets, models and results, are open source on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 475, + 345, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 345, + 489 + ], + "score": 1.0, + "content": "GitHub at https://github.com/ml4tp/gamepad.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 108, + 504, + 200, + 517 + ], + "lines": [ + { + "bbox": [ + 104, + 502, + 201, + 520 + ], + "spans": [ + { + "bbox": [ + 104, + 502, + 201, + 520 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 529, + 505, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "To provide context for the rest of this paper, we begin by illustrating the formal proof process and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "how it can be modeled as a game. We then walk through a simple example of a proof in Coq and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 433, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 433, + 565 + ], + "score": 1.0, + "content": "end by summarizing the constructs that we will encounter in the rest of the paper.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 575, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 506, + 588 + ], + "score": 1.0, + "content": "Theorem proving process When humans write a pencil-paper proof, we typically maintain a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "mental and/or written scratchpad that keeps track of (1) what we need to show and (2) the facts", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "we currently have at our disposal. As the proof progresses, the items in this scratchpad changes.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 610, + 504, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 504, + 621 + ], + "score": 1.0, + "content": "For instance, we might transform what we need to show (e.g., it suffices to show another statement)", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "and/or discover new facts (e.g., we derive an additional fact as a consequence of known facts). When", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "we move from the pencil-paper setting to an ITP such as Coq, we will still have such a scratchpad,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "score": 1.0, + "content": "but use the ITP’s term language to express mathematical statements and the rules of the ITP’s formal", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 654, + 257, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 257, + 665 + ], + "score": 1.0, + "content": "system to construct the proof instead.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "Formal theorem proving as a game A game serves as useful mental model of the proving pro-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "cess. The state of the game is a proof state, which looks like the scratchpad and is comprised of (1) a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "goal (expressed in the term language) stating what we need to prove and (2) a context containing the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "assumptions (also expressed in the term language). The starting state has as its goal the statement", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "of a theorem and an empty context, while a final state has as its goal a statement that is trivially", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 87, + 507, + 196 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 87, + 507, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 87, + 507, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 87, + 507, + 196 + ], + "score": 0.853, + "type": "image", + "image_path": "1fec650b161e90747296d39eababa690f2aa0ff7a0fc43bc4747eed66f79eeba.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 87, + 507, + 123.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 123.33333333333334, + 507, + 159.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 159.66666666666669, + 507, + 196.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 212, + 505, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "Figure 1: A proof script in Coq (left) and the resulting proof states, proof steps, and the complete", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 223, + 504, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 504, + 235 + ], + "score": 1.0, + "content": "proof tree (right). A proof state consists of a context (pink rectangles) and a goal (white rectangles).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 232, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 505, + 247 + ], + "score": 1.0, + "content": "The initial proof state has as its goal the statement we are trying to prove and an empty context. The", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 245, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 257 + ], + "score": 1.0, + "content": "arrows indicate what tactic the prover used. The final states of the proof are indicated by the red", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 255, + 463, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 463, + 268 + ], + "score": 1.0, + "content": "circles and can be transitioned to only when the goal in the previous state is trivially true.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 505, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "Coq is a mature system with an active developer community that has been used to formalize non-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "trivial theorems, including Feit-Thompson and CompCert. Second, Coq supports the extraction of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "verified software. Consequently, one can prove that a program is correct and then run the verified", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 322, + 452, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 452, + 334 + ], + "score": 1.0, + "content": "program. The ease of extraction makes Coq a popular choice for program verification.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 289, + 506, + 334 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "Our contributions are the following. First, we introduce GamePad, which provides a structured", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 350, + 507, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 507, + 363 + ], + "score": 1.0, + "content": "Python representation of Coq proofs (Section 3), including all the proof states encountered in a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "proof, the steps taken, and expression abstract syntax trees (ASTs). The tool also enables lightweight", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "score": 1.0, + "content": "interaction with Coq so that it can be used to dynamically build proofs (e.g., used as an environment", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 382, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 396 + ], + "score": 1.0, + "content": "for reinforcement learning). Tasks that can leverage this structured representation (Section 4) include", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "position evaluation (i.e., predict the number of proof steps left) and tactic prediction (i.e., predict", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "the next proof step to take). We also discuss how we can use the structured representation to embed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 414, + 507, + 429 + ], + "spans": [ + { + "bbox": [ + 104, + 414, + 175, + 429 + ], + "score": 1.0, + "content": "proof states into", + "type": "text" + }, + { + "bbox": [ + 175, + 415, + 191, + 426 + ], + "score": 0.88, + "content": "\\bar { \\mathbb { R } } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 414, + 507, + 429 + ], + "score": 1.0, + "content": "(Section 5). Second, we demonstrate the synthesis of Coq proof scripts that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 425, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 440 + ], + "score": 1.0, + "content": "makes use of a tactic prediction model for a hand-crafted algebraic rewriting problem (Section 6).", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "Third and finally, we apply baseline models for position evaluation and tactic prediction to the Feit-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 448, + 392, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 392, + 461 + ], + "score": 1.0, + "content": "Thompson formalization using data extracted by GamePad (Section 7).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 339, + 507, + 461 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 465, + 504, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 479 + ], + "score": 1.0, + "content": "The code for GamePad, as well as the associated data sets, models and results, are open source on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 475, + 345, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 345, + 489 + ], + "score": 1.0, + "content": "GitHub at https://github.com/ml4tp/gamepad.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 106, + 463, + 505, + 489 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 504, + 200, + 517 + ], + "lines": [ + { + "bbox": [ + 104, + 502, + 201, + 520 + ], + "spans": [ + { + "bbox": [ + 104, + 502, + 201, + 520 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 529, + 505, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "To provide context for the rest of this paper, we begin by illustrating the formal proof process and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "how it can be modeled as a game. We then walk through a simple example of a proof in Coq and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 433, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 433, + 565 + ], + "score": 1.0, + "content": "end by summarizing the constructs that we will encounter in the rest of the paper.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 529, + 505, + 565 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 575, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 506, + 588 + ], + "score": 1.0, + "content": "Theorem proving process When humans write a pencil-paper proof, we typically maintain a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "mental and/or written scratchpad that keeps track of (1) what we need to show and (2) the facts", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "we currently have at our disposal. As the proof progresses, the items in this scratchpad changes.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 610, + 504, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 504, + 621 + ], + "score": 1.0, + "content": "For instance, we might transform what we need to show (e.g., it suffices to show another statement)", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "and/or discover new facts (e.g., we derive an additional fact as a consequence of known facts). When", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "we move from the pencil-paper setting to an ITP such as Coq, we will still have such a scratchpad,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "score": 1.0, + "content": "but use the ITP’s term language to express mathematical statements and the rules of the ITP’s formal", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 654, + 257, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 257, + 665 + ], + "score": 1.0, + "content": "system to construct the proof instead.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 576, + 506, + 665 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "Formal theorem proving as a game A game serves as useful mental model of the proving pro-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "cess. The state of the game is a proof state, which looks like the scratchpad and is comprised of (1) a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "goal (expressed in the term language) stating what we need to prove and (2) a context containing the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "assumptions (also expressed in the term language). The starting state has as its goal the statement", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "of a theorem and an empty context, while a final state has as its goal a statement that is trivially", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "true (i.e., definitionally equal) given the context. The aim of a prover is to transform the starting", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "state into a collection of final states using only logically valid transitions, which can affect both the", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "context and the goal. It is possible to obtain multiple final states because some transitions may split", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "a goal into multiple subgoals. For a given (sub)goal, a successful proof corresponds to a proof tree,", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 363, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 363, + 139 + ], + "score": 1.0, + "content": "where the root is the start state and all the leaves are final states.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 677, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "true (i.e., definitionally equal) given the context. The aim of a prover is to transform the starting", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "state into a collection of final states using only logically valid transitions, which can affect both the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "context and the goal. It is possible to obtain multiple final states because some transitions may split", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "a goal into multiple subgoals. For a given (sub)goal, a successful proof corresponds to a proof tree,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 363, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 363, + 139 + ], + "score": 1.0, + "content": "where the root is the start state and all the leaves are final states.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 505, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 150, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 439, + 164 + ], + "score": 1.0, + "content": "An example Coq proof Consider showing that adding 0 to any natural number", + "type": "text" + }, + { + "bbox": [ + 440, + 153, + 447, + 161 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 150, + 458, + 164 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 459, + 153, + 466, + 161 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 150, + 505, + 164 + ], + "score": 1.0, + "content": "itself. A", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 162, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 104, + 162, + 312, + 174 + ], + "score": 1.0, + "content": "paper-pencil proof might proceed by induction on", + "type": "text" + }, + { + "bbox": [ + 313, + 164, + 320, + 172 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 162, + 506, + 174 + ], + "score": 1.0, + "content": ", where we check that the result holds on the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 166, + 186 + ], + "score": 1.0, + "content": "base case (i.e.,", + "type": "text" + }, + { + "bbox": [ + 166, + 173, + 192, + 183 + ], + "score": 0.86, + "content": "n = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 172, + 306, + 186 + ], + "score": 1.0, + "content": ") and the inductive case (i.e.,", + "type": "text" + }, + { + "bbox": [ + 306, + 173, + 350, + 184 + ], + "score": 0.86, + "content": "n = n + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 172, + 505, + 186 + ], + "score": 1.0, + "content": ". We can carry out a similar process in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 505, + 197 + ], + "score": 1.0, + "content": "Coq, using a sequence of commands called tactics in Coq’s tactic language Ltac that indicate what", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 207 + ], + "score": 1.0, + "content": "proofs steps to take. For instance, we can use the code induction n; simpl. to start a proof", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "by induction (on n) in Coq. The expressions induction and simpl are tactics (of arity 1 and 0", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 229 + ], + "score": 1.0, + "content": "respectively), the semicolon ; sequences two tactics, and the period . signals the end of one proof", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "step. The effect of the entire command (up to the period) is to run induction and then simplify all", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "resulting proof states and is shown in Figure 1 via the green arrows, which connect proof states 1", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 263 + ], + "score": 1.0, + "content": "with 2 and 1 with 3. After we perform the induction, we see that proof state 2 contains the base case", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "and proof state 3 contains the inductive case. The inductive case has an inductive hypothesis in the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 504, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 473, + 285 + ], + "score": 1.0, + "content": "context. Proofs in Coq are finished when the goal is trivially true given the context (e.g.,", + "type": "text" + }, + { + "bbox": [ + 473, + 272, + 504, + 282 + ], + "score": 0.86, + "content": "0 ~ = ~ 0", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 282, + 492, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 492, + 295 + ], + "score": 1.0, + "content": "given an empty context). The collection of proof states and tactic invocations forms a proof tree.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 504, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 504, + 319 + ], + "score": 1.0, + "content": "Formal definitions Coq provides three equi-expressive term languages that encode logical for-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 255, + 331 + ], + "score": 1.0, + "content": "mulas such as forall n: nat,", + "type": "text" + }, + { + "bbox": [ + 256, + 318, + 312, + 328 + ], + "score": 0.88, + "content": "{ \\mathrm { ~ ~ n ~ } } + 0 = { \\mathrm { ~ ~ n ~ } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 317, + 505, + 331 + ], + "score": 1.0, + "content": ". They differ in the amount of annotation they", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "support. Coq’s user-facing term language Gallina supports type-inference and other forms of no-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "tational convenience (e.g., syntactic sugar and extensible parsing). Coq’s mid-level term language", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "removes all forms of notational convenience. Finally, Coq’s kernel-level term language instantiates", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "all types and is the level at which proofs are checked. Thus, the mid-level and kernel-level languages", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "are similar, with the exception that the mid-level term language has a notion of an implicit argument.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 504, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 326, + 402 + ], + "score": 1.0, + "content": "Implicit arguments occur in application constructs—M", + "type": "text" + }, + { + "bbox": [ + 327, + 389, + 374, + 401 + ], + "score": 0.86, + "content": "M _ { 1 } \\dots M _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 389, + 438, + 402 + ], + "score": 1.0, + "content": "(apply function", + "type": "text" + }, + { + "bbox": [ + 438, + 389, + 450, + 399 + ], + "score": 0.8, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "to arguments", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 399, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 107, + 401, + 155, + 412 + ], + "score": 0.89, + "content": "M _ { 1 } \\ldots M _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 399, + 265, + 413 + ], + "score": 1.0, + "content": "at the kernel-level versus", + "type": "text" + }, + { + "bbox": [ + 266, + 401, + 330, + 412 + ], + "score": 0.92, + "content": "M M _ { 1 } ^ { \\iota _ { 1 } } \\dots M _ { n } ^ { \\iota _ { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 399, + 432, + 413 + ], + "score": 1.0, + "content": "at the mid-level, where", + "type": "text" + }, + { + "bbox": [ + 433, + 402, + 441, + 411 + ], + "score": 0.84, + "content": "\\iota _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 399, + 506, + 413 + ], + "score": 1.0, + "content": "marks implicit", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "versus non-implicit arguments. Implicit arguments take care of the book-keeping aspects of a formal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 422, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 436 + ], + "score": 1.0, + "content": "proof such as the types of all terms involved; a human prover would usually omit them in a paper-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "pencil proof. Hence, learning with mid-level terms without implicit arguments is closer to human-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 443, + 469, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 469, + 458 + ], + "score": 1.0, + "content": "level proving, whereas learning with kernel-level terms is closer to machine-level proving.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 218, + 473 + ], + "score": 1.0, + "content": "Coq represents a proof state", + "type": "text" + }, + { + "bbox": [ + 218, + 462, + 231, + 471 + ], + "score": 0.55, + "content": "P S", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 461, + 270, + 473 + ], + "score": 1.0, + "content": "as a tuple", + "type": "text" + }, + { + "bbox": [ + 270, + 461, + 310, + 473 + ], + "score": 0.93, + "content": "\\langle \\Gamma , M , n \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 461, + 380, + 473 + ], + "score": 1.0, + "content": "of a local context", + "type": "text" + }, + { + "bbox": [ + 381, + 462, + 388, + 471 + ], + "score": 0.52, + "content": "\\Gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 461, + 437, + 473 + ], + "score": 1.0, + "content": ", a goal term", + "type": "text" + }, + { + "bbox": [ + 437, + 462, + 449, + 471 + ], + "score": 0.8, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 461, + 505, + 473 + ], + "score": 1.0, + "content": ", and a unique", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 191, + 484 + ], + "score": 1.0, + "content": "proof state identifier", + "type": "text" + }, + { + "bbox": [ + 191, + 474, + 199, + 482 + ], + "score": 0.63, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 472, + 246, + 484 + ], + "score": 1.0, + "content": ". A context", + "type": "text" + }, + { + "bbox": [ + 247, + 473, + 255, + 482 + ], + "score": 0.66, + "content": "\\Gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "provides a list of identifiers and their types that expresses all", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 414, + 496 + ], + "score": 1.0, + "content": "the current assumptions. For instance, the proof state with identifier 3 has", + "type": "text" + }, + { + "bbox": [ + 414, + 484, + 445, + 493 + ], + "score": 0.85, + "content": "\\Gamma = \\mathrm { ~ n ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 482, + 506, + 496 + ], + "score": 1.0, + "content": ": nat, IHn :", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 162, + 505 + ], + "score": 0.86, + "content": "\\mathrm { ~ ~ n ~ } = \\mathrm { ~ ~ n ~ } + \\mathrm { ~ ~ 0 ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 494, + 505, + 507 + ], + "score": 1.0, + "content": ". A tactic invocation creates edges between the appropriate proof states, which forms", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 505, + 157, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 157, + 517 + ], + "score": 1.0, + "content": "a proof tree.