Add files using upload-large-folder tool

parse/train/AVx0r_GppCu/AVx0r_GppCu.md

@@ -0,0 +1,461 @@
+# Super-Acceleration with Cyclical Step-sizes
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Cyclical step-sizes are becoming increasingly popular in the optimization of deep   
+2 learning problems. Motivated by recent observations on the spectral gaps of   
+3 Hessians in machine learning, we show that these step-size schedules offer a   
+4 simple way to exploit them. More precisely, we develop a convergence rate   
+5 analysis for quadratic objectives that provides optimal parameters and shows that   
+6 cyclical learning rates can improve upon traditional lower complexity bounds.   
+7 We further propose a systematic approach to design optimal first order methods   
+8 for quadratic minimization with a given spectral structure. Finally, we provide a   
+9 local convergence rate analysis beyond quadratic minimization for the proposed   
+10 methods and illustrate our findings through benchmarks on least squares and   
+11 logistic regression problems.
+
+# 12 1 Introduction
+
+13 One of the most iconic methods in first order optimization is gradient descent with momentum, also   
+14 known as the heavy ball method [Polyak, 1964]. This method enjoys widespread popularity both in   
+15 its original formulation and in a stochastic variant that replaces the gradient by a stochastic estimate,   
+16 a method that is behind many of the recent breakthroughs in deep learning [Sutskever et al., 2013].   
+17 A variant of the stochastic heavy ball where the step-sizes are chosen in cyclical order has recently   
+18 come to the forefront of machine learning research, showing state-of-the art results on different deep   
+19 learning benchmarks [Loshchilov and Hutter, 2017, Smith, 2017]. Inspired by this empirical success,   
+20 we aim to study the convergence of the heavy ball algorithm where step-sizes $h _ { 0 } , h _ { 1 } , \ldots$ are not fixed   
+21 or decreasing but instead chosen in cyclical order:
+
+# Algorithm 1: Cyclical heavy ball $\mathrm { H B } _ { K } ( h _ { 0 } , \ldots , h _ { K - 1 } ; m )$
+
+Input: Initialization $x _ { 0 }$ , momentum $m \in ( 0 , 1 )$ , step-sizes $\{ h _ { 0 } , \ldots , h _ { K - 1 } \}$ $\begin{array} { r l } & { x _ { 1 } = x _ { 0 } - \frac { h _ { 0 } } { 1 + m } \nabla f ( x _ { 0 } ) } \\ & { { \bf f o r } t = 1 , 2 , \dots { \bf d o } x _ { t + 1 } = x _ { t } - h _ { \mathrm { m o d } ( t , K ) } \nabla f ( x _ { t } ) + m ( x _ { t } - x _ { t - 1 } ) } \end{array}$ 1   
+end
+
+22 The heavy ball method with constant step-sizes enjoys a mature theory, where it is known for example   
+23 to achieve optimal black-box worst-case complexity of quadratic convex optimization [Nemirovsky,   
+24 1992]. In stark contrast, little is known about the the convergence of the above variant with cyclical   
+25 step-sizes. Our main motivating question is
+
+Do cyclical step-sizes improve convergence of heavy ball?
+
+27 Our main contribution provides a positive answer to this question and, more importantly, quantifies   
+28 the speedup under different assumptions. In particular, we show that for quadratic problems, whenever   
+29 Hessian’s spectrum belongs to two or more disjoint intervals, the heavy ball method with cyclical step  
+30 sizes achieves a faster worst-case convergence rate. Recent works have shown that this assumption on   
+31 the spectrum is quite natural and occurs in many machine learning problems, including deep neural   
+32 networks [Sagun et al., 2017, Papyan, 2018, Ghorbani et al., 2019, Papyan, 2019]. More precisely,   
+33 we list our main contributions below.   
+34 • In sections 3 and 4, we provide a tight convergence rate analysis of the cyclical heavy ball method   
+35 (Theorems 3.1 and 3.2 for two step-sizes, and Theorem 4.8 for the general case). This analysis   
+36 highlights a regime under which this method achieves a faster worst-case rate than the accelerated   
+37 rate of heavy ball, a phenomenon we refer to as super-acceleration. Theorem 5.1 extends the (local)   
+38 convergence rate analysis results to non-quadratic objectives.   
+39 • As a byproduct of the convergence-rate analysis, we obtain an explicit expression for the optimal   
+40 parameters in in the case of cycles of length two (Algorithm 2) and an implicit expression in terms   
+41 of a system of $K$ equations in the general case.
+
+• Section 6 presents numerical benchmarks illustrating the improved convergence of the cyclical approach on 4 problems involving quadratic and logistic losses on both synthetic and a handwritten digits recognition dataset.
+
+• Finally, we conclude in Section 7 with a discussion of this work’s limitations.
+
+# 46 2 Notation and Problem Setting
+
+47 Throughout the paper, we consider the problem of minimizing quadratic functions of the form
+
+$$
+\operatorname* { m i n } _ { \alpha \in \mathbb { B } ^ { d } } f ( x ) , \mathrm { w i t h } f \in \mathcal { C } _ { \Lambda } \triangleq \left\{ f : f ( x ) = \frac { 1 } { 2 } ( x - x _ { * } ) ^ { T } H ( x - x _ { * } ) + f _ { * } , \mathrm { S p } ( H ) \subseteq \Lambda \right\} ,
+$$
+
+48 where $\mathcal { C } _ { \Lambda }$ is the class of quadratic functions whose spectrum $\operatorname { S p } ( H )$ is localized in $\Lambda \subseteq [ \mu , L ] \subseteq \mathbb { R } _ { > 0 }$   
+49 We discuss more general settings beyond quadratic minimization in Section 5.   
+50 The condition $\Lambda \subseteq [ \mu , L ]$ implies all quadratic functions under consideration are $L$ -smooth and   
+51 $\mu$ -strongly convex. For this function class, we define $\kappa$ , the (inverse) condition number, and $\rho$ , the   
+52 ratio between the center of $\Lambda$ and its radius, as
+
+$$
+\begin{array} { r l } { \kappa \triangleq \frac { \mu } { L } , \qquad \rho \triangleq \frac { L + \mu } { L - \mu } } & { = \left( \frac { 1 + \kappa } { 1 - \kappa } \right) . } \end{array}
+$$
+
+53 Finally, for a method solving (OPT) that generates a sequence of iterates $\{ x _ { t } \}$ , we define its worst-case   
+54 rate $r _ { t }$ and its asymptotic rate factor $\tau$ as
+
+$$
+r _ { t } \triangleq \operatorname* { s u p } _ { x _ { 0 } \in \mathbb { R } ^ { d } , f \in \mathcal { C } _ { \Lambda } } \frac { \Vert x _ { t } - x _ { * } \Vert } { \Vert x _ { 0 } - x _ { * } \Vert } , \qquad 1 - \tau \triangleq \operatorname* { l i m } _ { t \to \infty } \operatorname* { s u p } _ { \tau \in \mathcal { C } _ { \Lambda } } \sqrt { r _ { t } } .
+$$
+
+# 55 3 Super-acceleration with Cyclical Step-sizes
+
+<table><tr><td colspan="3">Algorithm 2: Cyclical (K = 2) heavy ball with with optimal parameters</td></tr><tr><td colspan="3">Input: Initialization xo,μ1&lt;L1 &lt; μ2&lt;L2 (whereL1-μ1=L2-μ2)</td></tr><tr><td>Set: p = L2+μ1 R= μ2-L1 L2-μ1 L2-μ1</td><td colspan="2">Vp²-R2-√p2-1 2 m= √1-R²</td></tr><tr><td colspan="3">χ1= xo-∀f(xo)</td></tr><tr><td>for t = 1, 2,... do ht= 1+m</td><td></td><td></td></tr><tr><td></td><td>(if t is even)， ht=1+m L1</td><td>(if t is odd) μ2</td></tr><tr><td></td><td>Xt+1= xt-htVf(xt)+m(xt-xt-1)</td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td>end</td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr></table>
+
+56 In this section we develop one of our main contri  
+57 butions, a convergence rate analysis of the cyclical   
+58 heavy ball method with cycles of length 2. This analy  
+59 sis crucially depends on the location of the Hessian’s   
+60 eigenvalues; we assume that these are contained in a   
+61 set $\Lambda$ that is the union of 2 intervals of the same size
+
+$$
+\Lambda = [ \mu _ { 1 } , L _ { 1 } ] \cup [ \mu _ { 2 } , L _ { 2 } ] , L _ { 1 } - \mu _ { 1 } = L _ { 2 } - \mu _ { 2 } .
+$$
+
+62 By symmetry, this set is alternatively described by
+
+$$
+\mu \triangleq \mu _ { 1 } , \quad L \triangleq L _ { 2 } \quad { \mathrm { a n d } } \quad R \triangleq { \frac { \mu _ { 2 } - L _ { 1 } } { L _ { 2 } - \mu _ { 1 } } } ,
+$$
+
+63 where $R$ is the relative length of the gap $\mu _ { 2 } - L _ { 1 }$   
+64 with respect to the diameter $L _ { \mathrm { 2 } } - \mu _ { \mathrm { 1 } }$ (see Figure 1).   
+65 This parametrization will reveal very convenient as   
+66 the relative gap will play a crucial role in the conver  
+67 gence rate analysis. Note also that the gap assumption   
+68 comes without loss of generality, as we allow $R = 0$ .   
+69 Through a correspondence between optimization   
+70 methods and polynomials that we expand upon in   
+71 Section 4, we can derive a worst-case analysis for the cyclical heavy ball method. The outcome of   
+72 this analysis is in the following theorem, that provides the asymptotic convergence rate of Algorithm   
+73 1 for cycles of length two. All proofs of results in this section can be found in Appendix D.3.   
+74 Theorem 3.1 (Rate factor of $\mathrm { H B } _ { 2 } ( h _ { 0 } , h _ { 1 } ; m ) )$ . Let $f \in { \mathcal { C } } _ { \Lambda }$ and $h _ { 0 } , h _ { 1 } , m \ge 0 .$ The asymptotic rate   
+75 factor of Algorithm 1 with cycles of length two is
+
+![](images/b6c3baaa1d848442fb38b5c37b7af9d8f5a72efb076794e6042b67b2ceac22b5.jpg)  
+Figure 1: Hessian eigenvalue histogram for a quadratic objective on MNIST. The outlier eigenvalue at $L _ { 2 }$ generates a non-zero relative gap $R = 0 . 7 7$ . Under these conditions, the 2-cycle heavy ball method has a faster asymptotic rate than the single-cycle one (see Section 3.1).