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 521, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "Proof states are interpreted in the presence of global state. The global state contains the interpre-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "tation of constructs such as constants and constructors that have global scope. It also contains the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "interpretation of constructs that have stateful effects such as existentials. In particular, it is possi-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "ble to posit the existence of an object in one proof state and then proceed to perform case analysis", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "resulting in multiple proof states that share the same existential. The GamePad tool (Section 3) ex-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "poses the constructs described above (with the exception of Gallina terms) as Python data structures,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 588, + 263, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 263, + 600 + ], + "score": 1.0, + "content": "which we can use for building models.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 108, + 615, + 211, + 628 + ], + "lines": [ + { + "bbox": [ + 104, + 613, + 214, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 613, + 214, + 631 + ], + "score": 1.0, + "content": "3 GAMEPAD TOOL", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 641, + 503, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "We briefly describe the GamePad tool with an emphasis on the data that it exposes and any other", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 652, + 306, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 306, + 665 + ], + "score": 1.0, + "content": "design decisions that affect the modeling process.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Obtaining proof traces We obtain the proof trace by implementing a patch to Coq (version 8.6.1)", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "that instruments the Coq Ltac interpreter to log the intermediate proof states. The patch also supports", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "SSreflect (Gonthier & Stephane Le, 2009), a popular Coq plugin for writing proofs about mathemat- ´", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "ics. Importantly, the patch does not touch the Coq proof checker, and hence, does not affect the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 362, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 362, + 734 + ], + "score": 1.0, + "content": "critical part of the system that verifies the correctness of proofs.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 81, + 506, + 139 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 505, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 150, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 439, + 164 + ], + "score": 1.0, + "content": "An example Coq proof Consider showing that adding 0 to any natural number", + "type": "text" + }, + { + "bbox": [ + 440, + 153, + 447, + 161 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 150, + 458, + 164 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 459, + 153, + 466, + 161 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 150, + 505, + 164 + ], + "score": 1.0, + "content": "itself. A", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 162, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 104, + 162, + 312, + 174 + ], + "score": 1.0, + "content": "paper-pencil proof might proceed by induction on", + "type": "text" + }, + { + "bbox": [ + 313, + 164, + 320, + 172 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 162, + 506, + 174 + ], + "score": 1.0, + "content": ", where we check that the result holds on the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 166, + 186 + ], + "score": 1.0, + "content": "base case (i.e.,", + "type": "text" + }, + { + "bbox": [ + 166, + 173, + 192, + 183 + ], + "score": 0.86, + "content": "n = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 172, + 306, + 186 + ], + "score": 1.0, + "content": ") and the inductive case (i.e.,", + "type": "text" + }, + { + "bbox": [ + 306, + 173, + 350, + 184 + ], + "score": 0.86, + "content": "n = n + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 172, + 505, + 186 + ], + "score": 1.0, + "content": ". We can carry out a similar process in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 505, + 197 + ], + "score": 1.0, + "content": "Coq, using a sequence of commands called tactics in Coq’s tactic language Ltac that indicate what", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 207 + ], + "score": 1.0, + "content": "proofs steps to take. For instance, we can use the code induction n; simpl. to start a proof", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "by induction (on n) in Coq. The expressions induction and simpl are tactics (of arity 1 and 0", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 229 + ], + "score": 1.0, + "content": "respectively), the semicolon ; sequences two tactics, and the period . signals the end of one proof", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "step. The effect of the entire command (up to the period) is to run induction and then simplify all", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "resulting proof states and is shown in Figure 1 via the green arrows, which connect proof states 1", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 263 + ], + "score": 1.0, + "content": "with 2 and 1 with 3. After we perform the induction, we see that proof state 2 contains the base case", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "and proof state 3 contains the inductive case. The inductive case has an inductive hypothesis in the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 504, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 473, + 285 + ], + "score": 1.0, + "content": "context. Proofs in Coq are finished when the goal is trivially true given the context (e.g.,", + "type": "text" + }, + { + "bbox": [ + 473, + 272, + 504, + 282 + ], + "score": 0.86, + "content": "0 ~ = ~ 0", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 282, + 492, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 492, + 295 + ], + "score": 1.0, + "content": "given an empty context). The collection of proof states and tactic invocations forms a proof tree.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 150, + 506, + 295 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 504, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 504, + 319 + ], + "score": 1.0, + "content": "Formal definitions Coq provides three equi-expressive term languages that encode logical for-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 255, + 331 + ], + "score": 1.0, + "content": "mulas such as forall n: nat,", + "type": "text" + }, + { + "bbox": [ + 256, + 318, + 312, + 328 + ], + "score": 0.88, + "content": "{ \\mathrm { ~ ~ n ~ } } + 0 = { \\mathrm { ~ ~ n ~ } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 317, + 505, + 331 + ], + "score": 1.0, + "content": ". They differ in the amount of annotation they", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "support. Coq’s user-facing term language Gallina supports type-inference and other forms of no-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "tational convenience (e.g., syntactic sugar and extensible parsing). Coq’s mid-level term language", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "removes all forms of notational convenience. Finally, Coq’s kernel-level term language instantiates", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "all types and is the level at which proofs are checked. Thus, the mid-level and kernel-level languages", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "are similar, with the exception that the mid-level term language has a notion of an implicit argument.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 307, + 505, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 504, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 326, + 402 + ], + "score": 1.0, + "content": "Implicit arguments occur in application constructs—M", + "type": "text" + }, + { + "bbox": [ + 327, + 389, + 374, + 401 + ], + "score": 0.86, + "content": "M _ { 1 } \\dots M _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 389, + 438, + 402 + ], + "score": 1.0, + "content": "(apply function", + "type": "text" + }, + { + "bbox": [ + 438, + 389, + 450, + 399 + ], + "score": 0.8, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "to arguments", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 399, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 107, + 401, + 155, + 412 + ], + "score": 0.89, + "content": "M _ { 1 } \\ldots M _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 399, + 265, + 413 + ], + "score": 1.0, + "content": "at the kernel-level versus", + "type": "text" + }, + { + "bbox": [ + 266, + 401, + 330, + 412 + ], + "score": 0.92, + "content": "M M _ { 1 } ^ { \\iota _ { 1 } } \\dots M _ { n } ^ { \\iota _ { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 399, + 432, + 413 + ], + "score": 1.0, + "content": "at the mid-level, where", + "type": "text" + }, + { + "bbox": [ + 433, + 402, + 441, + 411 + ], + "score": 0.84, + "content": "\\iota _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 399, + 506, + 413 + ], + "score": 1.0, + "content": "marks implicit", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "versus non-implicit arguments. Implicit arguments take care of the book-keeping aspects of a formal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 422, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 436 + ], + "score": 1.0, + "content": "proof such as the types of all terms involved; a human prover would usually omit them in a paper-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "pencil proof. Hence, learning with mid-level terms without implicit arguments is closer to human-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 443, + 469, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 469, + 458 + ], + "score": 1.0, + "content": "level proving, whereas learning with kernel-level terms is closer to machine-level proving.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 389, + 506, + 458 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 218, + 473 + ], + "score": 1.0, + "content": "Coq represents a proof state", + "type": "text" + }, + { + "bbox": [ + 218, + 462, + 231, + 471 + ], + "score": 0.55, + "content": "P S", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 461, + 270, + 473 + ], + "score": 1.0, + "content": "as a tuple", + "type": "text" + }, + { + "bbox": [ + 270, + 461, + 310, + 473 + ], + "score": 0.93, + "content": "\\langle \\Gamma , M , n \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 461, + 380, + 473 + ], + "score": 1.0, + "content": "of a local context", + "type": "text" + }, + { + "bbox": [ + 381, + 462, + 388, + 471 + ], + "score": 0.52, + "content": "\\Gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 461, + 437, + 473 + ], + "score": 1.0, + "content": ", a goal term", + "type": "text" + }, + { + "bbox": [ + 437, + 462, + 449, + 471 + ], + "score": 0.8, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 461, + 505, + 473 + ], + "score": 1.0, + "content": ", and a unique", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 191, + 484 + ], + "score": 1.0, + "content": "proof state identifier", + "type": "text" + }, + { + "bbox": [ + 191, + 474, + 199, + 482 + ], + "score": 0.63, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 472, + 246, + 484 + ], + "score": 1.0, + "content": ". A context", + "type": "text" + }, + { + "bbox": [ + 247, + 473, + 255, + 482 + ], + "score": 0.66, + "content": "\\Gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "provides a list of identifiers and their types that expresses all", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 414, + 496 + ], + "score": 1.0, + "content": "the current assumptions. For instance, the proof state with identifier 3 has", + "type": "text" + }, + { + "bbox": [ + 414, + 484, + 445, + 493 + ], + "score": 0.85, + "content": "\\Gamma = \\mathrm { ~ n ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 482, + 506, + 496 + ], + "score": 1.0, + "content": ": nat, IHn :", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 162, + 505 + ], + "score": 0.86, + "content": "\\mathrm { ~ ~ n ~ } = \\mathrm { ~ ~ n ~ } + \\mathrm { ~ ~ 0 ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 494, + 505, + 507 + ], + "score": 1.0, + "content": ". A tactic invocation creates edges between the appropriate proof states, which forms", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 505, + 157, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 157, + 517 + ], + "score": 1.0, + "content": "a proof tree.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 461, + 506, + 517 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 521, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "Proof states are interpreted in the presence of global state. The global state contains the interpre-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "tation of constructs such as constants and constructors that have global scope. It also contains the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "interpretation of constructs that have stateful effects such as existentials. In particular, it is possi-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "ble to posit the existence of an object in one proof state and then proceed to perform case analysis", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "resulting in multiple proof states that share the same existential. The GamePad tool (Section 3) ex-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "poses the constructs described above (with the exception of Gallina terms) as Python data structures,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 588, + 263, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 263, + 600 + ], + "score": 1.0, + "content": "which we can use for building models.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 522, + 505, + 600 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 615, + 211, + 628 + ], + "lines": [ + { + "bbox": [ + 104, + 613, + 214, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 613, + 214, + 631 + ], + "score": 1.0, + "content": "3 GAMEPAD TOOL", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 641, + 503, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "We briefly describe the GamePad tool with an emphasis on the data that it exposes and any other", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 652, + 306, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 306, + 665 + ], + "score": 1.0, + "content": "design decisions that affect the modeling process.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 640, + 505, + 665 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Obtaining proof traces We obtain the proof trace by implementing a patch to Coq (version 8.6.1)", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "that instruments the Coq Ltac interpreter to log the intermediate proof states. The patch also supports", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "SSreflect (Gonthier & Stephane Le, 2009), a popular Coq plugin for writing proofs about mathemat- ´", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "ics. Importantly, the patch does not touch the Coq proof checker, and hence, does not affect the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 362, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 362, + 734 + ], + "score": 1.0, + "content": "critical part of the system that verifies the correctness of proofs.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 677, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "This implementation choice affords us flexibility in deciding what proof steps to consider atomic.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "As a reminder, the sequenced tactic induction n; simpl. (Section 2) can be considered as a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "single proof step (read up to the delimiting .) as in the example, but it is comprised of two primitive", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "score": 1.0, + "content": "tactics—induction and simpl. As the granularity of a proof step has consequences for setting", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "up a tactic prediction task (Section 4), we made the choice to obtain the finer-grained proof states", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "by modifying Coq directly while maintaining the mapping to the human-level proof script. Thus,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "the tool records proof states at the granularity of primitive tactics (i.e., breaks up ;) as well as at the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 260, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 260, + 172 + ], + "score": 1.0, + "content": "granularity of the proof script (i.e., .)", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 184, + 505, + 261 + ], + "lines": [ + { + "bbox": [ + 106, + 183, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 505, + 198 + ], + "score": 1.0, + "content": "Representing Coq proofs We expose the proof trace obtained from Coq as Python data structures,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "including Coq’s mid-level and kernel-level term languages, proof states, proof steps, and proof trees", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "so that they can be manipulated with arbitrary Python code. Consequently, we can build models", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "score": 1.0, + "content": "using this structure. For efficiency reasons, we represent Coq terms in a shared form. Representing", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "terms in a shared form is a necessary technique to scale compilers and ITPs to real-world programs", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "and proofs. In our setting, it is also essential to scaling training/inference to real-world data sets (see", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 152, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 152, + 262 + ], + "score": 1.0, + "content": "Section 7).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 319 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "Light-weight interaction The API provides a thin wrapper around coqtop, the Coq repl, which", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "can be used to interactively construct proof scripts from within Python (see Section 6). This compo-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 298, + 504, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 504, + 309 + ], + "score": 1.0, + "content": "nent mediates all interaction with Coq, including proof state parsing into GamePad’s representation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 308, + 272, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 272, + 320 + ], + "score": 1.0, + "content": "of Coq proofs and sending tactic actions.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 108, + 336, + 160, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 162, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 162, + 352 + ], + "score": 1.0, + "content": "4 TASKS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "In the theorem proving process, we encounter two important tasks: position evaluation and tactic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 240, + 387 + ], + "score": 1.0, + "content": "prediction. A position evaluator", + "type": "text" + }, + { + "bbox": [ + 240, + 374, + 249, + 384 + ], + "score": 0.82, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "tells how easy it is to prove a given proof state, while a tactic", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 145, + 398 + ], + "score": 1.0, + "content": "predictor", + "type": "text" + }, + { + "bbox": [ + 145, + 385, + 154, + 395 + ], + "score": 0.8, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "tells how to take an action that can make our proof state easier to prove. As a reminder,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "proof states are interpreted in the presence of global state, which we have left implicit here. Note", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 169, + 419 + ], + "score": 1.0, + "content": "that in general,", + "type": "text" + }, + { + "bbox": [ + 169, + 407, + 178, + 417 + ], + "score": 0.8, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 406, + 227, + 419 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 228, + 407, + 237, + 417 + ], + "score": 0.81, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "; a good tactic predictor would find it easier to prove a given proof", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 231, + 430 + ], + "score": 1.0, + "content": "state. Also, an action taken by", + "type": "text" + }, + { + "bbox": [ + 232, + 418, + 241, + 428 + ], + "score": 0.82, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 417, + 436, + 430 + ], + "score": 1.0, + "content": "can lead to multiple child proof states, and thus", + "type": "text" + }, + { + "bbox": [ + 436, + 418, + 445, + 428 + ], + "score": 0.83, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "must consider", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 428, + 264, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 264, + 442 + ], + "score": 1.0, + "content": "the provability of all child proof states.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 310, + 458 + ], + "score": 1.0, + "content": "In this work, we aim to learn parametric functions", + "type": "text" + }, + { + "bbox": [ + 311, + 444, + 329, + 456 + ], + "score": 0.9, + "content": "\\mathcal { P } ^ { \\mathcal { T } _ { h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 444, + 347, + 458 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 348, + 446, + 360, + 456 + ], + "score": 0.87, + "content": "\\mathcal { T } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 444, + 505, + 458 + ], + "score": 1.0, + "content": "using supervised learning on a data", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 216, + 469 + ], + "score": 1.0, + "content": "set of human proofs, where", + "type": "text" + }, + { + "bbox": [ + 216, + 457, + 227, + 468 + ], + "score": 0.89, + "content": "\\mathcal { T } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "is the human tactic predictor implicit in the data set. In the future, one", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 138, + 481 + ], + "score": 1.0, + "content": "can use", + "type": "text" + }, + { + "bbox": [ + 138, + 468, + 147, + 478 + ], + "score": 0.81, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 466, + 165, + 481 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 165, + 468, + 175, + 478 + ], + "score": 0.82, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 466, + 505, + 481 + ], + "score": 1.0, + "content": "within an end-to-end prover, and learn them directly using reinforcement learning.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "score": 1.0, + "content": "This requires the ability to manipulate proof states by sending actions, which is possible using the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 490, + 296, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 296, + 502 + ], + "score": 1.0, + "content": "light-weight interaction provided by GamePad.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "Position evaluation The goal of the position evaluation task is to predict the approximate number", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 481, + 537 + ], + "score": 1.0, + "content": "of steps required to finish a proof given an input proof state. We define this as the function", + "type": "text" + }, + { + "bbox": [ + 481, + 524, + 497, + 536 + ], + "score": 0.75, + "content": "\\mathcal { P } ^ { \\mathcal { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 524, + 506, + 537 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 535, + 504, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 182, + 548 + ], + "score": 0.93, + "content": "P S \\stackrel { - } { } \\{ 1 , \\stackrel { - } { \\cdot } \\cdot \\cdot , K \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 535, + 294, + 548 + ], + "score": 1.0, + "content": "for a given tactic predictor", + "type": "text" + }, + { + "bbox": [ + 294, + 537, + 303, + 546 + ], + "score": 0.8, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 535, + 493, + 548 + ], + "score": 1.0, + "content": ", where we bin the sizes of the proof trees into", + "type": "text" + }, + { + "bbox": [ + 493, + 536, + 504, + 546 + ], + "score": 0.77, + "content": "K", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "classes to make learning easy. Given a data set of steps taken by a human prover, we aim to learn", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 555, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 124, + 568 + ], + "score": 0.88, + "content": "\\mathcal { P } ^ { \\mathcal { T } _ { h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 555, + 233, + 573 + ], + "score": 1.0, + "content": "by supervised learning on", + "type": "text" + }, + { + "bbox": [ + 233, + 559, + 243, + 568 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 555, + 306, + 573 + ], + "score": 1.0, + "content": "training tuples", + "type": "text" + }, + { + "bbox": [ + 306, + 558, + 378, + 570 + ], + "score": 0.93, + "content": "\\{ ( s _ { n } , d _ { n } ) \\} _ { 1 \\leq n \\leq N }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 555, + 430, + 573 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 430, + 560, + 442, + 569 + ], + "score": 0.86, + "content": "s _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 555, + 506, + 573 + ], + "score": 1.0, + "content": "is a proof state", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 567, + 412, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 144, + 583 + ], + "score": 1.0, + "content": "and each", + "type": "text" + }, + { + "bbox": [ + 144, + 569, + 156, + 580 + ], + "score": 0.89, + "content": "d _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 567, + 396, + 583 + ], + "score": 1.0, + "content": "is the binned size of the proof tree below the corresponding", + "type": "text" + }, + { + "bbox": [ + 396, + 571, + 407, + 580 + ], + "score": 0.85, + "content": "s _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 567, + 412, + 583 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 586, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "The position evaluator defined above provides a proxy for how difficult it is to complete the proof", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "from the current state—a lower number indicates that the goal is easy to deduce given the context.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "score": 1.0, + "content": "A model for position evaluation can be used to create a sequence of tasks for curriculum learning or", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "score": 1.0, + "content": "by human provers to measure the progress they are making in a proof. 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Thus,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "the tool records proof states at the granularity of primitive tactics (i.e., breaks up ;) as well as at the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 260, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 260, + 172 + ], + "score": 1.0, + "content": "granularity of the proof script (i.e., .)", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 82, + 506, + 172 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 184, + 505, + 261 + ], + "lines": [ + { + "bbox": [ + 106, + 183, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 505, + 198 + ], + "score": 1.0, + "content": "Representing Coq proofs We expose the proof trace obtained from Coq as Python data structures,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "including Coq’s mid-level and kernel-level term languages, proof states, proof steps, and proof trees", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "so that they can be manipulated with arbitrary Python code. Consequently, we can build models", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "score": 1.0, + "content": "using this structure. For efficiency reasons, we represent Coq terms in a shared form. Representing", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "terms in a shared form is a necessary technique to scale compilers and ITPs to real-world programs", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "and proofs. In our setting, it is also essential to scaling training/inference to real-world data sets (see", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 152, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 152, + 262 + ], + "score": 1.0, + "content": "Section 7).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 183, + 506, + 262 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 319 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "Light-weight interaction The API provides a thin wrapper around coqtop, the Coq repl, which", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "can be used to interactively construct proof scripts from within Python (see Section 6). This compo-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 298, + 504, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 504, + 309 + ], + "score": 1.0, + "content": "nent mediates all interaction with Coq, including proof state parsing into GamePad’s representation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 308, + 272, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 272, + 320 + ], + "score": 1.0, + "content": "of Coq proofs and sending tactic actions.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 275, + 505, + 320 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 336, + 160, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 162, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 162, + 352 + ], + "score": 1.0, + "content": "4 TASKS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "In the theorem proving process, we encounter two important tasks: position evaluation and tactic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 240, + 387 + ], + "score": 1.0, + "content": "prediction. A position evaluator", + "type": "text" + }, + { + "bbox": [ + 240, + 374, + 249, + 384 + ], + "score": 0.82, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "tells how easy it is to prove a given proof state, while a tactic", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 145, + 398 + ], + "score": 1.0, + "content": "predictor", + "type": "text" + }, + { + "bbox": [ + 145, + 385, + 154, + 395 + ], + "score": 0.8, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "tells how to take an action that can make our proof state easier to prove. As a reminder,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "proof states are interpreted in the presence of global state, which we have left implicit here. Note", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 169, + 419 + ], + "score": 1.0, + "content": "that in general,", + "type": "text" + }, + { + "bbox": [ + 169, + 407, + 178, + 417 + ], + "score": 0.8, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 406, + 227, + 419 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 228, + 407, + 237, + 417 + ], + "score": 0.81, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "; a good tactic predictor would find it easier to prove a given proof", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 231, + 430 + ], + "score": 1.0, + "content": "state. Also, an action taken by", + "type": "text" + }, + { + "bbox": [ + 232, + 418, + 241, + 428 + ], + "score": 0.82, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 417, + 436, + 430 + ], + "score": 1.0, + "content": "can lead to multiple child proof states, and thus", + "type": "text" + }, + { + "bbox": [ + 436, + 418, + 445, + 428 + ], + "score": 0.83, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "must consider", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 428, + 264, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 264, + 442 + ], + "score": 1.0, + "content": "the provability of all child proof states.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 362, + 506, + 442 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 310, + 458 + ], + "score": 1.0, + "content": "In this work, we aim to learn parametric functions", + "type": "text" + }, + { + "bbox": [ + 311, + 444, + 329, + 456 + ], + "score": 0.9, + "content": "\\mathcal { P } ^ { \\mathcal { T } _ { h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 444, + 347, + 458 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 348, + 446, + 360, + 456 + ], + "score": 0.87, + "content": "\\mathcal { T } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 444, + 505, + 458 + ], + "score": 1.0, + "content": "using supervised learning on a data", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 216, + 469 + ], + "score": 1.0, + "content": "set of human proofs, where", + "type": "text" + }, + { + "bbox": [ + 216, + 457, + 227, + 468 + ], + "score": 0.89, + "content": "\\mathcal { T } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "is the human tactic predictor implicit in the data set. In the future, one", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 138, + 481 + ], + "score": 1.0, + "content": "can use", + "type": "text" + }, + { + "bbox": [ + 138, + 468, + 147, + 478 + ], + "score": 0.81, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 466, + 165, + 481 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 165, + 468, + 175, + 478 + ], + "score": 0.82, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 466, + 505, + 481 + ], + "score": 1.0, + "content": "within an end-to-end prover, and learn them directly using reinforcement learning.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "score": 1.0, + "content": "This requires the ability to manipulate proof states by sending actions, which is possible using the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 490, + 296, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 296, + 502 + ], + "score": 1.0, + "content": "light-weight interaction provided by GamePad.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 444, + 505, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "Position evaluation The goal of the position evaluation task is to predict the approximate number", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 481, + 537 + ], + "score": 1.0, + "content": "of steps required to finish a proof given an input proof state. We define this as the function", + "type": "text" + }, + { + "bbox": [ + 481, + 524, + 497, + 536 + ], + "score": 0.75, + "content": "\\mathcal { P } ^ { \\mathcal { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 524, + 506, + 537 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 535, + 504, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 182, + 548 + ], + "score": 0.93, + "content": "P S \\stackrel { - } { } \\{ 1 , \\stackrel { - } { \\cdot } \\cdot \\cdot , K \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 535, + 294, + 548 + ], + "score": 1.0, + "content": "for a given tactic predictor", + "type": "text" + }, + { + "bbox": [ + 294, + 537, + 303, + 546 + ], + "score": 0.8, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 535, + 493, + 548 + ], + "score": 1.0, + "content": ", where we bin the sizes of the proof trees into", + "type": "text" + }, + { + "bbox": [ + 493, + 536, + 504, + 546 + ], + "score": 0.77, + "content": "K", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "classes to make learning easy. Given a data set of steps taken by a human prover, we aim to learn", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 555, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 124, + 568 + ], + "score": 0.88, + "content": "\\mathcal { P } ^ { \\mathcal { T } _ { h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 555, + 233, + 573 + ], + "score": 1.0, + "content": "by supervised learning on", + "type": "text" + }, + { + "bbox": [ + 233, + 559, + 243, + 568 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 555, + 306, + 573 + ], + "score": 1.0, + "content": "training tuples", + "type": "text" + }, + { + "bbox": [ + 306, + 558, + 378, + 570 + ], + "score": 0.93, + "content": "\\{ ( s _ { n } , d _ { n } ) \\} _ { 1 \\leq n \\leq N }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 555, + 430, + 573 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 430, + 560, + 442, + 569 + ], + "score": 0.86, + "content": "s _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 555, + 506, + 573 + ], + "score": 1.0, + "content": "is a proof state", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 567, + 412, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 144, + 583 + ], + "score": 1.0, + "content": "and each", + "type": "text" + }, + { + "bbox": [ + 144, + 569, + 156, + 580 + ], + "score": 0.89, + "content": "d _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 567, + 396, + 583 + ], + "score": 1.0, + "content": "is the binned size of the proof tree below the corresponding", + "type": "text" + }, + { + "bbox": [ + 396, + 571, + 407, + 580 + ], + "score": 0.85, + "content": "s _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 567, + 412, + 583 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 514, + 506, + 583 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 586, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "The position evaluator defined above provides a proxy for how difficult it is to complete the proof", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "from the current state—a lower number indicates that the goal is easy to deduce given the context.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "score": 1.0, + "content": "A model for position evaluation can be used to create a sequence of tasks for curriculum learning or", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "score": 1.0, + "content": "by human provers to measure the progress they are making in a proof. We also note that position", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "evaluation contains aspects of the premise selection problem (Irving et al., 2016) in that it should", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "assign a high number to proof states which do not yet contain the requisite hypotheses to prove the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 652, + 159, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 159, + 664 + ], + "score": 1.0, + "content": "current goal.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 585, + 506, + 664 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "Tactic prediction The goal of the tactic prediction task is to predict the next tactic to apply given", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 321, + 700 + ], + "score": 1.0, + "content": "an input proof state. We define this as the function", + "type": "text" + }, + { + "bbox": [ + 322, + 688, + 379, + 698 + ], + "score": 0.91, + "content": "\\mathcal { T } : P S T", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 688, + 411, + 700 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 411, + 688, + 419, + 698 + ], + "score": 0.69, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "represents the set of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "possible tactic actions. Given a data set of steps taken by a human prover, we aim to learn the human", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 168, + 723 + ], + "score": 1.0, + "content": "tactic predictor", + "type": "text" + }, + { + "bbox": [ + 168, + 710, + 180, + 721 + ], + "score": 0.88, + "content": "\\mathcal { T } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 709, + 204, + 723 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 205, + 710, + 215, + 720 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 709, + 275, + 723 + ], + "score": 1.0, + "content": "training tuples", + "type": "text" + }, + { + "bbox": [ + 275, + 709, + 345, + 722 + ], + "score": 0.93, + "content": "\\{ ( { \\bar { s } } _ { n } , t _ { n } ) \\} _ { 1 \\leq n \\leq N }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 709, + 375, + 723 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 376, + 711, + 386, + 721 + ], + "score": 0.87, + "content": "t _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "is the tactic the human prover", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 720, + 173, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 158, + 734 + ], + "score": 1.0, + "content": "used in state", + "type": "text" + }, + { + "bbox": [ + 158, + 722, + 169, + 732 + ], + "score": 0.87, + "content": "s _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 720, + 173, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 676, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "Tactic prediction is more localized to a single proof state compared to position evaluation, although", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "there are two additional challenges. First, we must choose the granularity of a proof step, which", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "ranges from considering only atomic tactics to human-level proof steps (i.e., compound tactics).", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "Currently, our tool provides access to atomic tactics and as well as the capability to treat sequences", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 263, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 263, + 140 + ], + "score": 1.0, + "content": "of atomic tactics as a single proof step.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "Second, tactic prediction may additionally require the synthesis of an argument. Some arguments to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 504, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 504, + 166 + ], + "score": 1.0, + "content": "tactics such as induction or rewrite require the user to select an identifier in the local or global", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 164, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 505, + 179 + ], + "score": 1.0, + "content": "context to apply—what to do induction on and what to equality to rewrite by respectively. This can", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "be considered a premise selection problem. Other arguments to tactics include synthesizing entire", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 304, + 200 + ], + "score": 1.0, + "content": "Coq terms. For example, the tactic have: x :", + "type": "text" + }, + { + "bbox": [ + 304, + 188, + 325, + 198 + ], + "score": 0.53, + "content": "\\qquad = ~ \\mathrm { ~ M ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "declares a local lemma that asserts that M is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 277, + 211 + ], + "score": 1.0, + "content": "true and to introduce it into the context as", + "type": "text" + }, + { + "bbox": [ + 277, + 200, + 285, + 208 + ], + "score": 0.67, + "content": "_ \\textrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "after it has been proven. Our tool provides the tactics,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 223 + ], + "score": 1.0, + "content": "the arguments, and extracts local and global identifiers referenced in the arguments for convenience.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "This decomposes the problem of synthesizing tactic arguments into (1) predicting the identifiers", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "involved and (2) constructing a term given a set of identifiers. The problem of synthesizing a term", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 241, + 419, + 256 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 419, + 256 + ], + "score": 1.0, + "content": "is difficult and a topic of research in itself—we do not address it in this paper.