+
+$$
+\begin{array} { r l } & { 1 - \tau = \left\{ \begin{array} { l l } { \sqrt { m } } & { \mathrm { i f ~ } \sigma _ { \mathrm { s u p } } \leq 1 , } \\ { \sqrt { m } \left( \sigma _ { \mathrm { s u p } } + \sqrt { \sigma _ { \mathrm { s u p } } ^ { 2 } - 1 } \right) ^ { \frac { 1 } { 2 } } } & { \mathrm { i f ~ } \sigma _ { \mathrm { s u p } } \in \left( 1 , \frac { 1 + m ^ { 2 } } { 2 m } \right) , } \\ { \geq 1 \left( n o c o n v e r g e n c e \right) } & { \mathrm { i f ~ } \frac { 1 + m ^ { 2 } } { 2 m } \leq \sigma _ { \mathrm { s u p } } , } \end{array} \right. } \\ & { \sigma _ { \mathrm { s u p } } = \underset { \lambda \in \left\{ \mu _ { 1 } , L _ { 1 } , \mu _ { 2 } , L _ { 2 } , \frac { h _ { 0 } + h _ { 1 } } { 2 h _ { 0 } h _ { 1 } } \right\} \cap \Lambda } { \operatorname* { s u p } } \Big | 2 \left( \frac { 1 + m - \lambda h _ { 0 } } { 2 \sqrt { m } } \right) \left( \frac { 1 + m - \lambda h _ { 1 } } { 2 \sqrt { m } } \right) - 1 \Big | \ . } \end{array}
+$$
+
+76
+
+77 This theorem gives the convergence rate for all triplets $( m , h _ { 0 } , h _ { 1 } )$ . By evaluating this expression over a grid of step-sizes, Figure 2 shows how the rate changes as a function of both step-sizes:
+
+![](images/7c66b4c0f2b433c2aa9f9f7518012f1f00e87a6154cce08e66608dfc0083e539.jpg)  
+Figure 2: Asymptotic rate of cyclical $K = 2$ ) heavy ball in terms of its step-sizes $h _ { 0 } , h _ { 1 }$ across 3 different values of the relative gap $R$ . In the left plot, the relative gap is zero, and so the step-sizes with smallest rate coincide $h _ { 0 } = h _ { 1 } \mathrm { , }$ ). For non-zero values of $R$ (center and right), the optimal method instead alternates between two different step-sizes. In all plots the momentum parameter $m$ is set according to Algorithm 2.
+
+79 From the asymptotic rate expression of Theorem 3.1 we can optimize over the parameters $( h _ { 0 } , h _ { 1 } , m )$   
+80 to obtain the method with smallest convergence rate. This leads to our other main contribution of this   
+81 section, the asymptotically optimal Algorithm 2. This algorithm enjoys the following rate:
+
+Corollary 3.2. The worst-case (asymptotic) rates 82 $r _ { t } ^ { A l g . 2 }$ and $1 - \tau ^ { A l g . 2 }$ of Algorithm 2 over $\mathcal { C } _ { \Lambda }$ are
+
+$$
+\begin{array} { r } { r _ { t } ^ { A l g . ~ 2 } = \left( 1 + t \sqrt { \frac { \rho ^ { 2 } - 1 } { \rho ^ { 2 } - R ^ { 2 } } } \right) \left( \frac { \sqrt { \rho ^ { 2 } - R ^ { 2 } } - \sqrt { \rho ^ { 2 } - 1 } } { \sqrt { 1 - R ^ { 2 } } } \right) ^ { t } , \quad 1 - \tau ^ { A l g . ~ 2 } = \frac { \sqrt { \rho ^ { 2 } - R ^ { 2 } } - \sqrt { \rho ^ { 2 } - 1 } } { \sqrt { 1 - R ^ { 2 } } } \quad f _ { 0 } = \tau ^ { A l g . ~ 2 } , } \end{array}
+$$
+
+# 83 3.1 Comparison with Polyak Heavy Ball
+
+84 In the absence of eigenvalue gap $R = 0$ and $\Lambda = [ \mu , L ] )$ , Algorithm 2 reduces to Polyak heavy   
+85 ball (PHB) [Polyak, 1964], whose worst-case rate is detailed in Appendix B. Since the asymptotic rate   
+86 of Algorithm 2 is monotonically decreasing in $R$ , it is always better or equal than PHB. Furthermore,   
+87 in the ill-conditioned regime (small $\kappa$ ), the comparison is particularly simple: the optimal 2-cycle   
+88 algorithm has a $\sqrt { 1 - R ^ { 2 } }$ relative improvement over PHB, as provided by the next proposition.   
+89 A more thorough comparison for different support sets $\Lambda$ is discussed in Table 1.
+
+Proposition 3.3. Let $R \in [ 0 , 1 )$ . The rate factors of respectively Algorithm 2 and PHB verify
+
+$$
+\begin{array} { r l r } { 1 - \tau ^ { A l g . 2 } \underset { \kappa  0 } { = } 1 - \frac { 2 \sqrt { \kappa } } { \sqrt { 1 - R ^ { 2 } } } + o ( \sqrt { \kappa } ) , } & { } & { 1 - \tau ^ { P H B } \underset { \kappa  0 } { = } 1 - 2 \sqrt { \kappa } + o ( \sqrt { \kappa } ) . } \end{array}
+$$
+
+<table><tr><td>Relative gap R</td><td>Set A</td><td></td><td>Rate factor TSpeedup T/7PHB</td></tr><tr><td>R ∈ [0,1)</td><td>[μ,μ+R(L-μ)]U[L-R(L-μ),L]</td><td>2√ 1-R</td><td>(1-R²)-</td></tr><tr><td>R=1-√K/2</td><td>[,μ+]U[-,]</td><td>2Vk</td><td>K</td></tr><tr><td>R=1-2γK</td><td>[μ,(1+γ)μ]U[L- γμ,L]</td><td> indep. of K</td><td>0（√）</td></tr></table>
+
+Table 1: Case study of the convergence of Algorithm 2 as a function of $R$ , in the regime $\kappa  0$ . The first line corresponds to the regime where $R$ is independent of $\kappa$ , and we observe a constant gain w.r.t. PHB. The second line considers a setting in which $R$ depends on $\sqrt { \kappa }$ , that is, the two intervals in $\Lambda$ are relatively small. The asymptotic rate reads $( 1 - 2 \sqrt [ 4 ] { \kappa } ) ^ { t }$ , beating the classical $( 1 - 2 \sqrt { \kappa } ) ^ { t }$ lower bound, unimprovable when $R = 0$ . Finally, in the third line, $R$ depends on $\kappa$ , the two intervals in $\Lambda$ are so small that the convergence becomes $O ( 1 )$ , i.e., is independent of $\kappa$ .
+
+# 91 4 A constructive Approach: Minimax Polynomials
+
+92 This section presents a generic framework (Algorithm 3) that allows designing optimal momentum   
+93 and step-size cycles for given sets $\Lambda$ and cycle length $K$ .
+
+# Algorithm 3: Optimal momentum method with cyclical step-sizes
+
+Input: Eigenvalue localization $\Lambda$ , cycle length $K$ , initialization $x _ { 0 }$ . Preprocessing:
+
+1. Find the polynomial $\sigma _ { K } ^ { \Lambda }$ such that it satisfies (16).
+
+2. Set step-sizes $\{ h _ { i } \} _ { i = 0 , \dots , K - 1 }$ and momentum $m$ that satisfy resp. equations (21) and (22).
+
+for $t = 1$ $x _ { 1 } = x _ { 0 } - \frac { h _ { 0 } } { 1 + m } \nabla f ( x _ { 0 } )$ , 2, . . . do $\begin{array} { r l } & { \nabla f ( x _ { 0 } ) } \\ & { \quad x _ { t + 1 } = x _ { t } - h _ { \mathrm { m o d } ( t , K ) } \nabla f ( x _ { t } ) + m ( x _ { t } - x _ { t - 1 } ) } \end{array}$ end
+
+94 We first recall classical results that link optimal first order methods on quadratics and Chebyshev   
+95 polynomials. Then, we generalize the approach by showing that optimal methods can be viewed as
+
+combinations of Chebyshev polynomials, and minimax polynomials 96 $\sigma _ { K } ^ { \Lambda }$ of degree $K$ over the set $\Lambda$ Finally, we show how to recover the step-size schedule from 97 $\sigma _ { K } ^ { \Lambda }$ .
+
+# 4.1 First Order Methods on Quadratics and Polynomials
+
+A key property that we will use extensively in the analysis is the following link between first order methods and polynomials (see [Hestenes and Stiefel, 1952]).
+
+101 Proposition 4.1. Let $f \in { \mathcal { C } } _ { \Lambda }$ . The iterates $x _ { t }$ satisfy
+
+$$
+x _ { t + 1 } \in x _ { 0 } + \operatorname { s p a n } \{ \nabla f ( x _ { 0 } ) , \ldots , \nabla f ( x _ { t } ) \} ,
+$$
+
+102 where $x _ { 0 }$ is the initial approximation of $x _ { * }$ , if and only if there exists a sequence of polynomials   
+103 $( P _ { t } ) _ { t \in \mathbb { N } }$ , each of degree at most $I$ more than the highest degree of all previous polynomials and $P _ { 0 }$ of   
+104 degree $O$ (hence the degree of $P _ { t }$ is at most $t$ ), such that
+
+$$
+\forall t \quad x _ { t } - x _ { * } = P _ { t } ( H ) ( x _ { 0 } - x _ { * } ) , \quad P _ { t } ( 0 ) = 1 .
+$$
+
+05 Example 4.2 (Gradient descent). Consider the gradient descent algorithm with fixed step-size $h$ ,   
+106 applied to problem (OPT). Then, after unrolling the update, we have
+
+$$
+x _ { t + 1 } - x _ { * } = x _ { t } - x _ { * } - h \nabla f ( x _ { t } ) = x _ { t } - x _ { * } - h H ( x _ { t } - x _ { * } ) = ( I - h H ) ^ { t + 1 } ( x _ { 0 } - x _ { * } ) .
+$$
+
+In this case, the polynomial associated to gradient descent is $P _ { t } ( \lambda ) = ( 1 - h \lambda ) ^ { t }$ .