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 269, + 284, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 286, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 286, + 284 + ], + "score": 1.0, + "content": "5 REPRESENTING PROOF STATES", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 295, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "score": 1.0, + "content": "As we have just seen, both position evaluation and tactic prediction require a representation of proof", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 305, + 494, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 494, + 320 + ], + "score": 1.0, + "content": "states. Hence, we now discuss the representation of proof states in a form amenable for learning.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 322, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "One manner in which the structured representation of proofs states provided by GamePad can be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "leveraged is to apply recurrent neural networks (RNNs) in a similar manner to how they are applied", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "to parse trees in natural language processing to obtain an embedding vector. We can embed terms", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 346, + 370 + ], + "score": 1.0, + "content": "using their structure with a recursive embedding function", + "type": "text" + }, + { + "bbox": [ + 347, + 356, + 424, + 367 + ], + "score": 0.87, + "content": "\\mathcal { E } : \\mathrm { T e r m } \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 354, + 506, + 370 + ], + "score": 1.0, + "content": ". For example, the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 367, + 408, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 244, + 379 + ], + "score": 1.0, + "content": "embedding of an application term", + "type": "text" + }, + { + "bbox": [ + 244, + 367, + 303, + 378 + ], + "score": 0.91, + "content": "M _ { 0 } M _ { 1 } \\ldots M _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 367, + 408, + 379 + ], + "score": 1.0, + "content": "is obtained recursively as", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 384, + 414, + 398 + ], + "lines": [ + { + "bbox": [ + 195, + 384, + 414, + 398 + ], + "spans": [ + { + "bbox": [ + 195, + 384, + 414, + 398 + ], + "score": 0.89, + "content": "\\mathcal { E } ( M _ { 0 } \\ldots \\mathit { M _ { r } } ) = \\mathbf { R } \\mathbf { N } \\mathbf { N } ( \\mathcal { E ^ { \\prime } } [ \\mathbb { A } \\mathrm { p p } ] , \\mathcal { E } ( M _ { 0 } ) , \\ldots , \\mathcal { E } ( M _ { r } ) )", + "type": "interline_equation", + "image_path": "c022d3a4b407faf827ee813709d84a0f0ed665288326d76bfcd476b29304c1a8.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 195, + 384, + 414, + 398 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 134, + 415 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 403, + 155, + 416 + ], + "score": 0.91, + "content": "\\mathcal { E } ^ { \\prime } [ [ \\cdot ] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "is a learnable embedding table indexed by the kind of the AST node (e.g., App for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "score": 1.0, + "content": "J Kapplication node). The leaves of the AST consist of constants, inductive types, constructors, exis-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "tentials, and variables. Each constant, inductive type, constructor, and existential is given an entry", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 437, + 361, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 361, + 449 + ], + "score": 1.0, + "content": "in a learnable embedding table, which encodes the global state.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "Towards interpreter-inspired embeddings One way to add inductive bias to the embedding is to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 471, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 485 + ], + "score": 1.0, + "content": "use the known reduction semantics of terms. 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For example, the meaning of the program expression", + "type": "text" + }, + { + "bbox": [ + 497, + 496, + 504, + 504 + ], + "score": 0.4, + "content": "_ \\textrm { x }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 505, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 517 + ], + "score": 1.0, + "content": "under an empty environment is a run-time error whereas the meaning of the same expression under", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 176, + 529 + ], + "score": 1.0, + "content": "the environment", + "type": "text" + }, + { + "bbox": [ + 177, + 515, + 223, + 527 + ], + "score": 0.93, + "content": "\\{ \\mathrm { x } \\mapsto \\underline { { 4 2 } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "is 42. We can apply this idea to obtain a new embedding function", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 524, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 107, + 527, + 210, + 537 + ], + "score": 0.88, + "content": "\\mathcal { E } : \\mathrm { T e r m } \\times \\mathrm { E n v } \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 524, + 369, + 540 + ], + "score": 1.0, + "content": "that additionally takes an environment", + "type": "text" + }, + { + "bbox": [ + 370, + 528, + 377, + 538 + ], + "score": 0.82, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 524, + 506, + 540 + ], + "score": 1.0, + "content": "of type Env which is a lookup", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "table mapping variables to embedding vectors. Whenever we encounter a binding form such as a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 183, + 561 + ], + "score": 1.0, + "content": "dependent product", + "type": "text" + }, + { + "bbox": [ + 183, + 549, + 240, + 559 + ], + "score": 0.9, + "content": "\\Pi x : M _ { 1 } . M _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 548, + 382, + 561 + ], + "score": 1.0, + "content": "(similar to an anonymous function", + "type": "text" + }, + { + "bbox": [ + 382, + 549, + 439, + 560 + ], + "score": 0.91, + "content": "\\lambda x : M _ { 1 } . M _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 548, + 506, + 561 + ], + "score": 1.0, + "content": ", we bind a new", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 559, + 503, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 559, + 158, + 572 + ], + "score": 1.0, + "content": "meaning for", + "type": "text" + }, + { + "bbox": [ + 158, + 561, + 165, + 569 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 559, + 289, + 572 + ], + "score": 1.0, + "content": "within its scope (i.e., the term", + "type": "text" + }, + { + "bbox": [ + 289, + 560, + 304, + 570 + ], + "score": 0.88, + "content": "M _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 559, + 466, + 572 + ], + "score": 1.0, + "content": "). To do so, we sample a random vector", + "type": "text" + }, + { + "bbox": [ + 466, + 559, + 503, + 570 + ], + "score": 0.91, + "content": "v \\sim \\mathcal { N } ^ { D }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 568, + 504, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 342, + 585 + ], + "score": 1.0, + "content": "according to a standard (multivariate) normal distribution2", + "type": "text" + }, + { + "bbox": [ + 342, + 570, + 360, + 580 + ], + "score": 0.86, + "content": "\\mathcal { N } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 568, + 497, + 585 + ], + "score": 1.0, + "content": "and extend the local environment", + "type": "text" + }, + { + "bbox": [ + 497, + 572, + 504, + 582 + ], + "score": 0.78, + "content": "\\rho", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 178, + 594 + ], + "score": 1.0, + "content": "with the mapping", + "type": "text" + }, + { + "bbox": [ + 179, + 583, + 207, + 592 + ], + "score": 0.88, + "content": "x \\mapsto v", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 581, + 241, + 594 + ], + "score": 1.0, + "content": ", written", + "type": "text" + }, + { + "bbox": [ + 241, + 581, + 280, + 593 + ], + "score": 0.93, + "content": "\\rho [ x \\mapsto v ]", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 581, + 411, + 594 + ], + "score": 1.0, + "content": ", so that mentions of the variable", + "type": "text" + }, + { + "bbox": [ + 412, + 583, + 419, + 591 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 581, + 430, + 594 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 430, + 581, + 446, + 592 + ], + "score": 0.89, + "content": "M _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "can be looked", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "up. Then the embedding of a binding form, such as a dependent product term (Prod), is obtained", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 603, + 164, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 164, + 617 + ], + "score": 1.0, + "content": "recursively as", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 34.5 + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 620, + 483, + 634 + ], + "lines": [ + { + "bbox": [ + 127, + 620, + 483, + 634 + ], + "spans": [ + { + "bbox": [ + 127, + 620, + 483, + 634 + ], + "score": 0.88, + "content": "\\mathcal { E } ( \\Pi x : M _ { 1 } . M _ { 2 } , \\rho ) = \\mathrm { R N N } ( \\mathcal { E } ^ { \\prime } [ \\mathrm { P r } \\mathrm { c o d } ] , \\mathcal { E } ( M _ { 1 } , \\rho ) , \\mathcal { E } ( M _ { 2 } , \\rho [ x \\mapsto v ] ) ) \\quad \\mathrm { w h e r e ~ } v \\sim \\mathcal { N } ^ { D } .", + "type": "interline_equation", + "image_path": "41ecd72483433f1646f3a316a196f391c4690e8f94f87afe8c2353afb701e364.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 127, + 620, + 483, + 634 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 639, + 502, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 501, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 435, + 653 + ], + "score": 1.0, + "content": "The embedding for the variable case (which occur at the leaves of the AST) is", + "type": "text" + }, + { + "bbox": [ + 435, + 639, + 501, + 652 + ], + "score": 0.92, + "content": "\\mathcal { E } ( x , \\rho ) = \\rho ( x )", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 651, + 356, + 663 + ], + "spans": [ + { + "bbox": [ + 107, + 651, + 356, + 663 + ], + "score": 1.0, + "content": "which corresponds to a variable lookup from the environment.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 234, + 681 + ], + "score": 1.0, + "content": "We resample the corresponding", + "type": "text" + }, + { + "bbox": [ + 234, + 671, + 240, + 678 + ], + "score": 0.67, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 667, + 263, + 681 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 264, + 667, + 281, + 678 + ], + "score": 0.9, + "content": "\\mathcal { N } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "every forward pass in training when we embed the term", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 107, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 107, + 679, + 162, + 690 + ], + "score": 0.74, + "content": "\\Pi x : M _ { 1 } . M _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 678, + 339, + 691 + ], + "score": 1.0, + "content": ". 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Note", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 117, + 721, + 437, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 718, + 439, + 735 + ], + "spans": [ + { + "bbox": [ + 118, + 718, + 264, + 735 + ], + "score": 1.0, + "content": "2We conjecture that we can use the type", + "type": "text" + }, + { + "bbox": [ + 265, + 723, + 278, + 731 + ], + "score": 0.84, + "content": "M _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 718, + 439, + 735 + ], + "score": 1.0, + "content": "to put a better prior on the vectors sampled.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "Tactic prediction is more localized to a single proof state compared to position evaluation, although", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "there are two additional challenges. 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Some arguments to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 504, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 504, + 166 + ], + "score": 1.0, + "content": "tactics such as induction or rewrite require the user to select an identifier in the local or global", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 164, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 505, + 179 + ], + "score": 1.0, + "content": "context to apply—what to do induction on and what to equality to rewrite by respectively. This can", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "be considered a premise selection problem. Other arguments to tactics include synthesizing entire", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 304, + 200 + ], + "score": 1.0, + "content": "Coq terms. 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Our tool provides the tactics,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 223 + ], + "score": 1.0, + "content": "the arguments, and extracts local and global identifiers referenced in the arguments for convenience.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "This decomposes the problem of synthesizing tactic arguments into (1) predicting the identifiers", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "involved and (2) constructing a term given a set of identifiers. The problem of synthesizing a term", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 241, + 419, + 256 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 419, + 256 + ], + "score": 1.0, + "content": "is difficult and a topic of research in itself—we do not address it in this paper.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 142, + 506, + 256 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 269, + 284, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 286, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 286, + 284 + ], + "score": 1.0, + "content": "5 REPRESENTING PROOF STATES", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 295, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "score": 1.0, + "content": "As we have just seen, both position evaluation and tactic prediction require a representation of proof", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 305, + 494, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 494, + 320 + ], + "score": 1.0, + "content": "states. Hence, we now discuss the representation of proof states in a form amenable for learning.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 295, + 505, + 320 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 322, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "One manner in which the structured representation of proofs states provided by GamePad can be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "leveraged is to apply recurrent neural networks (RNNs) in a similar manner to how they are applied", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "to parse trees in natural language processing to obtain an embedding vector. We can embed terms", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 346, + 370 + ], + "score": 1.0, + "content": "using their structure with a recursive embedding function", + "type": "text" + }, + { + "bbox": [ + 347, + 356, + 424, + 367 + ], + "score": 0.87, + "content": "\\mathcal { E } : \\mathrm { T e r m } \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 354, + 506, + 370 + ], + "score": 1.0, + "content": ". For example, the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 367, + 408, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 244, + 379 + ], + "score": 1.0, + "content": "embedding of an application term", + "type": "text" + }, + { + "bbox": [ + 244, + 367, + 303, + 378 + ], + "score": 0.91, + "content": "M _ { 0 } M _ { 1 } \\ldots M _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 367, + 408, + 379 + ], + "score": 1.0, + "content": "is obtained recursively as", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 322, + 506, + 379 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 384, + 414, + 398 + ], + "lines": [ + { + "bbox": [ + 195, + 384, + 414, + 398 + ], + "spans": [ + { + "bbox": [ + 195, + 384, + 414, + 398 + ], + "score": 0.89, + "content": "\\mathcal { E } ( M _ { 0 } \\ldots \\mathit { M _ { r } } ) = \\mathbf { R } \\mathbf { N } \\mathbf { N } ( \\mathcal { E ^ { \\prime } } [ \\mathbb { A } \\mathrm { p p } ] , \\mathcal { E } ( M _ { 0 } ) , \\ldots , \\mathcal { E } ( M _ { r } ) )", + "type": "interline_equation", + "image_path": "c022d3a4b407faf827ee813709d84a0f0ed665288326d76bfcd476b29304c1a8.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 195, + 384, + 414, + 398 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 134, + 415 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 403, + 155, + 416 + ], + "score": 0.91, + "content": "\\mathcal { E } ^ { \\prime } [ [ \\cdot ] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "is a learnable embedding table indexed by the kind of the AST node (e.g., App for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "score": 1.0, + "content": "J Kapplication node). The leaves of the AST consist of constants, inductive types, constructors, exis-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "tentials, and variables. Each constant, inductive type, constructor, and existential is given an entry", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 437, + 361, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 361, + 449 + ], + "score": 1.0, + "content": "in a learnable embedding table, which encodes the global state.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 403, + 505, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "Towards interpreter-inspired embeddings One way to add inductive bias to the embedding is to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 471, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 485 + ], + "score": 1.0, + "content": "use the known reduction semantics of terms. 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M _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 548, + 506, + 561 + ], + "score": 1.0, + "content": ", we bind a new", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 559, + 503, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 559, + 158, + 572 + ], + "score": 1.0, + "content": "meaning for", + "type": "text" + }, + { + "bbox": [ + 158, + 561, + 165, + 569 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 559, + 289, + 572 + ], + "score": 1.0, + "content": "within its scope (i.e., the term", + "type": "text" + }, + { + "bbox": [ + 289, + 560, + 304, + 570 + ], + "score": 0.88, + "content": "M _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 559, + 466, + 572 + ], + "score": 1.0, + "content": "). To do so, we sample a random vector", + "type": "text" + }, + { + "bbox": [ + 466, + 559, + 503, + 570 + ], + "score": 0.91, + "content": "v \\sim \\mathcal { N } ^ { D }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 568, + 504, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 342, + 585 + ], + "score": 1.0, + "content": "according to a standard (multivariate) normal distribution2", + "type": "text" + }, + { + "bbox": [ + 342, + 570, + 360, + 580 + ], + "score": 0.86, + "content": "\\mathcal { N } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 568, + 497, + 585 + ], + "score": 1.0, + "content": "and extend the local environment", + "type": "text" + }, + { + "bbox": [ + 497, + 572, + 504, + 582 + ], + "score": 0.78, + "content": "\\rho", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 178, + 594 + ], + "score": 1.0, + "content": "with the mapping", + "type": "text" + }, + { + "bbox": [ + 179, + 583, + 207, + 592 + ], + "score": 0.88, + "content": "x \\mapsto v", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 581, + 241, + 594 + ], + "score": 1.0, + "content": ", written", + "type": "text" + }, + { + "bbox": [ + 241, + 581, + 280, + 593 + ], + "score": 0.93, + "content": "\\rho [ x \\mapsto v ]", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 581, + 411, + 594 + ], + "score": 1.0, + "content": ", so that mentions of the variable", + "type": "text" + }, + { + "bbox": [ + 412, + 583, + 419, + 591 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 581, + 430, + 594 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 430, + 581, + 446, + 592 + ], + "score": 0.89, + "content": "M _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "can be looked", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "up. Then the embedding of a binding form, such as a dependent product term (Prod), is obtained", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 603, + 164, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 164, + 617 + ], + "score": 1.0, + "content": "recursively as", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 461, + 506, + 617 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 620, + 483, + 634 + ], + "lines": [ + { + "bbox": [ + 127, + 620, + 483, + 634 + ], + "spans": [ + { + "bbox": [ + 127, + 620, + 483, + 634 + ], + "score": 0.88, + "content": "\\mathcal { E } ( \\Pi x : M _ { 1 } . M _ { 2 } , \\rho ) = \\mathrm { R N N } ( \\mathcal { E } ^ { \\prime } [ \\mathrm { P r } \\mathrm { c o d } ] , \\mathcal { E } ( M _ { 1 } , \\rho ) , \\mathcal { E } ( M _ { 2 } , \\rho [ x \\mapsto v ] ) ) \\quad \\mathrm { w h e r e ~ } v \\sim \\mathcal { N } ^ { D } .", + "type": "interline_equation", + "image_path": "41ecd72483433f1646f3a316a196f391c4690e8f94f87afe8c2353afb701e364.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 127, + 620, + 483, + 634 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 639, + 502, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 501, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 435, + 653 + ], + "score": 1.0, + "content": "The embedding for the variable case (which occur at the leaves of the AST) is", + "type": "text" + }, + { + "bbox": [ + 435, + 639, + 501, + 652 + ], + "score": 0.92, + "content": "\\mathcal { E } ( x , \\rho ) = \\rho ( x )", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 651, + 356, + 663 + ], + "spans": [ + { + "bbox": [ + 107, + 651, + 356, + 663 + ], + "score": 1.0, + "content": "which corresponds to a variable lookup from the environment.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5, + "bbox_fs": [ + 106, + 638, + 501, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 234, + 681 + ], + "score": 1.0, + "content": "We resample the corresponding", + "type": "text" + }, + { + "bbox": [ + 234, + 671, + 240, + 678 + ], + "score": 0.67, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 667, + 263, + 681 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 264, + 667, + 281, + 678 + ], + "score": 0.9, + "content": "\\mathcal { N } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "every forward pass in training when we embed the term", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 107, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 107, + 679, + 162, + 690 + ], + "score": 0.74, + "content": "\\Pi x : M _ { 1 } . M _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 678, + 339, + 691 + ], + "score": 1.0, + "content": ". By doing so, we encode the semantics that", + "type": "text" + }, + { + "bbox": [ + 340, + 681, + 347, + 689 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "is just a placeholder and we should get", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 690, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 296, + 702 + ], + "score": 1.0, + "content": "the same result if we had used a different vector", + "type": "text" + }, + { + "bbox": [ + 296, + 693, + 302, + 700 + ], + "score": 0.77, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 690, + 505, + 702 + ], + "score": 1.0, + "content": "to embed it, while also preserving the environment", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 701, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 257, + 713 + ], + "score": 1.0, + "content": "lookup semantics that the embedding", + "type": "text" + }, + { + "bbox": [ + 257, + 704, + 263, + 711 + ], + "score": 0.76, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 701, + 279, + 713 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 279, + 705, + 285, + 710 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 701, + 409, + 713 + ], + "score": 1.0, + "content": "is constant within the scope of", + "type": "text" + }, + { + "bbox": [ + 410, + 703, + 416, + 711 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 701, + 505, + 713 + ], + "score": 1.0, + "content": "in a single pass. Note", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "that the embedding is invariant by construction to variable renaming. Wang et al. (2017) propose", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "another structured approach based on a De Bruijn term representation that is invariant under variable", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "renaming that would be interesting to compare against. It would also be interesting to extend the", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "entire embedding to more closely follow the structure of an interpreter so that it better reflects the", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 453, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 453, + 138 + ], + "score": 1.0, + "content": "semantics as opposed to the syntax, although we leave these extensions to future work.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 667, + 506, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "that the embedding is invariant by construction to variable renaming. Wang et al. (2017) propose", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "another structured approach based on a De Bruijn term representation that is invariant under variable", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "renaming that would be interesting to compare against. It would also be interesting to extend the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "entire embedding to more closely follow the structure of an interpreter so that it better reflects the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 453, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 453, + 138 + ], + "score": 1.0, + "content": "semantics as opposed to the syntax, although we leave these extensions to future work.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 505, + 206 + ], + "lines": [ + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "score": 1.0, + "content": "Embedding proof states After obtaining the embedding for each type3 in the proof state and the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 162, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 505, + 174 + ], + "score": 1.0, + "content": "embedding for the goal, we use another RNN over the embeddings. Note that the proof state context", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "score": 1.0, + "content": "must be traversed in-order because the types of items later in the context may depend on the types", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 197 + ], + "score": 1.0, + "content": "of items that appear earlier in the context. As expected, the result of embedding a proof state is a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 191, + 163, + 207 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 144, + 207 + ], + "score": 1.0, + "content": "vector in", + "type": "text" + }, + { + "bbox": [ + 144, + 194, + 159, + 205 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 191, + 163, + 207 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 223, + 447, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 449, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 449, + 238 + ], + "score": 1.0, + "content": "6 END-TO-END PROOF GENERATION FOR ALGEBRAIC REWRITES", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 305 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 504, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 504, + 261 + ], + "score": 1.0, + "content": "In this section, we walk through a basic setup that uses GamePad to learn a simple algebraic rewriter.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "score": 1.0, + "content": "First, we use a deterministic procedure to synthesize Coq proofs for our domain and use GamePad to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "extract the resulting proof trees. Second, we train a tactic predictor and then deploy it to synthesize", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "score": 1.0, + "content": "end-to-end proofs using GamePad’s interactive mechanisms. This setup applies to any other domain", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 294, + 456, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 456, + 306 + ], + "score": 1.0, + "content": "of interest, provided we have a method of generating Coq proof scripts for that domain.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 107, + 318, + 306, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 306, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 306, + 331 + ], + "score": 1.0, + "content": "6.1 SIMPLE ALGEBRAIC REWRITE PROBLEM", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 504, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "We consider a problem that involves showing that two algebraic expressions are equivalent to one", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 351, + 356, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 356, + 363 + ], + "score": 1.0, + "content": "another. More concretely, we consider statements of the form:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 271, + 369, + 339, + 382 + ], + "lines": [ + { + "bbox": [ + 271, + 369, + 339, + 382 + ], + "spans": [ + { + "bbox": [ + 271, + 369, + 339, + 382 + ], + "score": 0.9, + "content": "\\forall b \\in G , X = b ,", + "type": "interline_equation", + "image_path": "1d78beec4a0cf37ab5484fbb39cdaa5eec604fa1cc1da24e30030e79fac30b2b.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 271, + 369, + 339, + 382 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 504, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 135, + 401 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 389, + 145, + 398 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 387, + 335, + 401 + ], + "score": 1.0, + "content": "is an arbitrary expression from the grammar", + "type": "text" + }, + { + "bbox": [ + 335, + 388, + 447, + 401 + ], + "score": 0.91, + "content": "X \\ { \\mathrel { \\mathop : } } { = } \\ b \\ | \\ e \\ | \\ m \\ | \\ X \\oplus X", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 387, + 506, + 401 + ], + "score": 1.0, + "content": "composed of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 216, + 412 + ], + "score": 1.0, + "content": "elements with left-identity", + "type": "text" + }, + { + "bbox": [ + 216, + 402, + 222, + 410 + ], + "score": 0.73, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 399, + 281, + 412 + ], + "score": 1.0, + "content": ", right-identity", + "type": "text" + }, + { + "bbox": [ + 281, + 402, + 291, + 410 + ], + "score": 0.77, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 399, + 378, + 412 + ], + "score": 1.0, + "content": ", and binary operator", + "type": "text" + }, + { + "bbox": [ + 378, + 401, + 387, + 410 + ], + "score": 0.83, + "content": "\\oplus", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 399, + 505, + 412 + ], + "score": 1.0, + "content": ". We have two simplification", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 410, + 423, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 131, + 423 + ], + "score": 1.0, + "content": "rules:", + "type": "text" + }, + { + "bbox": [ + 132, + 411, + 210, + 422 + ], + "score": 0.91, + "content": "\\forall b \\in G , b \\oplus m = b", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 410, + 289, + 423 + ], + "score": 1.0, + "content": "(right identity) and", + "type": "text" + }, + { + "bbox": [ + 289, + 411, + 363, + 422 + ], + "score": 0.92, + "content": "\\forall b \\in G , e \\oplus b = b", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 410, + 423, + 423 + ], + "score": 1.0, + "content": "(left identity).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 427, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "Although the problem is simple, it involves bits of non-trivial reasoning. For instance, consider", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 208, + 451 + ], + "score": 1.0, + "content": "showing the equivalence", + "type": "text" + }, + { + "bbox": [ + 208, + 438, + 318, + 451 + ], + "score": 0.92, + "content": "\\forall b \\in G , b \\oplus ( e \\oplus m ) = b", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 438, + 478, + 451 + ], + "score": 1.0, + "content": ". Here, we can choose to eliminate the", + "type": "text" + }, + { + "bbox": [ + 478, + 441, + 484, + 449 + ], + "score": 0.61, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "(left", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 170, + 462 + ], + "score": 1.0, + "content": "identity) or the", + "type": "text" + }, + { + "bbox": [ + 171, + 451, + 181, + 460 + ], + "score": 0.72, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 449, + 378, + 462 + ], + "score": 1.0, + "content": "(right identity). Notably, choosing to eliminate", + "type": "text" + }, + { + "bbox": [ + 379, + 451, + 389, + 460 + ], + "score": 0.65, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "does not progress the proof", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 281, + 473 + ], + "score": 1.0, + "content": "because we cannot simplify the proof state", + "type": "text" + }, + { + "bbox": [ + 281, + 461, + 323, + 471 + ], + "score": 0.91, + "content": "b \\oplus e = b", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 461, + 506, + 473 + ], + "score": 1.0, + "content": ". Note that the proof is not stuck because we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 154, + 483 + ], + "score": 1.0, + "content": "can expand", + "type": "text" + }, + { + "bbox": [ + 154, + 472, + 177, + 482 + ], + "score": 0.9, + "content": "b \\oplus e", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 472, + 210, + 483 + ], + "score": 1.0, + "content": "back to", + "type": "text" + }, + { + "bbox": [ + 210, + 471, + 261, + 483 + ], + "score": 0.92, + "content": "b \\oplus ( e \\oplus m )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 472, + 338, + 483 + ], + "score": 1.0, + "content": "and then get rid of", + "type": "text" + }, + { + "bbox": [ + 339, + 473, + 345, + 481 + ], + "score": 0.76, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 472, + 506, + 483 + ], + "score": 1.0, + "content": "the second time around. Thus, a prover", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "has at least two choices in solving such problems: (1) maintain a global perspective to choose which", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "parts of the goal to rewrite or (2) learn to expand terms. For this problem, we write a deterministic", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "procedure that generates proofs of the first form that selects a position and an identity law, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "attempt to learn this algorithm. We do not generate proofs that require backtracking (e.g., due to a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 527, + 411, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 411, + 538 + ], + "score": 1.0, + "content": "greedy rewrite) although it would be an interesting direction of future work.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 552, + 270, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 271, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 271, + 565 + ], + "score": 1.0, + "content": "6.2 END-TO-END PROOF SYNTHESIS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 504, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 415, + 586 + ], + "score": 1.0, + "content": "Tactic prediction The tactic prediction model embeds a proof state into", + "type": "text" + }, + { + "bbox": [ + 415, + 572, + 430, + 583 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 572, + 505, + 586 + ], + "score": 1.0, + "content": "and uses a fully-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 597 + ], + "score": 1.0, + "content": "connected layer for prediction. We can model the proofs for this problem as a tactic prediction", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 609 + ], + "score": 1.0, + "content": "problem where the predicted category is a pair of the position in the AST and the identity to apply.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "We convert the position in the AST into a number using a preorder traversal of the Coq AST. For", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 190, + 630 + ], + "score": 1.0, + "content": "example, the second", + "type": "text" + }, + { + "bbox": [ + 190, + 618, + 200, + 628 + ], + "score": 0.81, + "content": "\\oplus", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 617, + 272, + 630 + ], + "score": 1.0, + "content": "in the expression", + "type": "text" + }, + { + "bbox": [ + 272, + 617, + 324, + 629 + ], + "score": 0.91, + "content": "b \\oplus ( e \\oplus m )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "has position 2. We can encode each identity", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 628, + 423, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 423, + 641 + ], + "score": 1.0, + "content": "as a single number. We obtain the prediction class as the pair of both numbers.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "We implement end-to-end proof synthesis by combining a tactic prediction model that is trained", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "offline with GamePad’s lightweight interaction module. The interaction module takes care of reading", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "in the current Coq proof state and sending tactic calls to advance the proof. We use the trained model", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "to do inference on the current Coq proof state to obtain a position to rewrite and the identity law to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "apply. We then translate this to a tactic and take the corresponding action in Coq. For the current", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 279, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 279, + 713 + ], + "score": 1.0, + "content": "problem, we do not consider backtracking.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 118, + 721, + 363, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 719, + 365, + 734 + ], + "spans": [ + { + "bbox": [ + 119, + 719, + 365, + 734 + ], + "score": 1.0, + "content": "3Recall that Coq uses the same language to encode types and terms.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 83, + 506, + 138 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 505, + 206 + ], + "lines": [ + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "score": 1.0, + "content": "Embedding proof states After obtaining the embedding for each type3 in the proof state and the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 162, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 505, + 174 + ], + "score": 1.0, + "content": "embedding for the goal, we use another RNN over the embeddings. Note that the proof state context", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "score": 1.0, + "content": "must be traversed in-order because the types of items later in the context may depend on the types", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 197 + ], + "score": 1.0, + "content": "of items that appear earlier in the context. As expected, the result of embedding a proof state is a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 191, + 163, + 207 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 144, + 207 + ], + "score": 1.0, + "content": "vector in", + "type": "text" + }, + { + "bbox": [ + 144, + 194, + 159, + 205 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 191, + 163, + 207 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 151, + 506, + 207 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 223, + 447, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 449, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 449, + 238 + ], + "score": 1.0, + "content": "6 END-TO-END PROOF GENERATION FOR ALGEBRAIC REWRITES", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 305 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 504, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 504, + 261 + ], + "score": 1.0, + "content": "In this section, we walk through a basic setup that uses GamePad to learn a simple algebraic rewriter.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "score": 1.0, + "content": "First, we use a deterministic procedure to synthesize Coq proofs for our domain and use GamePad to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "extract the resulting proof trees. Second, we train a tactic predictor and then deploy it to synthesize", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "score": 1.0, + "content": "end-to-end proofs using GamePad’s interactive mechanisms. This setup applies to any other domain", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 294, + 456, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 456, + 306 + ], + "score": 1.0, + "content": "of interest, provided we have a method of generating Coq proof scripts for that domain.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 249, + 505, + 306 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 318, + 306, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 306, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 306, + 331 + ], + "score": 1.0, + "content": "6.1 SIMPLE ALGEBRAIC REWRITE PROBLEM", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 504, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "We consider a problem that involves showing that two algebraic expressions are equivalent to one", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 351, + 356, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 356, + 363 + ], + "score": 1.0, + "content": "another. More concretely, we consider statements of the form:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 106, + 339, + 505, + 363 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 271, + 369, + 339, + 382 + ], + "lines": [ + { + "bbox": [ + 271, + 369, + 339, + 382 + ], + "spans": [ + { + "bbox": [ + 271, + 369, + 339, + 382 + ], + "score": 0.9, + "content": "\\forall b \\in G , X = b ,", + "type": "interline_equation", + "image_path": "1d78beec4a0cf37ab5484fbb39cdaa5eec604fa1cc1da24e30030e79fac30b2b.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 271, + 369, + 339, + 382 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 504, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 135, + 401 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 389, + 145, + 398 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 387, + 335, + 401 + ], + "score": 1.0, + "content": "is an arbitrary expression from the grammar", + "type": "text" + }, + { + "bbox": [ + 335, + 388, + 447, + 401 + ], + "score": 0.91, + "content": "X \\ { \\mathrel { \\mathop : } } { = } \\ b \\ | \\ e \\ | \\ m \\ | \\ X \\oplus X", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 387, + 506, + 401 + ], + "score": 1.0, + "content": "composed of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 216, + 412 + ], + "score": 1.0, + "content": "elements with left-identity", + "type": "text" + }, + { + "bbox": [ + 216, + 402, + 222, + 410 + ], + "score": 0.73, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 399, + 281, + 412 + ], + "score": 1.0, + "content": ", right-identity", + "type": "text" + }, + { + "bbox": [ + 281, + 402, + 291, + 410 + ], + "score": 0.77, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 399, + 378, + 412 + ], + "score": 1.0, + "content": ", and binary operator", + "type": "text" + }, + { + "bbox": [ + 378, + 401, + 387, + 410 + ], + "score": 0.83, + "content": "\\oplus", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 399, + 505, + 412 + ], + "score": 1.0, + "content": ". We have two simplification", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 410, + 423, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 131, + 423 + ], + "score": 1.0, + "content": "rules:", + "type": "text" + }, + { + "bbox": [ + 132, + 411, + 210, + 422 + ], + "score": 0.91, + "content": "\\forall b \\in G , b \\oplus m = b", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 410, + 289, + 423 + ], + "score": 1.0, + "content": "(right identity) and", + "type": "text" + }, + { + "bbox": [ + 289, + 411, + 363, + 422 + ], + "score": 0.92, + "content": "\\forall b \\in G , e \\oplus b = b", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 410, + 423, + 423 + ], + "score": 1.0, + "content": "(left identity).