+
+108 The above proposition can be used to obtain worst-case rates for first order methods by bounding   
+109 their associated polynomials. Indeed, using the Cauchy-Schwartz inequality in (9) leads to
+
+$$
+\| x _ { t } - x _ { * } \| \leq \operatorname* { s u p } _ { \lambda \in \Lambda } | P _ { t } ( \lambda ) | \ \| x _ { 0 } - x _ { * } \| \quad \Longrightarrow \quad r _ { t } = \operatorname* { s u p } _ { \lambda \in \Lambda } | P _ { t } ( \lambda ) | , \quad \mathrm { w h e r e } \ P ( 0 ) = 1 .
+$$
+
+110 Therefore, finding the algorithm with the fastest worst-case rate can be equivalently framed as the   
+111 problem of finding the polynomial with smallest value on the eigenvalue support $\Lambda$ , subject to the   
+112 normalization condition $\dot { P _ { t } } ( 0 ) = 1$ . Such polynomials are referred to as minimax. Throughout the   
+113 paper, we use this polynomial-based approach to find methods with optimal rates.   
+114 An important property of minimax polynomials is their equioscillation on $\Lambda$ (see Theorem C.1 and   
+115 its proof for a formal statement).   
+116 Definition 4.3. (Equioscillation) A polynomial $P _ { t }$ equioscillates on $\Lambda$ if it verifies $P _ { t } ( 0 ) = 1$ and   
+117 there exist $\lambda _ { 0 } < \lambda _ { 1 } < . . . < \lambda _ { t } \in \Lambda$ such that
+
+$$
+P _ { t } ( \lambda _ { i } ) = ( - 1 ) ^ { i } \operatorname* { m a x } _ { \lambda \in \Lambda } | P _ { t } ( \Lambda ) | .
+$$
+
+118 Example 4.4 $\Lambda$ is an interval). The $t { \cdot }$ -th order Chebyshev polynomials of the first kind $T _ { t }$ satisfy   
+119 the equioscillation property on $[ - 1 , 1 ]$ . It follows that minimax polynomials on $\Lambda = [ \mu , L ]$ can be   
+120 obtained by composing the Chebyshev polynomial $T _ { t }$ with the linear transformation $\sigma _ { 1 } ^ { \Lambda }$ :
+
+$$
+\frac { T _ { t } \left( \sigma _ { 1 } ^ { \Lambda } ( \lambda ) \right) } { T _ { t } \left( \sigma _ { 1 } ^ { \Lambda } ( 0 ) \right) } = \operatorname * { a r g m i n } _ { P \in \mathbb { R } _ { t } [ X ] , P ( 0 ) = 1 } \operatorname * { s u p } _ { \lambda \in \Lambda } \lvert P ( \lambda ) \rvert , \mathrm { w i t h } \sigma _ { 1 } ^ { \Lambda } ( \lambda ) = \frac { L + \mu } { L - \mu } - \frac { 2 } { L - \mu } \lambda ,
+$$
+
+121 where $\sigma _ { 1 } ^ { \Lambda }$ maps the interval $[ \mu , L ]$ to $[ - 1 , 1 ]$ . The optimization method associated with this minimax   
+122 polynomial is the Chebyshev semi-terative method [Flanders and Shortley, 1950, Golub and Varga,   
+123 1961] (described also in Appendix B.1). This method achieves the lower complexity bound for   
+124 smooth strongly convex quadratic minimization, see for instance [Nemirovsky, 1995, Chapter 12] or   
+125 [Nemirovsky, 1992, Nesterov, 2003].   
+126 The next proposition provides the main results in this subsection, which is key for obtaining Algo  
+127 rithm 2. It characterizes the even degree minimax polynomial in the setting of Section 3, that is,   
+128 129 when Cheb $\Lambda$ is the union of 2 intervals of same size. In this case, the minihev polynomials, but composed with a degree-two polynomial $\sigma _ { 2 } ^ { \Lambda }$ solution is also based on.
+
+130 Proposition 4.5. Let $\Lambda = [ \mu _ { 1 } , L _ { 1 } ] \cup [ \mu _ { 2 } , L _ { 2 } ]$ be an union of two intervals of the same size 131 $( L _ { 1 } - \mu _ { 1 } = L _ { 2 } - \mu _ { 2 } )$ and let m be as defined in Algorithm 2. Then the minimax polynomial (solution to (12)) is, for all 132 $t = 2 n$ , $n \in \mathbb { N } _ { 0 } ^ { + }$ ,
+
+$$
+{ \frac { T _ { n } \left( \sigma _ { 2 } ^ { \Lambda } ( \lambda ) \right) } { T _ { n } \left( \sigma _ { 2 } ^ { \Lambda } ( 0 ) \right) } } = \underset { P ( 0 ) = 1 } { \operatorname { a r g m i n } } \ \underset { \lambda \in \Lambda } { \operatorname* { s u p } } | P ( \lambda ) | , \ w i t h \ \ \sigma _ { 2 } ^ { \Lambda } ( \lambda ) = 2 \left( { \frac { 1 + m } { 2 \sqrt { m } } } \right) ^ { 2 } \left( 1 - { \frac { \lambda } { L _ { 1 } } } \right) \left( 1 - { \frac { \lambda } { \mu _ { 2 } } } \right) - 1 .
+$$
+
+134 The polynomial in Example 4.4 uses a linear link function $\sigma _ { 1 } ^ { \Lambda }$ to map $\Lambda$ to $[ - 1 , 1 ]$ . In Proposition 4.5,   
+135 we see that a degree two link function $\sigma _ { 2 } ^ { \Lambda }$ can be used to find the minimax polynomial when $\Lambda$ is the   
+136 union of two intervals. This section generalizes this approach and considers higher-order polynomials   
+137 for $\sigma _ { K }$ . We start with the following parametrization, with an arbitrary polynomial $\sigma _ { K }$ of degree $K$ ,
+
+$$
+P _ { t } ( \lambda ; \sigma _ { K } ) \triangleq \frac { T _ { n } \left( \sigma _ { K } ( \lambda ) \right) } { T _ { n } \left( \sigma _ { K } ( 0 ) \right) } , \quad \forall t = K n , n \in \mathbb { N } _ { 0 } ^ { + } .
+$$
+
+138 As we will see in the next subsection, this parametrization allows considering cycles of step-sizes.   
+139 Our goal now is to find the $\sigma _ { K }$ that obtains the fastest convergence rate possible. The next proposition   
+140 quantifies its impact on the asymptotic rate and its proof can be found in Appendix D.1.   
+141 Proposition 4.6. For a given $\sigma _ { K }$ such that $\operatorname* { s u p } _ { \lambda \in \Lambda } | \sigma _ { K } ( \lambda ) | = 1$ , the asymptotic rate factor $\tau ^ { \sigma _ { K } }$ of   
+142 the method associated to the polynomial (14) is
+
+$$
+1 - \tau ^ { \sigma \kappa } = \operatorname * { l i m } _ { t  \infty } \sqrt { \smash [ b ] { \operatorname { s u p } _ { \lambda \in \Lambda } | P _ { t } ( \lambda ; \sigma _ { K } ) | } } = ( \sigma _ { 0 } - \sqrt { \sigma _ { 0 } ^ { 2 } - 1 } ) ^ { \frac { 1 } { \kappa } } , \quad w i t h \ \sigma _ { 0 } \triangleq \sigma _ { K } ( 0 ) .
+$$
+
+143 For a fixed $K$ , the asymptotic rate (15) is a decreasing function of $\sigma _ { 0 }$ . This motivates the introduction of the “optimal” degree 144 $K$ polynomial $\sigma _ { K } ^ { \Lambda }$ as the one that solves
+
+$$
+\sigma _ { K } ^ { \Lambda } \triangleq \underset { \sigma \in \mathbb { R } _ { K } [ X ] } { \arg \operatorname* { m a x } } \sigma ( 0 ) \quad \mathrm { s . t . } ~ \underset { \lambda \in \Lambda } { \operatorname* { s u p } } | \sigma ( \lambda ) | = 1 .
+$$
+
+Using the above definition, we recover the 145 $\sigma _ { 1 } ^ { \Lambda }$ and $\sigma _ { 2 } ^ { \Lambda }$ from Example 4.4 and Proposition 4.5.
+
+146 Finding the polynomial. Finding an exact and explicit solution for the general $K$ and $\Lambda$ case   
+147 148 equations. Here we describe an approximate approach. Let σΛK (x) = PKi=0 σixi . We propose to   
+149
+
+$$
+\begin{array} { r } { \underset { \sigma _ { i } } { \operatorname* { m a x } } \sigma _ { 0 } \quad \mathrm { ~ s . t . ~ } - 1 \leq \sum _ { i = 0 } ^ { K } \sigma _ { i } \lambda _ { j } ^ { i } \leq 1 , \quad \forall j = 1 , \ldots , N . } \end{array}
+$$
+
+To check the optimality, it suffices to verify that the polynomial 50 $\sigma _ { K } ^ { \Lambda }$ satisfies the equioscillation 151 property (Definition 4.3), as depicted in Figure 3.
+
+152 Remark 4.7 (Relationship between optimal and minimax polynomials). For later reference, we note   
+153 that the optimal polynomial $\sigma _ { K } ^ { \Lambda }$ is equivalent to finding a minimax polynomial on $\Lambda$ and to rescale it.   
+154 More precisely, $\sigma _ { K } ^ { \Lambda }$ is optimal if and only if $\sigma _ { K } ^ { \Lambda } / \sigma _ { K } ^ { \Lambda } ( 0 )$ is minimax.
+
+# 155 4.3 Cyclical Heavy Ball and (Non-)asymptotic Rates of Convergence
+
+We now describe the link between 156 $\sigma _ { K } ^ { \Lambda }$ and Algorithm 3. Using the recurrence for Chebyshev polynomials of the first kind in (14), we have 157 $\forall t = K n$ , $n \in \mathbb { N } _ { 0 } ^ { + }$ ,
+
+$$
+\frac { T _ { n + 1 } ( \sigma _ { K } ^ { \Lambda } ( \lambda ) ) } { T _ { n + 1 } ( \sigma _ { K } ^ { \Lambda } ( 0 ) ) } = 2 \sigma _ { K } ^ { \Lambda } ( \lambda ) \left[ \frac { T _ { n } ( \sigma _ { K } ^ { \Lambda } ( \lambda ) ) } { T _ { n } ( \sigma _ { K } ^ { \Lambda } ( 0 ) ) } \right] \underbrace { \left[ \frac { T _ { n } ( \sigma _ { K } ^ { \Lambda } ( 0 ) ) } { T _ { n + 1 } ( \sigma _ { K } ^ { \Lambda } ( 0 ) ) } \right] } _ { = a _ { n } } - \left[ \frac { T _ { n - 1 } ( \sigma _ { K } ^ { \Lambda } ( \lambda ) ) } { T _ { n - 1 } ( \sigma _ { K } ^ { \Lambda } ( 0 ) ) } \right] \underbrace { \left[ \frac { T _ { n - 1 } ( \sigma _ { K } ^ { \Lambda } ( 0 ) ) } { T _ { n + 1 } ( \sigma _ { K } ^ { \Lambda } ( 0 ) ) } \right] } _ { = b _ { n } } .
+$$
+
+158 It still remains to find an algorithm associated with this polynomial. To obtain one in the form of   
+159 Algorithm 1, one can use the stationary behavior of the recurrence. From [Scieur and Pedregosa,   
+160 2020], the coefficients $a _ { n }$ and $b _ { n }$ converge as $n \to \infty$ to their fixed-points $a _ { \infty }$ and $b _ { \infty }$ . We therefore   
+161 consider here an asymptotic polynomial $\big ( \bar { P } _ { t } ( \lambda ; \sigma _ { K } ^ { \Lambda } )$ , whose recurrence satisfies
+
+$$
+\bar { P } _ { t } ( \lambda ; \sigma _ { K } ^ { \Lambda } ) = 2 a _ { \infty } \sigma _ { K } ^ { \Lambda } ( \lambda ) \bar { P } _ { t - K } ( \lambda ; \sigma _ { K } ^ { \Lambda } ) - b _ { \infty } \bar { P } _ { t - 2 K } ( \lambda ; \sigma _ { K } ^ { \Lambda } ) .
+$$
+
+162 Similarly to $K = 1$ , where this limit recursion corresponds to PHB, this recursion corresponds to   
+163 an instance of Algorithm 3 (see Proposition 4.9 below), further motivating the cyclical heavy ball   
+164 algorithm.   
+165 The following theorem is the main result of this section and characterizes the convergence rate of   
+166 Algorithm 1 for arbitrary momentum and step-size sequences $\{ h _ { i } \} _ { i \in [ [ 1 , K ] ] }$ . By optimizing over these   
+167 J Kparameters, we obtain a method associated to (18), whose rate is described in Proposition 4.9. All   
+168 proofs can be found in Appendix D.2.   
+169 Theorem 4.8. The worst-case rate of convergence of Algorithm $^ { l }$ on $\mathcal { C } _ { \Lambda }$ with an arbitrary momentum   
+170 $m$ and an arbitrary sequence of step-sizes $\{ h _ { i } \}$ is
+
+![](images/9240a9958eaa348db8f2ba977a8a2708f567c10acd0a1745f092f5ee93f5b674.jpg)  
+Figure 3: Examples of optimal polynomials $\sigma _ { K } ^ { \Lambda }$ from (16), all of them verifying the equioscillation property (Definition 4.3). The $" \star "$ symbol highlights the degree of $\sigma _ { K } ^ { \Lambda }$ that achieves the best asymptotic rate $\tau ^ { \sigma _ { K } ^ { \Lambda } }$ in (15) amongst all $K$ (see Section 4.4). (Left) When $\Lambda$ is an unique interval, all 3 polynomials are equivalently optimal $\begin{array} { r l r } { \tau ^ { \sigma _ { 1 } ^ { \Lambda } } = } & { { } } & { = \tau ^ { \sigma _ { 3 } ^ { \Lambda } } } \end{array}$ . (Center) When $\Lambda$ is the union of two intervals of the same size, the degree 2 polynomial is optimal $\tau ^ { \sigma _ { 2 } ^ { \Lambda } } > \tau ^ { \sigma _ { 3 } ^ { \Lambda } } > \tau ^ { \sigma _ { 1 } ^ { \Lambda } }$ . This is expected given the result in Proposition 4.5. (Right) When $\Lambda$ is the union of two unbalanced intervals, the degree 3 polynomial instead achieves the best asymptotic rate $\tau ^ { \sigma _ { 3 } ^ { \Lambda } } > > > \tau ^ { \sigma _ { 1 } ^ { \Lambda } }$ (see Section 4.4).
+
+$$
+1 - \tau = \left\{ \begin{array} { l l } { \sqrt { m } , } & { \mathrm { i f ~ } \sigma _ { \operatorname* { s u p } } \leq 1 } \\ { \sqrt { m } \left( \sigma _ { \operatorname* { s u p } } + \sqrt { \sigma _ { \operatorname* { s u p } } ^ { 2 } - 1 } \right) ^ { 1 / K } , } & { \mathrm { i f ~ } \sigma _ { \operatorname* { s u p } } \in \left( 1 , \displaystyle \frac { 1 + m ^ { K } } { 2 \left( \sqrt { m } \right) ^ { K } } \right) } \\ { \geq 1 \left( n o c o n v e r g e n c e \right) } & { \mathrm { i f ~ } \sigma _ { \operatorname* { s u p } } \geq \displaystyle \frac { 1 + m ^ { K } } { 2 \left( \sqrt { m } \right) ^ { K } } } \end{array} \right. ,
+$$
+
+where 171 $\sigma _ { \operatorname* { s u p } } \triangleq \operatorname* { s u p } _ { \lambda \in \Lambda } \vert \sigma ( \lambda ; \{ h _ { i } \} , m ) \vert$ , and $\sigma ( \lambda ; \{ h _ { i } \} , m )$ is the $K$ -degree polynomial
+
+$$
+\sigma ( \lambda ; \{ h _ { i } \} , m ) \triangleq \frac { 1 } { 2 } \mathrm { T r } \left( \left[ \begin{array} { c c c } { \frac { 1 + m - h _ { K - 1 } \lambda } { \sqrt { m } } } & { - 1 } \\ { 1 } & { 0 } \end{array} \right] \left[ \begin{array} { c c c } { \frac { 1 + m - h _ { K - 2 } \lambda } { \sqrt { m } } } & { - 1 } \\ { 1 } & { 0 } \end{array} \right] \cdots \left[ \begin{array} { c c } { \frac { 1 + m - h _ { 0 } \lambda } { \sqrt { m } } } & { - 1 } \\ { 1 } & { 0 } \end{array} \right] \right) .
+$$
+
+172 Proposition 4.9. Let $\sigma ( \lambda ; \{ h _ { i } \} , m )$ be the polynomial defined by (20), and $\sigma _ { K } ^ { \Lambda }$ be the optimal link   
+173 function of degree $K$ defined by (16). If the momentum $m$ and the sequence of step-sizes $\{ h _ { i } \}$ satisfy
+
+$$
+\sigma ( \lambda ; \{ h _ { i } \} , m ) = \sigma _ { K } ^ { \Lambda } ( \lambda ) ,
+$$
+
+174 then 1) the parameters are optimal, in the sense that they minimize the asymptotic rate factor from   
+175 Theorem 4.8, 2) the optimal momentum parameter is
+
+$$
+m = \left( \sigma _ { 0 } - \sqrt { \sigma _ { 0 } ^ { 2 } - 1 } \right) ^ { 2 / K } , \quad w h e r e \sigma _ { 0 } = \sigma _ { K } ^ { \Lambda } ( 0 ) ,
+$$
+
+176 3) the iterates from Algo. 3 with parameters $\{ h _ { i } \}$ and $m$ form a polynomial with recurrence (18), and 4) Algorithm 3 achieves the worst-case rate 177 $r _ { t } ^ { A l g . \ 3 }$ and the asymptotic rate factor $1 - \tau ^ { A l g }$ . 3
+
+$$
+r _ { t } ^ { A l g . ~ 3 } = O \left( t \left( \sigma _ { 0 } - \sqrt { \sigma _ { 0 } ^ { 2 } - 1 } \right) ^ { t / K } \right) , \qquad 1 - \tau ^ { A l g . ~ 3 } = \left( \sigma _ { 0 } - \sqrt { \sigma _ { 0 } ^ { 2 } - 1 } \right) ^ { 1 / K } .
+$$
+
+178 Solving the system (21) The system is constructed by identification of the coefficients in both   
+179 polynomials $\bar { \sigma } _ { K } ^ { \Lambda }$ and $\sigma ( \lambda ; \{ h _ { i } \} , m )$ , which can be solved using a naive grid-search for instance. We   
+180 are not aware of any efficient algorithm to solve this system exactly, although it is possible to use   
+181 iterative methods such as steepest descent or Newton’s method.
+
+This section discusses the (asymptotic) optimality of Algorithm 3. In Section 4.2, the polynomial $P _ { t } ( \cdot ; \sigma _ { K } ^ { \Lambda } )$ was written as a composition of Chebyshev polynomials with $\sigma _ { K } ^ { \Lambda }$ , defined in (16). The best the $K$ is chosen as follows: we solve (16) for several values of imizers of (15). However, following such steps does not $K$ , then pick the smallest arantee that the polyno $K$ aal $P _ { t , K } ^ { \Lambda }$ is minimax, as it is not guaranteed to minimize the worst-case rate $\operatorname* { s u p } _ { \lambda \in \Lambda } \left| P _ { t } ( \lambda ) \right|$ (see (11)).
+
+We give here an optimality certificate, linked to a generalized version of equioscillation. In short, if we can find $K$ non overlapping intervals (more formally, whose interiors are disjoint) $\Lambda _ { i }$ in $\Lambda$ such that $\sigma _ { K } ^ { \Lambda } ( \Lambda _ { i } ) = [ - 1 , 1 ]$ then $P _ { t , K } ^ { \tilde { \Lambda } }$ is minimax for all $t = n K$ , $n \in \mathbb { N } _ { 0 } ^ { + }$ . The detailed result is provided by Theorem C.2. A direct consequence of this result is the asymptotic optimality of Algorithm 3, i.e., there exists no first order algorithm with a better asymptotic rate $1 - \tau$ for the function class $\mathcal { C } _ { \Lambda }$ .