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 387, + 506, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 427, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "Although the problem is simple, it involves bits of non-trivial reasoning. For instance, consider", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 208, + 451 + ], + "score": 1.0, + "content": "showing the equivalence", + "type": "text" + }, + { + "bbox": [ + 208, + 438, + 318, + 451 + ], + "score": 0.92, + "content": "\\forall b \\in G , b \\oplus ( e \\oplus m ) = b", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 438, + 478, + 451 + ], + "score": 1.0, + "content": ". Here, we can choose to eliminate the", + "type": "text" + }, + { + "bbox": [ + 478, + 441, + 484, + 449 + ], + "score": 0.61, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "(left", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 170, + 462 + ], + "score": 1.0, + "content": "identity) or the", + "type": "text" + }, + { + "bbox": [ + 171, + 451, + 181, + 460 + ], + "score": 0.72, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 449, + 378, + 462 + ], + "score": 1.0, + "content": "(right identity). Notably, choosing to eliminate", + "type": "text" + }, + { + "bbox": [ + 379, + 451, + 389, + 460 + ], + "score": 0.65, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "does not progress the proof", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 281, + 473 + ], + "score": 1.0, + "content": "because we cannot simplify the proof state", + "type": "text" + }, + { + "bbox": [ + 281, + 461, + 323, + 471 + ], + "score": 0.91, + "content": "b \\oplus e = b", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 461, + 506, + 473 + ], + "score": 1.0, + "content": ". Note that the proof is not stuck because we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 154, + 483 + ], + "score": 1.0, + "content": "can expand", + "type": "text" + }, + { + "bbox": [ + 154, + 472, + 177, + 482 + ], + "score": 0.9, + "content": "b \\oplus e", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 472, + 210, + 483 + ], + "score": 1.0, + "content": "back to", + "type": "text" + }, + { + "bbox": [ + 210, + 471, + 261, + 483 + ], + "score": 0.92, + "content": "b \\oplus ( e \\oplus m )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 472, + 338, + 483 + ], + "score": 1.0, + "content": "and then get rid of", + "type": "text" + }, + { + "bbox": [ + 339, + 473, + 345, + 481 + ], + "score": 0.76, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 472, + 506, + 483 + ], + "score": 1.0, + "content": "the second time around. Thus, a prover", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "has at least two choices in solving such problems: (1) maintain a global perspective to choose which", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "parts of the goal to rewrite or (2) learn to expand terms. For this problem, we write a deterministic", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "procedure that generates proofs of the first form that selects a position and an identity law, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "attempt to learn this algorithm. We do not generate proofs that require backtracking (e.g., due to a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 527, + 411, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 411, + 538 + ], + "score": 1.0, + "content": "greedy rewrite) although it would be an interesting direction of future work.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 427, + 506, + 538 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 552, + 270, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 271, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 271, + 565 + ], + "score": 1.0, + "content": "6.2 END-TO-END PROOF SYNTHESIS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 504, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 415, + 586 + ], + "score": 1.0, + "content": "Tactic prediction The tactic prediction model embeds a proof state into", + "type": "text" + }, + { + "bbox": [ + 415, + 572, + 430, + 583 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 572, + 505, + 586 + ], + "score": 1.0, + "content": "and uses a fully-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 597 + ], + "score": 1.0, + "content": "connected layer for prediction. We can model the proofs for this problem as a tactic prediction", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 609 + ], + "score": 1.0, + "content": "problem where the predicted category is a pair of the position in the AST and the identity to apply.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "We convert the position in the AST into a number using a preorder traversal of the Coq AST. For", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 190, + 630 + ], + "score": 1.0, + "content": "example, the second", + "type": "text" + }, + { + "bbox": [ + 190, + 618, + 200, + 628 + ], + "score": 0.81, + "content": "\\oplus", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 617, + 272, + 630 + ], + "score": 1.0, + "content": "in the expression", + "type": "text" + }, + { + "bbox": [ + 272, + 617, + 324, + 629 + ], + "score": 0.91, + "content": "b \\oplus ( e \\oplus m )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "has position 2. We can encode each identity", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 628, + 423, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 423, + 641 + ], + "score": 1.0, + "content": "as a single number. We obtain the prediction class as the pair of both numbers.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 572, + 506, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "We implement end-to-end proof synthesis by combining a tactic prediction model that is trained", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "offline with GamePad’s lightweight interaction module. The interaction module takes care of reading", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "in the current Coq proof state and sending tactic calls to advance the proof. We use the trained model", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "to do inference on the current Coq proof state to obtain a position to rewrite and the identity law to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "apply. We then translate this to a tactic and take the corresponding action in Coq. For the current", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 279, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 279, + 713 + ], + "score": 1.0, + "content": "problem, we do not consider backtracking.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 644, + 506, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 148 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 403, + 95 + ], + "score": 1.0, + "content": "Results For this problem, we generate 400 unique theorems of the form", + "type": "text" + }, + { + "bbox": [ + 403, + 83, + 466, + 94 + ], + "score": 0.91, + "content": "\\forall b \\in G , X = b", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "and their", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 162, + 105 + ], + "score": 1.0, + "content": "proofs where", + "type": "text" + }, + { + "bbox": [ + 162, + 94, + 172, + 104 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 93, + 478, + 105 + ], + "score": 1.0, + "content": "is a randomly generated algebraic expression of length 10 that evaluates to", + "type": "text" + }, + { + "bbox": [ + 478, + 94, + 484, + 104 + ], + "score": 0.63, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 93, + 505, + 105 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 146, + 117 + ], + "score": 1.0, + "content": "construct", + "type": "text" + }, + { + "bbox": [ + 146, + 105, + 156, + 114 + ], + "score": 0.78, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 104, + 394, + 117 + ], + "score": 1.0, + "content": "by recursively expanding the left and right expressions of", + "type": "text" + }, + { + "bbox": [ + 395, + 105, + 404, + 115 + ], + "score": 0.83, + "content": "\\oplus", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "subject to the constraint", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 311, + 128 + ], + "score": 1.0, + "content": "that only one side, chosen at random, reduces to", + "type": "text" + }, + { + "bbox": [ + 311, + 116, + 317, + 125 + ], + "score": 0.67, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "and the other side reduces to the appropriate", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "identity (left or right). We stop when the length of the expression is 10. We then extract the proof", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 345, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 345, + 149 + ], + "score": 1.0, + "content": "states using GamePad and train the tactic prediction model.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "To test the model, we generate a distinct set of 50 randomly generated algebraic expressions of length", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "10 and test how many proofs our model can complete using a greedy approach. That is, at each proof", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "state, we use the trained model to perform inference and choose the (rewrite position, identity law)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "tuple with the highest probability as our action. We find that such an approach can generate 14", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "complete proofs, where we score a proof as a failure if any proof step fails. For expressions of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "length 10, there are 9 proof steps. We can relax this setting so that when any single proof step fails,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "we use the deterministic procedure to supply a proof step. This approach completes all 50 proofs", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 327, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 327, + 244 + ], + "score": 1.0, + "content": "with an average failure rate of 1 proof step per a proof.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 504, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "We have observed cases where a good position is selected but the wrong identity law is paired with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "it. As we predict the position and rewrite jointly, this behavior is somewhat surprising. We also", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 416, + 282 + ], + "score": 1.0, + "content": "observe that the accuracy on the same test set for the tactic prediction task is", + "type": "text" + }, + { + "bbox": [ + 416, + 270, + 435, + 280 + ], + "score": 0.89, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 269, + 506, + 282 + ], + "score": 1.0, + "content": ". Thus, while the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "imitation learning model has good generalization in the traditional sense, more work needs to be", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 361, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 361, + 304 + ], + "score": 1.0, + "content": "done to leverage the policy when synthesizing complete proofs.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 325, + 257, + 338 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 259, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 259, + 340 + ], + "score": 1.0, + "content": "7 REAL-WORLD DATA SETS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 504, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "score": 1.0, + "content": "We can also apply GamePad to data extracted from real-world formalizations. In this section, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 366, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 378 + ], + "score": 1.0, + "content": "apply baseline position evaluation and tactic prediction models to data extracted from the Feit-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 378, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 389 + ], + "score": 1.0, + "content": "Thompson formalization. We encounter difficulties not present in the simple algebraic rewrite do-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 407, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 407, + 401 + ], + "score": 1.0, + "content": "main here, including scaling and a more difficult tactic prediction problem.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "The Feit-Thompson data set The Feit-Thompson theorem states that every finite group of odd-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 428, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 443 + ], + "score": 1.0, + "content": "order is solvable, and is a deep result in group theory. The formalization has the interesting property", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "that the researchers attempted to follow the book proofs as closely as possible. The extracted data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "set consists of 1602 lemmas and expands into 83478 proof states. For our tasks, we split the lemmas", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 358, + 475 + ], + "score": 1.0, + "content": "in the data set into a training, validation and test set in the ratio", + "type": "text" + }, + { + "bbox": [ + 358, + 463, + 391, + 473 + ], + "score": 0.9, + "content": "8 : 1 : 1", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 463, + 505, + 475 + ], + "score": 1.0, + "content": ", and ensure that the number", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "of proof states in the splits are in a similar ratio. Note that we split by lemmas and not by proof states", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "because predictions for proof states within the proof of the same lemma are not independent and can", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "lead to a more optimistic evaluation of the generalization capability of the models (particularly for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 506, + 234, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 234, + 518 + ], + "score": 1.0, + "content": "the case of position evaluation).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "As a reminder, each proof state consists of a context and an associated goal. Each context contains", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 504, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 504, + 546 + ], + "score": 1.0, + "content": "on average 60 terms or identifiers, and on average 4000 nodes at the kernel level (and 1000 at the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "mid level without implicit arguments). The most common nodes include constants, applications,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "and variables, and as such, we focus on those during the design of our embedding. A formalization", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "such as CompCert would contain a different distribution of AST nodes. In particular, as it concerns", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "program verification, there would be more AST nodes involving fixed-points4, case analysis, and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 590, + 219, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 219, + 602 + ], + "score": 1.0, + "content": "inductive type constructors.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 606, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "The most prevalent tactics include rewrite and have. As a reminder, a rewrite tactic requires", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 618, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 628 + ], + "score": 1.0, + "content": "the user to indicate which equality in the current local or global context to apply and is akin to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "premise selection. The have tactic introduces an intermediate lemma in the proof and is the hardest", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 637, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 506, + 653 + ], + "score": 1.0, + "content": "to learn as the user usually uses some mathematical insight about the problem before conjecturing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 650, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 662 + ], + "score": 1.0, + "content": "a statement. We currently do not synthesize arguments for have tactics, although our tool extracts", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 661, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 673 + ], + "score": 1.0, + "content": "such data. We believe a generative model for synthesizing them would be a great direction for future", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 672, + 132, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 132, + 684 + ], + "score": 1.0, + "content": "work.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 701, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 118, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 118, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "4One difficulty with embedding fixed-points is that we need the embedding of the body in order to embed", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 324, + 724 + ], + "score": 1.0, + "content": "the body. An interesting approach here is to introduce a loss", + "type": "text" + }, + { + "bbox": [ + 324, + 711, + 393, + 723 + ], + "score": 0.92, + "content": "| \\mathcal { E } ( f ( x ) ) - \\mathcal { E } ( x ) | ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 711, + 506, + 724 + ], + "score": 1.0, + "content": ", where we start with learnable", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 249, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 249, + 733 + ], + "score": 1.0, + "content": "random embeddings for the base cases.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 148 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 403, + 95 + ], + "score": 1.0, + "content": "Results For this problem, we generate 400 unique theorems of the form", + "type": "text" + }, + { + "bbox": [ + 403, + 83, + 466, + 94 + ], + "score": 0.91, + "content": "\\forall b \\in G , X = b", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "and their", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 162, + 105 + ], + "score": 1.0, + "content": "proofs where", + "type": "text" + }, + { + "bbox": [ + 162, + 94, + 172, + 104 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 93, + 478, + 105 + ], + "score": 1.0, + "content": "is a randomly generated algebraic expression of length 10 that evaluates to", + "type": "text" + }, + { + "bbox": [ + 478, + 94, + 484, + 104 + ], + "score": 0.63, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 93, + 505, + 105 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 146, + 117 + ], + "score": 1.0, + "content": "construct", + "type": "text" + }, + { + "bbox": [ + 146, + 105, + 156, + 114 + ], + "score": 0.78, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 104, + 394, + 117 + ], + "score": 1.0, + "content": "by recursively expanding the left and right expressions of", + "type": "text" + }, + { + "bbox": [ + 395, + 105, + 404, + 115 + ], + "score": 0.83, + "content": "\\oplus", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "subject to the constraint", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 311, + 128 + ], + "score": 1.0, + "content": "that only one side, chosen at random, reduces to", + "type": "text" + }, + { + "bbox": [ + 311, + 116, + 317, + 125 + ], + "score": 0.67, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "and the other side reduces to the appropriate", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "identity (left or right). We stop when the length of the expression is 10. We then extract the proof", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 345, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 345, + 149 + ], + "score": 1.0, + "content": "states using GamePad and train the tactic prediction model.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 149 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "To test the model, we generate a distinct set of 50 randomly generated algebraic expressions of length", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "10 and test how many proofs our model can complete using a greedy approach. That is, at each proof", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "state, we use the trained model to perform inference and choose the (rewrite position, identity law)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "tuple with the highest probability as our action. We find that such an approach can generate 14", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "complete proofs, where we score a proof as a failure if any proof step fails. For expressions of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "length 10, there are 9 proof steps. We can relax this setting so that when any single proof step fails,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "we use the deterministic procedure to supply a proof step. This approach completes all 50 proofs", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 327, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 327, + 244 + ], + "score": 1.0, + "content": "with an average failure rate of 1 proof step per a proof.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 154, + 506, + 244 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 504, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "We have observed cases where a good position is selected but the wrong identity law is paired with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "it. As we predict the position and rewrite jointly, this behavior is somewhat surprising. We also", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 416, + 282 + ], + "score": 1.0, + "content": "observe that the accuracy on the same test set for the tactic prediction task is", + "type": "text" + }, + { + "bbox": [ + 416, + 270, + 435, + 280 + ], + "score": 0.89, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 269, + 506, + 282 + ], + "score": 1.0, + "content": ". Thus, while the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "imitation learning model has good generalization in the traditional sense, more work needs to be", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 361, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 361, + 304 + ], + "score": 1.0, + "content": "done to leverage the policy when synthesizing complete proofs.