+
+It is possible that such $\sigma _ { K } ^ { \Lambda }$ does not exist for a given $\Lambda$ . A complete characterization of the set $\Lambda$ for which there exists such $\sigma _ { K } ^ { \Lambda }$ is out of the scope of this paper. A partial answer is given in [Fischer, 2011] when $\Lambda$ is the union of two intervals. However, the problem remains open in the general case.
+
+# 96 5 Local Convergence for Non-Quadratic Functions
+
+197 When $f$ is twice-differentiable, it is possible to show local convergence rates when $x _ { 0 }$ is close   
+198 enough to $x _ { * }$ [Polyak, 1964]. We give here a similar result that applies to Algorithm 1 (see proof in   
+199 Appendix E). Those results are only local, as it is possible to find pathological counter-examples for   
+200 which even PHB does not converge globally, for some specific initialization [Lessard et al., 2016].   
+201 Theorem 5.1 (Local convergence). Let $f : \mathbb { R } ^ { d } \mapsto \mathbb { R }$ be a (potentially non-quadratic) twice continu  
+202 ously differentiable function, $x _ { * }$ a local minimizer, and $H$ be the Hessian of $f$ at $x _ { * }$ with $S p ( H ) \subseteq \Lambda$   
+203 Let $x _ { t }$ denote the result of running Algorithm 1 with parameters $h _ { 1 } , h _ { 2 } , \cdot \cdot \cdot , h _ { K } , m$ , and let $1 - \tau$ be   
+204 the linear convergence rate on the quadratic objective (OPT). Then we have
+
+$$
+\forall \varepsilon > 0 , \exists \mathrm { o p e n ~ s e t } \ : V _ { \varepsilon } : x _ { 0 } , x _ { * } \in V _ { \varepsilon } \implies \| x _ { t } - x _ { * } \| = O ( ( 1 - \tau + \varepsilon ) ^ { t } ) \| x _ { 0 } - x _ { * } \| .
+$$
+
+205 In short, when Algorithm 1 is guaranteed to converge at rate $1 - \tau$ on (OPT), then the convergence   
+206 rate on a nonlinear functions can be arbitrary close to $1 - \tau$ when $x _ { 0 }$ is sufficiently close to $x _ { * }$ .
+
+# 6 Experiments
+
+In this section we present an empirical comparison of the cyclical heavy ball method for different length cycles across 4 different problems. We consider two different problems, quadratic and logistic regression, each applied on two datasets, the MNIST handwritten digits [Le Cun et al., 2010] and a synthetic dataset. The results of these experiments, together with a histogram of the Hessian’s eigenvalues are presented in Figure 4 (see caption for a discussion).
+
+Dataset description. The MNIST dataset consists of a data matrix $A$ with 60000 images of handwritten digits each one with $2 8 \times 2 8 = 7 8 4$ pixels. The synthetic dataset is generated according to a spiked covariance model [Johnstone, 2001], which has been shown to be an accurate model of covariance matrices arising for instance in spectral clustering [Couillet and Benaych-Georges, 2016] and deep networks [Pennington and Worah, 2017, Granziol et al., 2020]. In this model, the data matrix $A = X Z$ is generated from a $m \times n$ random Gaussian matrix $X$ and an $m \times m$ deterministic matrix $Z$ . In our case, we take $n = 1 0 0 0 , m = 1 2 0 0$ and $Z$ is the identity where the first three entries are multiplied by 100 (this will lead to three outlier eigenvalues). We also generate an $n$ -dimensional target vector $b$ as $b = A x$ or $b = { \mathrm { s i g n } } ( A x )$ for the quadratic and logistic problem respectively.
+
+Objective function For each dataset, we consider a quadratic and a logistic regression problem, leadin to 4 different problems. All pro ms are of the form $\begin{array} { r } { \operatorname* { m i n } _ { x \in \mathbb { R } ^ { p } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ( \breve { A } _ { i } ^ { \top } x , b _ { i } ) \stackrel { \cdot } { + } \lambda \| x \| ^ { 2 } } \end{array}$ , $\ell$ $A$ is the data matrix and $b$ are the target values. We set the regularization parameter to $\lambda = 1 0 ^ { - 3 } \Vert A \Vert ^ { 2 }$ . For logistic regression, since guarantees only hold at a neighborhood of the solution (even for the 1-cycle algorithm), we initialize the first iterate as the result of 100 iteration of gradient descent. In the case of logistic regression, the Hessian eigenvalues are computed at the optimum.
+
+![](images/1e22ce525180458a522848876892c69e42308b6733154fc2bd6a5e5629064183.jpg)  
+Figure 4: Hessian Eigenvalue histogram (top row) and Benchmarks (bottom row). The top row shows the Hessian eigenvalue histogram at optimum for the 4 problems consider, together with the interval boundaries $\mu _ { 1 } < L _ { 1 } < \mu _ { 2 } < L _ { 2 }$ for the two-interval split of the eigenvalue support described in Section 3. In all cases, there’s a non-zero gap radius $R$ . This is shown in the bottom row, where we compare the suboptimality in terms of gradient norm as a function of the number of iterations. As predicted by the theory, the non-zero gap radius translates into a faster convergence of the cyclical approach, compared to PHB in all cases. The improvement is observed on both quadratic and logistic regression problems, even through the theory for the latter is limited to local convergence.
+
+# 229 7 Conclusion
+
+This work is motivated by two recent observations from the optimization practice of machine learning. First, cyclical step-sizes have been shown to enjoy excellent empirical convergence [Loshchilov and Hutter, 2017, Smith, 2017]. Second, spectral gaps are pervasive in the Hessian spectrum of deep learning models [Sagun et al., 2017, Papyan, 2018, Ghorbani et al., 2019, Papyan, 2019]. Based on the simpler context of quadratic convex minimization, we develop a convergence-rate analysis and optimal parameters for the heavy ball method with cyclical step-sizes. This analysis highlights the regimes under which cyclical step-sizes have faster rates than classical accelerated methods. Finally, we illustrate these findings through numerical benchmarks.
+
+Main Limitations. In Section 3 we gave explicit formulas for the optimal parameters in the case of the 2-cycle heavy ball algorithm. These formulas depend not only on extremal eigenvalues—as is usual for accelerated methods—but also on the spectral gap $R$ . The gap can sometimes be computed after computed the top eigenvalues (e.g. top-2 eigenvalue for MNIST). However, in general, there is no guarantee on how many eigenvalues are needed to estimate it. Moreover, global convergence result rely heavily on the quadratic assumption.
+
+244 Another limitation regards long cycles. For cycles longer than 2, we have only given an implicit   
+245 formula to set the optimal parameters (Proposition 4.9). This involves solving a set of non-linear   
+246 equations whose complexity increases with the cycle length. That being said, cyclical step-sizes   
+247 might significantly enhance convergence speeds both in terms of worst-case rates and empirically,   
+248 and this work advocates that new tuning practices involving different cycle lengths might be relevant.
+
+Broader Impact. This work is mostly theoretical, and as such we believe it does not present direct societal consequences. However, the methods described in this paper can be used to train machine learning models which could themselves have societal consequences. For example, the deployment of machine learning models in decision-making has been shown to suffer from gender and racial bias and to amplify existing inequalities, see for instance [Hutchinson and Mitchell, 2019, Barocas et al., 2017, Obermeyer et al., 2019].
+
+# References
+
+256 Solon Barocas, Moritz Hardt, and Arvind Narayanan. Fairness in machine learning. Nips tutorial, 2017.   
+258 Dimitri P. Bertsekas. Nonlinear programming. Journal of the Operational Research Society, 1997.   
+59 Romain Couillet and Florent Benaych-Georges. Kernel spectral clustering of large dimensional data. Electronic Journal of Statistics, 2016. Bernd Fischer. Polynomial based iteration methods for symmetric linear systems. SIAM, 2011. Donald A. Flanders and George Shortley. Numerical determination of fundamental modes. Journal of Applied Physics, 1950. Behrooz Ghorbani, Shankar Krishnan, and Ying Xiao. An investigation into neural net optimization via hessian eigenvalue density. In International Conference on Machine Learning (ICML), 2019. Gene H. Golub and Richard S. Varga. Chebyshev semi-iterative methods, successive overrelaxation iterative methods, and second order Richardson iterative methods. Numerische Mathematik, 1961. Diego Granziol, Xingchen Wan, Samuel Albanie, and Stephen Roberts. Explaining the Adaptive Generalisation Gap. arXiv preprint arXiv:2011.08181, 2020.   
+70 Magnus R. Hestenes and Eduard Stiefel. Methods of conjugate gradients for solving linear systems, volume 49. NBS Washington, DC, 1952. Ben Hutchinson and Margaret Mitchell. 50 years of test (un) fairness: Lessons for machine learning. In Proceedings of the Conference on Fairness, Accountability, and Transparency, 2019.   
+274 Iain M. Johnstone. On the distribution of the largest eigenvalue in principal components analysis. Annals of statistics, 2001.   
+76 Yann Le Cun, Corinna Cortes, and Chris Burges. MNIST handwritten digit database. ATT Labs [Online], 2010. Laurent Lessard, Benjamin Recht, and Andrew Packard. Analysis and design of optimization algorithms via integral quadratic constraints. SIAM Journal on Optimization, 2016. Ilya Loshchilov and Frank Hutter. SGDR: stochastic gradient descent with warm restarts. In International Conference on Learning Representations (ICLR), 2017.   
+282 Arkadi S. Nemirovsky. Information-based complexity of linear operator equations. Journal of Complexity, 1992.   
+284 Arkadi S. Nemirovsky. Information-based complexity of convex programming. Lecture Notes, 1995.   
+285 Yurii Nesterov. Introductory Lectures on Convex Optimization. Springer, 2003.   
+286 Ziad Obermeyer, Brian Powers, Christine Vogeli, and Sendhil Mullainathan. Dissecting racial bias in an algorithm used to manage the health of populations. Science, 2019. Vardan Papyan. The full spectrum of deepnet hessians at scale: Dynamics with SGD training and sample size. arXiv preprint arXiv:1811.07062, 2018. Vardan Papyan. Measurements of Three-Level Hierarchical Structure in the Outliers in the Spectrum of Deepnet Hessians. In International Conference on Machine Learning (ICML), 2019.   
+92 Jeffrey Pennington and Pratik Worah. Nonlinear random matrix theory for deep learning. In Advances on Neural Information Processing Systems (NIPS), 2017. Boris T. Polyak. Some methods of speeding up the convergence of iteration methods. USSR computational mathematics and mathematical physics, 1964. Levent Sagun, Utku Evci, V. Ugur Guney, Yann Dauphin, and Leon Bottou. Empirical analysis of the Hessian of over-parametrized neural networks. arXiv preprint arXiv:1706.04454, 2017.   
+98 Damien Scieur and Fabian Pedregosa. Universal Asymptotic Optimality of Polyak Momentum. In International Conference on Machine Learning (ICML), 2020. Leslie N. Smith. Cyclical learning rates for training neural networks. In 2017 IEEE Winter Conference on Applications of Computer Vision (WACV). IEEE, 2017.   
+02 Ilya Sutskever, James Martens, George Dahl, and Geoffrey Hinton. On the importance of initialization and momentum in deep learning. In International Conference on Machine Learning (ICML), 2013.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] The Introduction (Section 1) details where all results can be found.   
+(b) Did you describe the limitations of your work? [Yes] There is a paragraph "Main Limitations" in the Conclusion (Section 7).   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] As stated in the "Broader Impact" section, this work is mostly theoretical and as such we believe it doesn’t present a direct societal impact.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] In each and every statement we make.   
+(b) Did you include complete proofs of all theoretical results? [Yes] All proofs are available in the supplementary material.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] URLs are provided in the supplementary material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] In the experiments section (Section 6).   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] .   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] , in Appendix F
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] In the "Dataset description" paragraph of the Experiments section 6.   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] There is no new asset. We used MNIST existing Dataset as well as a synthetic dataset whose construction is described in papers cited in the Experiments section 6.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/AVx0r_GppCu/AVx0r_GppCu_content_list.json

@@ -0,0 +1,1622 @@
+[
+    {
+        "type": "text",
+        "text": "Super-Acceleration with Cyclical Step-sizes ",
+        "text_level": 1,
+        "bbox": [
+            233,
+            122,
+            761,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            200,
+            580,
+            256
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            292,
+            535,
+            309
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Cyclical step-sizes are becoming increasingly popular in the optimization of deep   \n2 learning problems. Motivated by recent observations on the spectral gaps of   \n3 Hessians in machine learning, we show that these step-size schedules offer a   \n4 simple way to exploit them. More precisely, we develop a convergence rate   \n5 analysis for quadratic objectives that provides optimal parameters and shows that   \n6 cyclical learning rates can improve upon traditional lower complexity bounds.   \n7 We further propose a systematic approach to design optimal first order methods   \n8 for quadratic minimization with a given spectral structure. Finally, we provide a   \n9 local convergence rate analysis beyond quadratic minimization for the proposed   \n10 methods and illustrate our findings through benchmarks on least squares and   \n11 logistic regression problems. ",
+        "bbox": [
+            148,
+            323,
+            766,
+            477
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "12 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            501,
+            312,
+            518
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "13 One of the most iconic methods in first order optimization is gradient descent with momentum, also   \n14 known as the heavy ball method [Polyak, 1964]. This method enjoys widespread popularity both in   \n15 its original formulation and in a stochastic variant that replaces the gradient by a stochastic estimate,   \n16 a method that is behind many of the recent breakthroughs in deep learning [Sutskever et al., 2013].   \n17 A variant of the stochastic heavy ball where the step-sizes are chosen in cyclical order has recently   \n18 come to the forefront of machine learning research, showing state-of-the art results on different deep   \n19 learning benchmarks [Loshchilov and Hutter, 2017, Smith, 2017]. Inspired by this empirical success,   \n20 we aim to study the convergence of the heavy ball algorithm where step-sizes $h _ { 0 } , h _ { 1 } , \\ldots$ are not fixed   \n21 or decreasing but instead chosen in cyclical order: ",
+        "bbox": [
+            147,
+            532,
+            825,
+            588
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            148,
+            594,
+            825,
+            665
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: Cyclical heavy ball $\\mathrm { H B } _ { K } ( h _ { 0 } , \\ldots , h _ { K - 1 } ; m )$ ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            683,
+            563,
+            699
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Input: Initialization $x _ { 0 }$ , momentum $m \\in ( 0 , 1 )$ , step-sizes $\\{ h _ { 0 } , \\ldots , h _ { K - 1 } \\}$ $\\begin{array} { r l } & { x _ { 1 } = x _ { 0 } - \\frac { h _ { 0 } } { 1 + m } \\nabla f ( x _ { 0 } ) } \\\\ & { { \\bf f o r } t = 1 , 2 , \\dots { \\bf d o } x _ { t + 1 } = x _ { t } - h _ { \\mathrm { m o d } ( t , K ) } \\nabla f ( x _ { t } ) + m ( x _ { t } - x _ { t - 1 } ) } \\end{array}$ 1   \nend ",
+        "bbox": [
+            173,
+            703,
+            669,
+            780
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "22 The heavy ball method with constant step-sizes enjoys a mature theory, where it is known for example   \n23 to achieve optimal black-box worst-case complexity of quadratic convex optimization [Nemirovsky,   \n24 1992]. In stark contrast, little is known about the the convergence of the above variant with cyclical   \n25 step-sizes. Our main motivating question is ",
+        "bbox": [
+            147,
+            799,
+            825,
+            856
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Do cyclical step-sizes improve convergence of heavy ball? ",
+        "bbox": [
+            307,
+            876,
+            689,
+            891
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "27 Our main contribution provides a positive answer to this question and, more importantly, quantifies   \n28 the speedup under different assumptions. In particular, we show that for quadratic problems, whenever   \n29 Hessian’s spectrum belongs to two or more disjoint intervals, the heavy ball method with cyclical step  \n30 sizes achieves a faster worst-case convergence rate. Recent works have shown that this assumption on   \n31 the spectrum is quite natural and occurs in many machine learning problems, including deep neural   \n32 networks [Sagun et al., 2017, Papyan, 2018, Ghorbani et al., 2019, Papyan, 2019]. More precisely,   \n33 we list our main contributions below.   \n34 • In sections 3 and 4, we provide a tight convergence rate analysis of the cyclical heavy ball method   \n35 (Theorems 3.1 and 3.2 for two step-sizes, and Theorem 4.8 for the general case). This analysis   \n36 highlights a regime under which this method achieves a faster worst-case rate than the accelerated   \n37 rate of heavy ball, a phenomenon we refer to as super-acceleration. Theorem 5.1 extends the (local)   \n38 convergence rate analysis results to non-quadratic objectives.   \n39 • As a byproduct of the convergence-rate analysis, we obtain an explicit expression for the optimal   \n40 parameters in in the case of cycles of length two (Algorithm 2) and an implicit expression in terms   \n41 of a system of $K$ equations in the general case. ",
+        "bbox": [
+            145,
+            92,
+            826,
+            189
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            199,
+            825,
+            270
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            148,
+            273,
+            825,
+            315
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Section 6 presents numerical benchmarks illustrating the improved convergence of the cyclical approach on 4 problems involving quadratic and logistic losses on both synthetic and a handwritten digits recognition dataset. ",
+        "bbox": [
+            161,
+            320,
+            826,
+            363
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Finally, we conclude in Section 7 with a discussion of this work’s limitations. ",
+        "bbox": [
+            161,
+            366,
+            700,
+            382
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "46 2 Notation and Problem Setting ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            398,
+            455,
+            417
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "47 Throughout the paper, we consider the problem of minimizing quadratic functions of the form ",
+        "bbox": [
+            155,
+            431,
+            789,
+            446
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aacf97b0d71146e57fa9b04ee9c15406ea8da2d9c5e67fc64c313e621188455a.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\alpha \\in \\mathbb { B } ^ { d } } f ( x ) , \\mathrm { w i t h } f \\in \\mathcal { C } _ { \\Lambda } \\triangleq \\left\\{ f : f ( x ) = \\frac { 1 } { 2 } ( x - x _ { * } ) ^ { T } H ( x - x _ { * } ) + f _ { * } , \\mathrm { S p } ( H ) \\subseteq \\Lambda \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            452,
+            767,
+            474
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "48 where $\\mathcal { C } _ { \\Lambda }$ is the class of quadratic functions whose spectrum $\\operatorname { S p } ( H )$ is localized in $\\Lambda \\subseteq [ \\mu , L ] \\subseteq \\mathbb { R } _ { > 0 }$   \n49 We discuss more general settings beyond quadratic minimization in Section 5.   \n50 The condition $\\Lambda \\subseteq [ \\mu , L ]$ implies all quadratic functions under consideration are $L$ -smooth and   \n51 $\\mu$ -strongly convex. For this function class, we define $\\kappa$ , the (inverse) condition number, and $\\rho$ , the   \n52 ratio between the center of $\\Lambda$ and its radius, as ",
+        "bbox": [
+            148,
+            484,
+            825,
+            515
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            520,
+            826,
+            563