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 247, + 506, + 304 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 325, + 257, + 338 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 259, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 259, + 340 + ], + "score": 1.0, + "content": "7 REAL-WORLD DATA SETS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 504, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "score": 1.0, + "content": "We can also apply GamePad to data extracted from real-world formalizations. In this section, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 366, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 378 + ], + "score": 1.0, + "content": "apply baseline position evaluation and tactic prediction models to data extracted from the Feit-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 378, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 389 + ], + "score": 1.0, + "content": "Thompson formalization. We encounter difficulties not present in the simple algebraic rewrite do-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 407, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 407, + 401 + ], + "score": 1.0, + "content": "main here, including scaling and a more difficult tactic prediction problem.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 354, + 506, + 401 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "The Feit-Thompson data set The Feit-Thompson theorem states that every finite group of odd-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 428, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 443 + ], + "score": 1.0, + "content": "order is solvable, and is a deep result in group theory. The formalization has the interesting property", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "that the researchers attempted to follow the book proofs as closely as possible. The extracted data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "set consists of 1602 lemmas and expands into 83478 proof states. For our tasks, we split the lemmas", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 358, + 475 + ], + "score": 1.0, + "content": "in the data set into a training, validation and test set in the ratio", + "type": "text" + }, + { + "bbox": [ + 358, + 463, + 391, + 473 + ], + "score": 0.9, + "content": "8 : 1 : 1", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 463, + 505, + 475 + ], + "score": 1.0, + "content": ", and ensure that the number", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "of proof states in the splits are in a similar ratio. Note that we split by lemmas and not by proof states", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "because predictions for proof states within the proof of the same lemma are not independent and can", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "lead to a more optimistic evaluation of the generalization capability of the models (particularly for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 506, + 234, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 234, + 518 + ], + "score": 1.0, + "content": "the case of position evaluation).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 418, + 506, + 518 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "As a reminder, each proof state consists of a context and an associated goal. Each context contains", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 504, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 504, + 546 + ], + "score": 1.0, + "content": "on average 60 terms or identifiers, and on average 4000 nodes at the kernel level (and 1000 at the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "mid level without implicit arguments). The most common nodes include constants, applications,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "and variables, and as such, we focus on those during the design of our embedding. A formalization", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "such as CompCert would contain a different distribution of AST nodes. In particular, as it concerns", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "program verification, there would be more AST nodes involving fixed-points4, case analysis, and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 590, + 219, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 219, + 602 + ], + "score": 1.0, + "content": "inductive type constructors.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 523, + 505, + 602 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 606, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "The most prevalent tactics include rewrite and have. As a reminder, a rewrite tactic requires", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 618, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 628 + ], + "score": 1.0, + "content": "the user to indicate which equality in the current local or global context to apply and is akin to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "premise selection. The have tactic introduces an intermediate lemma in the proof and is the hardest", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 637, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 506, + 653 + ], + "score": 1.0, + "content": "to learn as the user usually uses some mathematical insight about the problem before conjecturing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 650, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 662 + ], + "score": 1.0, + "content": "a statement. We currently do not synthesize arguments for have tactics, although our tool extracts", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 661, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 673 + ], + "score": 1.0, + "content": "such data. We believe a generative model for synthesizing them would be a great direction for future", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 672, + 132, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 132, + 684 + ], + "score": 1.0, + "content": "work.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 606, + 506, + 684 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 110, + 111, + 497, + 136 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 80, + 504, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 505, + 93 + ], + "score": 1.0, + "content": "Table 1: Training time speedups for the GRU model with state size of 128 on the position evaluation", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 473, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 473, + 103 + ], + "score": 1.0, + "content": "task obtained compared to an un-optimized baseline. ∗ indicates CPU and † indicates GPU.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 110, + 111, + 497, + 136 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 111, + 497, + 136 + ], + "spans": [ + { + "bbox": [ + 110, + 111, + 497, + 136 + ], + "score": 0.925, + "html": "
ModelBase*Embedding Sharing*Dynamic Batching*Both*Both†
Speedup (approx.)110×10×130×190×
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ModelPostPostTactTactTact t arguments
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SVM57.3757.5248.9449.45=
GRU65.3065.7458.2357.7025.98
TreeLSTM68.4466.3060.6360.5523.91
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We can", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "apply two optimizations to scale to larger trees. The first optimization involves embedding sharing,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "where we memoize the embedding and the associated computation graph for any expression or", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 308, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 319 + ], + "score": 1.0, + "content": "subtree when it appears another time in the same proof state. 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ModelBase*Embedding Sharing*Dynamic Batching*Both*Both†
Speedup (approx.)110×10×130×190×
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ModelPostPostTactTactTact t arguments
Constant53.6653.6644.7544.751
SVM57.3757.5248.9449.45=
GRU65.3065.7458.2357.7025.98
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We can", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "apply two optimizations to scale to larger trees. The first optimization involves embedding sharing,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "where we memoize the embedding and the associated computation graph for any expression or", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 308, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 319 + ], + "score": 1.0, + "content": "subtree when it appears another time in the same proof state. For instance, if the context has terms", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 107, + 318, + 138, + 329 + ], + "score": 0.9, + "content": "M _ { 1 } M _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 317, + 156, + 331 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 156, + 318, + 172, + 329 + ], + "score": 0.89, + "content": "M _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 317, + 252, + 331 + ], + "score": 1.0, + "content": ", the embedding for", + "type": "text" + }, + { + "bbox": [ + 253, + 318, + 268, + 329 + ], + "score": 0.89, + "content": "M _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 317, + 506, + 331 + ], + "score": 1.0, + "content": "is computed once. A single forward and backward pass is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "thus performed on this computation graph, with the gradients automatically accumulating from all", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 341, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 352 + ], + "score": 1.0, + "content": "places where the embedding was used. It thus helps save both memory and computation time of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "our models. The second optimization involves dynamic batching (Looks et al., 2017; Polosukhin", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "& Zavershynskyi, 2018), which enables us to efficiently batch the computation of ops that perform", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "the same operation albeit on different inputs, and thus make better use of the parallel acceleration", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 383, + 286, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 383, + 286, + 395 + ], + "score": 1.0, + "content": "provided by multi-core CPU’s and GPU’s. 5", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 274, + 506, + 395 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 400, + 504, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 414 + ], + "score": 1.0, + "content": "Table 1 shows the approximate speedups obtained from applying embedding sharing and dynamic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 412, + 502, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 502, + 424 + ], + "score": 1.0, + "content": "batching to representing proof states. Note that the GPU speedup will increase with larger models.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 399, + 505, + 424 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 439, + 340, + 451 + ], + "lines": [ + { + "bbox": [ + 107, + 439, + 342, + 452 + ], + "spans": [ + { + "bbox": [ + 107, + 439, + 342, + 452 + ], + "score": 1.0, + "content": "7.1 POSITION EVALUATION AND TACTIC PREDICTION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 396, + 474 + ], + "score": 1.0, + "content": "We use the interpreter-inspired embeddings to embed proof states into", + "type": "text" + }, + { + "bbox": [ + 396, + 461, + 412, + 472 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 461, + 505, + 474 + ], + "score": 1.0, + "content": ", and then aim to train", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "models for position evaluation and tactic prediction tasks. For the position evaluation task, we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 239, + 497 + ], + "score": 1.0, + "content": "binned the target predictions into", + "type": "text" + }, + { + "bbox": [ + 240, + 484, + 268, + 494 + ], + "score": 0.9, + "content": "K = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 483, + 333, + 497 + ], + "score": 1.0, + "content": "classes—close (", + "type": "text" + }, + { + "bbox": [ + 333, + 484, + 350, + 494 + ], + "score": 0.83, + "content": "\\mathit { \\Theta } _ { \\prec } 5", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 483, + 452, + 497 + ], + "score": 1.0, + "content": "steps), medium (between", + "type": "text" + }, + { + "bbox": [ + 452, + 484, + 479, + 494 + ], + "score": 0.54, + "content": "6 - 1 9", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 483, + 505, + 497 + ], + "score": 1.0, + "content": "steps,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 494, + 421, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 183, + 507 + ], + "score": 1.0, + "content": "inclusive), and far", + "type": "text" + }, + { + "bbox": [ + 184, + 495, + 206, + 505 + ], + "score": 0.79, + "content": "> 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 494, + 421, + 507 + ], + "score": 1.0, + "content": "steps), and perform a simple three way classification.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 461, + 505, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "The tactic prediction is more complicated. First, we group tactics into equivalence classes if they", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "perform the same mathematical operation. For example, we group the tactics reflexivity and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "done together because they are applied at the end of proofs to show that a statement is trivially true.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "Second, we now also have tactic arguments. We train two models, one that predicts only the tactic", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 555, + 504, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 504, + 568 + ], + "score": 1.0, + "content": "and another that additionally predicts the arguments. The first is a 23 way classification problem.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "For the second, the arguments could be (1) a term from the local context, (2) a term from the global", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "context (e.g., a lemma), or (3) a term created by the human user. As arguments in the third category", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 588, + 504, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 504, + 601 + ], + "score": 1.0, + "content": "can be any Coq term, the space of arguments is potentially infinite. As a reminder, the data set makes", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "extensive use of have tactics—in essence, this would require the model to conjecture a statement.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "For our baseline models, we focus on arguments in the first category and predict the presence or", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 620, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 634 + ], + "score": 1.0, + "content": "absence of each term in the context at any position in the arguments. For each term, we use the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 630, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 646 + ], + "score": 1.0, + "content": "final hidden state and the embedding of each term followed by a linear layer to produce a two way", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "prediction. Note that the distribution of labels is skewed towards the absent category. Thus, we are", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "score": 1.0, + "content": "more interested in the precision-recall curve. We weigh the cross-entropy loss higher for presence", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "class, and also try to balance the distribution of labels by randomly sampling only a subset of the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 676, + 230, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 230, + 689 + ], + "score": 1.0, + "content": "negative class at training time.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 511, + 506, + 689 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Results We first start by training a simple SVM (Cortes & Vapnik, 1995) on a set of heuristic", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "features like context size, goal size, number of hypothesis in context and the smallest edit distance", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 116 + ], + "score": 1.0, + "content": "between a hypothesis and the context. The SVM performs better than the constant baseline of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 114, + 507, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 114, + 507, + 129 + ], + "score": 1.0, + "content": "guessing the most common class. We then train RNN models to utilise our embedding strategy.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "We train GRU (Cho et al., 2014) and TreeLSTM (Tai et al., 2015) models using mini-batches of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "32 proof states, set the RNN state size to 128, and use the Adam optimizer Kingma & Ba (2015)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "with a learning rate of 0.001. We use input (Srivastava et al., 2014) and weight (Merity et al., 2017)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "dropout with a rate of 0.1 for the TreeLSTM models (higher rates led to much slower learning). All", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "neural net models were trained using PyTorch (Paszke et al., 2017). Table 2 show the results for the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "score": 1.0, + "content": "tasks. We were able to improve upon the SVM baseline, which indicates that it is possible to learn", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "score": 1.0, + "content": "useful representations using the human supervision data and utilizing our proof state embeddings.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "score": 1.0, + "content": "We then experiment with removing the bookkeeping aspects of the prover by switching from kernel", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "level to mid level proof states without implicit arguments. We obtain similar accuracies, indicating", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 225, + 246, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 246, + 237 + ], + "score": 1.0, + "content": "that most of that data is redundant.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 108, + 252, + 210, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 251, + 213, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 213, + 267 + ], + "score": 1.0, + "content": "8 RELATED WORK", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 108, + 277, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "score": 1.0, + "content": "The level of abstraction and representation of proofs that learning is applied to are salient points of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "comparison between work on learning and theorem proving. These choices inform the setup and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 299, + 303, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 303, + 313 + ], + "score": 1.0, + "content": "challenges associated with the learning problem.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "As in our work, there are systems that experiment with learning in ITPs. Duncan (2002) explores", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 328, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 339 + ], + "score": 1.0, + "content": "how to learn (user-defined) tactics from a corpus of proofs in Isabelle so that they can be applied", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 337, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 352 + ], + "score": 1.0, + "content": "to future proofs. ML4PG (Komendantskaya et al., 2012) interfaces to ITPs at the level of the user", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "interface for entering proof scripts. Thus, ML4PG is applicable to multiple ITPs although it ob-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "tains less granular proof states. Holphrasm (Whalen, 2016) uses string encodings of proof states", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 371, + 504, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 504, + 383 + ], + "score": 1.0, + "content": "and focuses on tactic argument synthesis (there is essentially only one tactic in the underlying ITP", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 380, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 396 + ], + "score": 1.0, + "content": "MetaMath (Megill, 2007)). HolStep (Kaliszyk et al., 2017) addresses premise selection using string-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 407 + ], + "score": 1.0, + "content": "level encodings of proof states. Wang et al. (2017) extend the HolStep work to show the advantage", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "of using a DeBruijn representation of proof terms as opposed to string-level encodings for premise", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "selection. The structured representation provided by GamePad would support experimenting with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 424, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 440 + ], + "score": 1.0, + "content": "such extensions. Gauthier et al. (2017) explores learning tactic-level proof search in Isabelle using", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "hand-crafted features on string encodings of proof states. It would be interesting to experiment with", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "their algorithm to our algebraic rewrite problem. Nagashima & He (2018) looks at explainable tactic", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 459, + 195, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 195, + 470 + ], + "score": 1.0, + "content": "prediction in Isabelle.