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8bb46c296ca18926ae61feb2bccae93e1a1f0a9a962878fb7aca97e81398253a.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\kappa \\triangleq \\frac { \\mu } { L } , \\qquad \\rho \\triangleq \\frac { L + \\mu } { L - \\mu } } & { = \\left( \\frac { 1 + \\kappa } { 1 - \\kappa } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            568,
+            627,
+            595
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "53 Finally, for a method solving (OPT) that generates a sequence of iterates $\\{ x _ { t } \\}$ , we define its worst-case   \n54 rate $r _ { t }$ and its asymptotic rate factor $\\tau$ as ",
+        "bbox": [
+            145,
+            608,
+            826,
+            637
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dd9345c0c6401056bc663d47bd51e6344d82d4a7b3bdf22373dbb0798bbf6a85.jpg",
+        "text": "$$\nr _ { t } \\triangleq \\operatorname* { s u p } _ { x _ { 0 } \\in \\mathbb { R } ^ { d } , f \\in \\mathcal { C } _ { \\Lambda } } \\frac { \\Vert x _ { t } - x _ { * } \\Vert } { \\Vert x _ { 0 } - x _ { * } \\Vert } , \\qquad 1 - \\tau \\triangleq \\operatorname* { l i m } _ { t \\to \\infty } \\operatorname* { s u p } _ { \\tau \\in \\mathcal { C } _ { \\Lambda } } \\sqrt { r _ { t } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            643,
+            686,
+            680
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "55 3 Super-acceleration with Cyclical Step-sizes ",
+        "text_level": 1,
+        "bbox": [
+            145,
+            695,
+            563,
+            714
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "table",
+        "img_path": "images/6bcf5f7610cdd67095016c85739e02be5e3913853b93bbef9b0a58fa76c42347.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">Algorithm 2: Cyclical (K = 2) heavy ball with with optimal parameters</td></tr><tr><td colspan=\"3\">Input: Initialization xo,μ1&lt;L1 &lt; μ2&lt;L2 (whereL1-μ1=L2-μ2)</td></tr><tr><td>Set: p = L2+μ1 R= μ2-L1 L2-μ1 L2-μ1</td><td colspan=\"2\">Vp²-R2-√p2-1 2 m= √1-R²</td></tr><tr><td colspan=\"3\">χ1= xo-∀f(xo)</td></tr><tr><td>for t = 1, 2,... do ht= 1+m</td><td></td><td></td></tr><tr><td></td><td>(if t is even)， ht=1+m L1</td><td>(if t is odd) μ2</td></tr><tr><td></td><td>Xt+1= xt-htVf(xt)+m(xt-xt-1)</td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td>end</td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr></table>",
+        "bbox": [
+            171,
+            739,
+            772,
+            898
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "56 In this section we develop one of our main contri  \n57 butions, a convergence rate analysis of the cyclical   \n58 heavy ball method with cycles of length 2. This analy  \n59 sis crucially depends on the location of the Hessian’s   \n60 eigenvalues; we assume that these are contained in a   \n61 set $\\Lambda$ that is the union of 2 intervals of the same size ",
+        "bbox": [
+            145,
+            90,
+            517,
+            175
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4192bfd89590bd08402b5d4eaf3d5cc6565956702b84feca4bd180a69df2552b.jpg",
+        "text": "$$\n\\Lambda = [ \\mu _ { 1 } , L _ { 1 } ] \\cup [ \\mu _ { 2 } , L _ { 2 } ] , L _ { 1 } - \\mu _ { 1 } = L _ { 2 } - \\mu _ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            190,
+            493,
+            209
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "62 By symmetry, this set is alternatively described by ",
+        "bbox": [
+            147,
+            227,
+            504,
+            241
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4140de8552768ddacce2cbad750d4d3ee43a43cb8f807cd8dc8c637a7367535e.jpg",
+        "text": "$$\n\\mu \\triangleq \\mu _ { 1 } , \\quad L \\triangleq L _ { 2 } \\quad { \\mathrm { a n d } } \\quad R \\triangleq { \\frac { \\mu _ { 2 } - L _ { 1 } } { L _ { 2 } - \\mu _ { 1 } } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            189,
+            250,
+            480,
+            284
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "63 where $R$ is the relative length of the gap $\\mu _ { 2 } - L _ { 1 }$   \n64 with respect to the diameter $L _ { \\mathrm { 2 } } - \\mu _ { \\mathrm { 1 } }$ (see Figure 1).   \n65 This parametrization will reveal very convenient as   \n66 the relative gap will play a crucial role in the conver  \n67 gence rate analysis. Note also that the gap assumption   \n68 comes without loss of generality, as we allow $R = 0$ .   \n69 Through a correspondence between optimization   \n70 methods and polynomials that we expand upon in   \n71 Section 4, we can derive a worst-case analysis for the cyclical heavy ball method. The outcome of   \n72 this analysis is in the following theorem, that provides the asymptotic convergence rate of Algorithm   \n73 1 for cycles of length two. All proofs of results in this section can be found in Appendix D.3.   \n74 Theorem 3.1 (Rate factor of $\\mathrm { H B } _ { 2 } ( h _ { 0 } , h _ { 1 } ; m ) )$ . Let $f \\in { \\mathcal { C } } _ { \\Lambda }$ and $h _ { 0 } , h _ { 1 } , m \\ge 0 .$ The asymptotic rate   \n75 factor of Algorithm 1 with cycles of length two is ",
+        "bbox": [
+            145,
+            292,
+            517,
+            377
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/b6c3baaa1d848442fb38b5c37b7af9d8f5a72efb076794e6042b67b2ceac22b5.jpg",
+        "image_caption": [
+            "Figure 1: Hessian eigenvalue histogram for a quadratic objective on MNIST. The outlier eigenvalue at $L _ { 2 }$ generates a non-zero relative gap $R = 0 . 7 7$ . Under these conditions, the 2-cycle heavy ball method has a faster asymptotic rate than the single-cycle one (see Section 3.1). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            532,
+            93,
+            818,
+            268
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            382,
+            517,
+            411
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            411,
+            826,
+            452
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            148,
+            455,
+            823,
+            484
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a1ac22572c97efdf7e11a132fb2a7837dde76c50385e5fb101288ae76cf0d887.jpg",
+        "text": "$$\n\\begin{array} { r l } & { 1 - \\tau = \\left\\{ \\begin{array} { l l } { \\sqrt { m } } & { \\mathrm { i f ~ } \\sigma _ { \\mathrm { s u p } } \\leq 1 , } \\\\ { \\sqrt { m } \\left( \\sigma _ { \\mathrm { s u p } } + \\sqrt { \\sigma _ { \\mathrm { s u p } } ^ { 2 } - 1 } \\right) ^ { \\frac { 1 } { 2 } } } & { \\mathrm { i f ~ } \\sigma _ { \\mathrm { s u p } } \\in \\left( 1 , \\frac { 1 + m ^ { 2 } } { 2 m } \\right) , } \\\\ { \\geq 1 \\left( n o c o n v e r g e n c e \\right) } & { \\mathrm { i f ~ } \\frac { 1 + m ^ { 2 } } { 2 m } \\leq \\sigma _ { \\mathrm { s u p } } , } \\end{array} \\right. } \\\\ & { \\sigma _ { \\mathrm { s u p } } = \\underset { \\lambda \\in \\left\\{ \\mu _ { 1 } , L _ { 1 } , \\mu _ { 2 } , L _ { 2 } , \\frac { h _ { 0 } + h _ { 1 } } { 2 h _ { 0 } h _ { 1 } } \\right\\} \\cap \\Lambda } { \\operatorname* { s u p } } \\Big | 2 \\left( \\frac { 1 + m - \\lambda h _ { 0 } } { 2 \\sqrt { m } } \\right) \\left( \\frac { 1 + m - \\lambda h _ { 1 } } { 2 \\sqrt { m } } \\right) - 1 \\Big | \\ . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            489,
+            776,
+            599
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "76 ",
+        "bbox": [
+            147,
+            553,
+            161,
+            564
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "77 This theorem gives the convergence rate for all triplets $( m , h _ { 0 } , h _ { 1 } )$ . By evaluating this expression over a grid of step-sizes, Figure 2 shows how the rate changes as a function of both step-sizes: ",
+        "bbox": [
+            145,
+            608,
+            826,
+            637
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/7c66b4c0f2b433c2aa9f9f7518012f1f00e87a6154cce08e66608dfc0083e539.jpg",
+        "image_caption": [
+            "Figure 2: Asymptotic rate of cyclical $K = 2$ ) heavy ball in terms of its step-sizes $h _ { 0 } , h _ { 1 }$ across 3 different values of the relative gap $R$ . In the left plot, the relative gap is zero, and so the step-sizes with smallest rate coincide $h _ { 0 } = h _ { 1 } \\mathrm { , }$ ). For non-zero values of $R$ (center and right), the optimal method instead alternates between two different step-sizes. In all plots the momentum parameter $m$ is set according to Algorithm 2. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            650,
+            825,
+            808
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "79 From the asymptotic rate expression of Theorem 3.1 we can optimize over the parameters $( h _ { 0 } , h _ { 1 } , m )$   \n80 to obtain the method with smallest convergence rate. This leads to our other main contribution of this   \n81 section, the asymptotically optimal Algorithm 2. This algorithm enjoys the following rate: ",
+        "bbox": [
+            148,
+            90,
+            826,
+            133
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Corollary 3.2. The worst-case (asymptotic) rates 82 $r _ { t } ^ { A l g . 2 }$ and $1 - \\tau ^ { A l g . 2 }$ of Algorithm 2 over $\\mathcal { C } _ { \\Lambda }$ are ",
+        "bbox": [
+            155,
+            136,
+            820,
+            155
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b96b01d1e6212a8df92e31b4f636e903e802a97da3af31723bebf8ca53eecd0f.jpg",
+        "text": "$$\n\\begin{array} { r } { r _ { t } ^ { A l g . ~ 2 } = \\left( 1 + t \\sqrt { \\frac { \\rho ^ { 2 } - 1 } { \\rho ^ { 2 } - R ^ { 2 } } } \\right) \\left( \\frac { \\sqrt { \\rho ^ { 2 } - R ^ { 2 } } - \\sqrt { \\rho ^ { 2 } - 1 } } { \\sqrt { 1 - R ^ { 2 } } } \\right) ^ { t } , \\quad 1 - \\tau ^ { A l g . ~ 2 } = \\frac { \\sqrt { \\rho ^ { 2 } - R ^ { 2 } } - \\sqrt { \\rho ^ { 2 } - 1 } } { \\sqrt { 1 - R ^ { 2 } } } \\quad f _ { 0 } = \\tau ^ { A l g . ~ 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            174,
+            159,
+            759,
+            189
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "83 3.1 Comparison with Polyak Heavy Ball ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            202,
+            465,
+            218
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "84 In the absence of eigenvalue gap $R = 0$ and $\\Lambda = [ \\mu , L ] )$ , Algorithm 2 reduces to Polyak heavy   \n85 ball (PHB) [Polyak, 1964], whose worst-case rate is detailed in Appendix B. Since the asymptotic rate   \n86 of Algorithm 2 is monotonically decreasing in $R$ , it is always better or equal than PHB. Furthermore,   \n87 in the ill-conditioned regime (small $\\kappa$ ), the comparison is particularly simple: the optimal 2-cycle   \n88 algorithm has a $\\sqrt { 1 - R ^ { 2 } }$ relative improvement over PHB, as provided by the next proposition.   \n89 A more thorough comparison for different support sets $\\Lambda$ is discussed in Table 1. ",
+        "bbox": [
+            145,
+            227,
+            826,
+            314