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "Other approaches focus on automated theorem provers, which are designed to prove theorems with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "no human interaction. Irving et al. (2016) describes the premise selection problem and trains neural", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "network models on proof traces obtained from applying E (Schulz, 2002) to the Mizar Mathemat-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "ical Corpus. Loos et al. (2017), in addition to addressing premise selection, also address a clause", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "selection task by applying neural network models to this problem in E. Kaliszyk & Urban (2014)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 353, + 543 + ], + "score": 1.0, + "content": "demonstrate that similar learning based methods can prove", + "type": "text" + }, + { + "bbox": [ + 354, + 531, + 374, + 541 + ], + "score": 0.87, + "content": "3 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "of the lemmas in the Flyspeck", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 542, + 246, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 246, + 554 + ], + "score": 1.0, + "content": "project (Kaliszyk & Urban, 2014).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 504, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "Another take on learning and theorem proving is to replace an entire theorem proving (sub)routine", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "with a learned algorithm instead of using learning for heuristics. For instance, end-to-end differ-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 580, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 593 + ], + "score": 1.0, + "content": "entiable proving (Rocktaschel & Riedel, 2017) replaces traditional ¨ unification with a trained neu-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "ral network and demonstrates the efficacy of this approach for knowledge base completion. Neu-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "rosat (Selsam et al., 2018) applies a neural network model to predict satisfaction problems and shows", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 613, + 266, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 266, + 626 + ], + "score": 1.0, + "content": "how to recover a satisfying assignment.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 107, + 641, + 195, + 654 + ], + "lines": [ + { + "bbox": [ + 104, + 638, + 198, + 657 + ], + "spans": [ + { + "bbox": [ + 104, + 638, + 198, + 657 + ], + "score": 1.0, + "content": "9 CONCLUSION", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 663, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 663, + 506, + 680 + ], + "score": 1.0, + "content": "In this work, we look at theorem proving problem through the lens of a system that enables learning", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "with proofs constructed with human supervision. We highlight three key aspects of the problem", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "at this level. The first concerns obtaining inputs to a learning algorithm that approximate the level", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "of abstraction faced by a human prover. For this, we use an ITP, as it retains aspects of human", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "supervision. GamePad preserves the structure of the proofs (e.g., annotations regarding implicit", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "arguments) so they can be used for building models. The second involves building models that", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Results We first start by training a simple SVM (Cortes & Vapnik, 1995) on a set of heuristic", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "features like context size, goal size, number of hypothesis in context and the smallest edit distance", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 116 + ], + "score": 1.0, + "content": "between a hypothesis and the context. The SVM performs better than the constant baseline of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 114, + 507, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 114, + 507, + 129 + ], + "score": 1.0, + "content": "guessing the most common class. We then train RNN models to utilise our embedding strategy.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "We train GRU (Cho et al., 2014) and TreeLSTM (Tai et al., 2015) models using mini-batches of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "32 proof states, set the RNN state size to 128, and use the Adam optimizer Kingma & Ba (2015)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "with a learning rate of 0.001. We use input (Srivastava et al., 2014) and weight (Merity et al., 2017)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "dropout with a rate of 0.1 for the TreeLSTM models (higher rates led to much slower learning). All", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "neural net models were trained using PyTorch (Paszke et al., 2017). Table 2 show the results for the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "score": 1.0, + "content": "tasks. We were able to improve upon the SVM baseline, which indicates that it is possible to learn", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "score": 1.0, + "content": "useful representations using the human supervision data and utilizing our proof state embeddings.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "score": 1.0, + "content": "We then experiment with removing the bookkeeping aspects of the prover by switching from kernel", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "level to mid level proof states without implicit arguments. We obtain similar accuracies, indicating", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 225, + 246, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 246, + 237 + ], + "score": 1.0, + "content": "that most of that data is redundant.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 82, + 507, + 237 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 252, + 210, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 251, + 213, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 213, + 267 + ], + "score": 1.0, + "content": "8 RELATED WORK", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 108, + 277, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "score": 1.0, + "content": "The level of abstraction and representation of proofs that learning is applied to are salient points of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "comparison between work on learning and theorem proving. These choices inform the setup and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 299, + 303, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 303, + 313 + ], + "score": 1.0, + "content": "challenges associated with the learning problem.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 277, + 506, + 313 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "As in our work, there are systems that experiment with learning in ITPs. Duncan (2002) explores", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 328, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 339 + ], + "score": 1.0, + "content": "how to learn (user-defined) tactics from a corpus of proofs in Isabelle so that they can be applied", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 337, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 352 + ], + "score": 1.0, + "content": "to future proofs. ML4PG (Komendantskaya et al., 2012) interfaces to ITPs at the level of the user", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "interface for entering proof scripts. Thus, ML4PG is applicable to multiple ITPs although it ob-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "tains less granular proof states. Holphrasm (Whalen, 2016) uses string encodings of proof states", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 371, + 504, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 504, + 383 + ], + "score": 1.0, + "content": "and focuses on tactic argument synthesis (there is essentially only one tactic in the underlying ITP", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 380, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 396 + ], + "score": 1.0, + "content": "MetaMath (Megill, 2007)). HolStep (Kaliszyk et al., 2017) addresses premise selection using string-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 407 + ], + "score": 1.0, + "content": "level encodings of proof states. Wang et al. (2017) extend the HolStep work to show the advantage", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "of using a DeBruijn representation of proof terms as opposed to string-level encodings for premise", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "selection. The structured representation provided by GamePad would support experimenting with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 424, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 440 + ], + "score": 1.0, + "content": "such extensions. Gauthier et al. (2017) explores learning tactic-level proof search in Isabelle using", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "hand-crafted features on string encodings of proof states. It would be interesting to experiment with", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "their algorithm to our algebraic rewrite problem. Nagashima & He (2018) looks at explainable tactic", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 459, + 195, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 195, + 470 + ], + "score": 1.0, + "content": "prediction in Isabelle.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 317, + 506, + 470 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "Other approaches focus on automated theorem provers, which are designed to prove theorems with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "no human interaction. Irving et al. (2016) describes the premise selection problem and trains neural", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "network models on proof traces obtained from applying E (Schulz, 2002) to the Mizar Mathemat-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "ical Corpus. Loos et al. (2017), in addition to addressing premise selection, also address a clause", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "selection task by applying neural network models to this problem in E. Kaliszyk & Urban (2014)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 353, + 543 + ], + "score": 1.0, + "content": "demonstrate that similar learning based methods can prove", + "type": "text" + }, + { + "bbox": [ + 354, + 531, + 374, + 541 + ], + "score": 0.87, + "content": "3 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "of the lemmas in the Flyspeck", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 542, + 246, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 246, + 554 + ], + "score": 1.0, + "content": "project (Kaliszyk & Urban, 2014).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 475, + 505, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 504, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "Another take on learning and theorem proving is to replace an entire theorem proving (sub)routine", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "with a learned algorithm instead of using learning for heuristics. For instance, end-to-end differ-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 580, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 593 + ], + "score": 1.0, + "content": "entiable proving (Rocktaschel & Riedel, 2017) replaces traditional ¨ unification with a trained neu-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "ral network and demonstrates the efficacy of this approach for knowledge base completion. Neu-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "rosat (Selsam et al., 2018) applies a neural network model to predict satisfaction problems and shows", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 613, + 266, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 266, + 626 + ], + "score": 1.0, + "content": "how to recover a satisfying assignment.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 558, + 506, + 626 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 641, + 195, + 654 + ], + "lines": [ + { + "bbox": [ + 104, + 638, + 198, + 657 + ], + "spans": [ + { + "bbox": [ + 104, + 638, + 198, + 657 + ], + "score": 1.0, + "content": "9 CONCLUSION", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 663, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 663, + 506, + 680 + ], + "score": 1.0, + "content": "In this work, we look at theorem proving problem through the lens of a system that enables learning", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "with proofs constructed with human supervision. We highlight three key aspects of the problem", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "at this level. The first concerns obtaining inputs to a learning algorithm that approximate the level", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "of abstraction faced by a human prover. For this, we use an ITP, as it retains aspects of human", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "supervision. GamePad preserves the structure of the proofs (e.g., annotations regarding implicit", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "arguments) so they can be used for building models. The second involves building models that", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "employ the game-like structure of ITP proofs. Here, we experiment with tactic prediction for toy", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "and real world data sets. Finally, as a consequence of theorem proving at a higher-level (compared to", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "SMT solvers), we will need to be careful to distinguish the syntax from the semantics of terms. Our", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "current approach is to provide structured representations of terms so that more semantic structure", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "can be exploited. While our results are preliminary, our hope is that GamePad provides an accessible", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 149 + ], + "score": 1.0, + "content": "starting point to explore the application of machine learning in the context of interactive theorem", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 143, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 143, + 163 + ], + "score": 1.0, + "content": "proving.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 48.5, + "bbox_fs": [ + 104, + 663, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "employ the game-like structure of ITP proofs. Here, we experiment with tactic prediction for toy", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "and real world data sets. Finally, as a consequence of theorem proving at a higher-level (compared to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "SMT solvers), we will need to be careful to distinguish the syntax from the semantics of terms. Our", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "current approach is to provide structured representations of terms so that more semantic structure", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "can be exploited. While our results are preliminary, our hope is that GamePad provides an accessible", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 149 + ], + "score": 1.0, + "content": "starting point to explore the application of machine learning in the context of interactive theorem", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 143, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 143, + 163 + ], + "score": 1.0, + "content": "proving.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 108, + 177, + 211, + 190 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 213, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 213, + 192 + ], + "score": 1.0, + "content": "10 FUTURE WORK", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "We end by suggesting a few avenues for extending our work. The first concerns the design of new", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "benchmarks for human-level proofs. In this paper, we designed a relatively simple algebraic rewrite", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "problem to test the system end-to-end. Designing more difficult problems that still admit tractable", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "learning (such as solving infinite sums or integrals) would be a great direction for future work.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "Note that you can define a new domain inside Coq and use GamePad to build provers that learn", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "from example proofs. A second direction concerns building models that conjecture and explore", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "the space of true statements, in addition to proving statements. This is particularly important in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "synthesizing arguments to tactics like have. Currently, we only predict the tactic identifiers and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "do not synthesize the entire term. Building a generative model would be a great next step. Lastly,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 300, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 104, + 300, + 505, + 316 + ], + "score": 1.0, + "content": "it would be interesting to see if using end-to-end training with reinforcement learning and utilizing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "Monte-Carlo tree search to efficiently explore the search space can be effectively applied to human-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 322, + 163, + 336 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 163, + 336 + ], + "score": 1.0, + "content": "level proofs.6", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 109, + 353, + 250, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 252, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 252, + 367 + ], + "score": 1.0, + "content": "11 ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 504, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "We would like to thank Diederik Kingma, Tim Salimans, and Geoffrey Irving for reviewing initial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "drafts of the work. We thank Daniel Selsam for suggesting that we consider removing implicit", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 400, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 414 + ], + "score": 1.0, + "content": "arguments, and Jonathan Cai for discussions about the toy problem. Daniel Huang was supported", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 411, + 236, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 236, + 423 + ], + "score": 1.0, + "content": "by DARPA FA8750-17-2-0091.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 441, + 175, + 453 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 176, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 176, + 455 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "Kyunghyun Cho, Bart van Merrienboer, C¸ alar G ¨ ulc¸ehre, Dzmitry Bahdanau, Fethi Bougares, Holger ¨", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 115, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "Schwenk, and Yoshua Bengio. 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The first concerns the design of new", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "benchmarks for human-level proofs. In this paper, we designed a relatively simple algebraic rewrite", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "problem to test the system end-to-end. Designing more difficult problems that still admit tractable", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "learning (such as solving infinite sums or integrals) would be a great direction for future work.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "Note that you can define a new domain inside Coq and use GamePad to build provers that learn", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "from example proofs. A second direction concerns building models that conjecture and explore", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "the space of true statements, in addition to proving statements. This is particularly important in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "synthesizing arguments to tactics like have. Currently, we only predict the tactic identifiers and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "do not synthesize the entire term. Building a generative model would be a great next step. Lastly,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 300, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 104, + 300, + 505, + 316 + ], + "score": 1.0, + "content": "it would be interesting to see if using end-to-end training with reinforcement learning and utilizing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "Monte-Carlo tree search to efficiently explore the search space can be effectively applied to human-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 322, + 163, + 336 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 163, + 336 + ], + "score": 1.0, + "content": "level proofs.6", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13.5, + "bbox_fs": [ + 104, + 203, + 505, + 336 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 353, + 250, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 252, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 252, + 367 + ], + "score": 1.0, + "content": "11 ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 504, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "We would like to thank Diederik Kingma, Tim Salimans, and Geoffrey Irving for reviewing initial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "drafts of the work. We thank Daniel Selsam for suggesting that we consider removing implicit", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 400, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 414 + ], + "score": 1.0, + "content": "arguments, and Jonathan Cai for discussions about the toy problem. Daniel Huang was supported", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 411, + 236, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 236, + 423 + ], + "score": 1.0, + "content": "by DARPA FA8750-17-2-0091.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 378, + 506, + 423 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 441, + 175, + 453 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 176, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 176, + 455 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "Kyunghyun Cho, Bart van Merrienboer, C¸ alar G ¨ ulc¸ehre, Dzmitry Bahdanau, Fethi Bougares, Holger ¨", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 115, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "Schwenk, and Yoshua Bengio. 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