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.3. Let $R \\in [ 0 , 1 )$ . The rate factors of respectively Algorithm 2 and PHB verify ",
+        "bbox": [
+            161,
+            316,
+            787,
+            332
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1312400b19b557cd8383ce7d8305474c1550c2806e3f078955558a124f1c884d.jpg",
+        "text": "$$\n\\begin{array} { r l r } { 1 - \\tau ^ { A l g . 2 } \\underset { \\kappa  0 } { = } 1 - \\frac { 2 \\sqrt { \\kappa } } { \\sqrt { 1 - R ^ { 2 } } } + o ( \\sqrt { \\kappa } ) , } & { } & { 1 - \\tau ^ { P H B } \\underset { \\kappa  0 } { = } 1 - 2 \\sqrt { \\kappa } + o ( \\sqrt { \\kappa } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            335,
+            756,
+            362
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/86e8649dbf4004249dfc92b4ad95c4677314025ea6d2bb1ef8c156dc682c0c9a.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Relative gap R</td><td>Set A</td><td></td><td>Rate factor TSpeedup T/7PHB</td></tr><tr><td>R ∈ [0,1)</td><td>[μ,μ+R(L-μ)]U[L-R(L-μ),L]</td><td>2√ 1-R</td><td>(1-R²)-</td></tr><tr><td>R=1-√K/2</td><td>[,μ+]U[-,]</td><td>2Vk</td><td>K</td></tr><tr><td>R=1-2γK</td><td>[μ,(1+γ)μ]U[L- γμ,L]</td><td> indep. of K</td><td>0（√）</td></tr></table>",
+        "bbox": [
+            181,
+            381,
+            812,
+            496
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Case study of the convergence of Algorithm 2 as a function of $R$ , in the regime $\\kappa  0$ . The first line corresponds to the regime where $R$ is independent of $\\kappa$ , and we observe a constant gain w.r.t. PHB. The second line considers a setting in which $R$ depends on $\\sqrt { \\kappa }$ , that is, the two intervals in $\\Lambda$ are relatively small. The asymptotic rate reads $( 1 - 2 \\sqrt [ 4 ] { \\kappa } ) ^ { t }$ , beating the classical $( 1 - 2 \\sqrt { \\kappa } ) ^ { t }$ lower bound, unimprovable when $R = 0$ . Finally, in the third line, $R$ depends on $\\kappa$ , the two intervals in $\\Lambda$ are so small that the convergence becomes $O ( 1 )$ , i.e., is independent of $\\kappa$ . ",
+        "bbox": [
+            173,
+            518,
+            826,
+            603
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "91 4 A constructive Approach: Minimax Polynomials ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            630,
+            609,
+            647
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "92 This section presents a generic framework (Algorithm 3) that allows designing optimal momentum   \n93 and step-size cycles for given sets $\\Lambda$ and cycle length $K$ . ",
+        "bbox": [
+            150,
+            660,
+            825,
+            690
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 3: Optimal momentum method with cyclical step-sizes ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            707,
+            609,
+            722
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Input: Eigenvalue localization $\\Lambda$ , cycle length $K$ , initialization $x _ { 0 }$ . Preprocessing: ",
+        "bbox": [
+            174,
+            724,
+            614,
+            753
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1. Find the polynomial $\\sigma _ { K } ^ { \\Lambda }$ such that it satisfies (16). ",
+        "bbox": [
+            215,
+            755,
+            557,
+            770
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2. Set step-sizes $\\{ h _ { i } \\} _ { i = 0 , \\dots , K - 1 }$ and momentum $m$ that satisfy resp. equations (21) and (22). ",
+        "bbox": [
+            202,
+            773,
+            825,
+            790
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "for $t = 1$ $x _ { 1 } = x _ { 0 } - \\frac { h _ { 0 } } { 1 + m } \\nabla f ( x _ { 0 } )$ , 2, . . . do $\\begin{array} { r l } & { \\nabla f ( x _ { 0 } ) } \\\\ & { \\quad x _ { t + 1 } = x _ { t } - h _ { \\mathrm { m o d } ( t , K ) } \\nabla f ( x _ { t } ) + m ( x _ { t } - x _ { t - 1 } ) } \\end{array}$ end ",
+        "bbox": [
+            171,
+            792,
+            660,
+            866
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "94 We first recall classical results that link optimal first order methods on quadratics and Chebyshev   \n95 polynomials. Then, we generalize the approach by showing that optimal methods can be viewed as ",
+        "bbox": [
+            143,
+            882,
+            826,
+            912
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "combinations of Chebyshev polynomials, and minimax polynomials 96 $\\sigma _ { K } ^ { \\Lambda }$ of degree $K$ over the set $\\Lambda$ Finally, we show how to recover the step-size schedule from 97 $\\sigma _ { K } ^ { \\Lambda }$ . ",
+        "bbox": [
+            156,
+            90,
+            825,
+            122
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 First Order Methods on Quadratics and Polynomials ",
+        "text_level": 1,
+        "bbox": [
+            160,
+            136,
+            581,
+            151
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A key property that we will use extensively in the analysis is the following link between first order methods and polynomials (see [Hestenes and Stiefel, 1952]). ",
+        "bbox": [
+            166,
+            161,
+            823,
+            190
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "101 Proposition 4.1. Let $f \\in { \\mathcal { C } } _ { \\Lambda }$ . The iterates $x _ { t }$ satisfy ",
+        "bbox": [
+            147,
+            193,
+            519,
+            208
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ea70a70afedf6560eb31aef7c53e05b33ef4790644e7fae73b1d72489e93af3c.jpg",
+        "text": "$$\nx _ { t + 1 } \\in x _ { 0 } + \\operatorname { s p a n } \\{ \\nabla f ( x _ { 0 } ) , \\ldots , \\nabla f ( x _ { t } ) \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            209,
+            642,
+            227
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "102 where $x _ { 0 }$ is the initial approximation of $x _ { * }$ , if and only if there exists a sequence of polynomials   \n103 $( P _ { t } ) _ { t \\in \\mathbb { N } }$ , each of degree at most $I$ more than the highest degree of all previous polynomials and $P _ { 0 }$ of   \n104 degree $O$ (hence the degree of $P _ { t }$ is at most $t$ ), such that ",
+        "bbox": [
+            145,
+            228,
+            830,
+            271
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c5126ec343daa4db4c556d5176c425b9ee59708da911035f43a87951f0fb1276.jpg",
+        "text": "$$\n\\forall t \\quad x _ { t } - x _ { * } = P _ { t } ( H ) ( x _ { 0 } - x _ { * } ) , \\quad P _ { t } ( 0 ) = 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            275,
+            660,
+            294
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "05 Example 4.2 (Gradient descent). Consider the gradient descent algorithm with fixed step-size $h$ ,   \n106 applied to problem (OPT). Then, after unrolling the update, we have ",
+        "bbox": [
+            150,
+            308,
+            825,
+            335
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3b9ddad9bf3471f50970bbd280d0e6fa6ff3a9105fd7994e0a71c3af9923241e.jpg",
+        "text": "$$\nx _ { t + 1 } - x _ { * } = x _ { t } - x _ { * } - h \\nabla f ( x _ { t } ) = x _ { t } - x _ { * } - h H ( x _ { t } - x _ { * } ) = ( I - h H ) ^ { t + 1 } ( x _ { 0 } - x _ { * } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            200,
+            338,
+            767,
+            356
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this case, the polynomial associated to gradient descent is $P _ { t } ( \\lambda ) = ( 1 - h \\lambda ) ^ { t }$ . ",
+        "bbox": [
+            181,
+            357,
+            705,
+            372
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "108 The above proposition can be used to obtain worst-case rates for first order methods by bounding   \n109 their associated polynomials. Indeed, using the Cauchy-Schwartz inequality in (9) leads to ",
+        "bbox": [
+            143,
+            381,
+            825,
+            410
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e0b5a7b577fd2542cca97fb127087f004c08c651afa0df4cae48319dc1cfc36.jpg",
+        "text": "$$\n\\| x _ { t } - x _ { * } \\| \\leq \\operatorname* { s u p } _ { \\lambda \\in \\Lambda } | P _ { t } ( \\lambda ) | \\ \\| x _ { 0 } - x _ { * } \\| \\quad \\Longrightarrow \\quad r _ { t } = \\operatorname* { s u p } _ { \\lambda \\in \\Lambda } | P _ { t } ( \\lambda ) | , \\quad \\mathrm { w h e r e } \\ P ( 0 ) = 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            205,
+            411,
+            764,
+            438
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "110 Therefore, finding the algorithm with the fastest worst-case rate can be equivalently framed as the   \n111 problem of finding the polynomial with smallest value on the eigenvalue support $\\Lambda$ , subject to the   \n112 normalization condition $\\dot { P _ { t } } ( 0 ) = 1$ . Such polynomials are referred to as minimax. Throughout the   \n113 paper, we use this polynomial-based approach to find methods with optimal rates.   \n114 An important property of minimax polynomials is their equioscillation on $\\Lambda$ (see Theorem C.1 and   \n115 its proof for a formal statement).   \n116 Definition 4.3. (Equioscillation) A polynomial $P _ { t }$ equioscillates on $\\Lambda$ if it verifies $P _ { t } ( 0 ) = 1$ and   \n117 there exist $\\lambda _ { 0 } < \\lambda _ { 1 } < . . . < \\lambda _ { t } \\in \\Lambda$ such that ",
+        "bbox": [
+            140,
+            439,
+            825,
+            496
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            501,
+            825,
+            530
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            532,
+            825,
+            560
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ca3db97a8b341d7d4544c4e1a54be009cc60e720286e780e0d42d5006e775435.jpg",
+        "text": "$$\nP _ { t } ( \\lambda _ { i } ) = ( - 1 ) ^ { i } \\operatorname* { m a x } _ { \\lambda \\in \\Lambda } | P _ { t } ( \\Lambda ) | .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            564,
+            601,
+            589
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "118 Example 4.4 $\\Lambda$ is an interval). The $t { \\cdot }$ -th order Chebyshev polynomials of the first kind $T _ { t }$ satisfy   \n119 the equioscillation property on $[ - 1 , 1 ]$ . It follows that minimax polynomials on $\\Lambda = [ \\mu , L ]$ can be   \n120 obtained by composing the Chebyshev polynomial $T _ { t }$ with the linear transformation $\\sigma _ { 1 } ^ { \\Lambda }$ : ",
+        "bbox": [
+            142,
+            597,
+            825,
+            640
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c03cc855317decf435e5eca5157cc2ab30d620c0e756d5bdd9c68a3bb1e7b770.jpg",
+        "text": "$$\n\\frac { T _ { t } \\left( \\sigma _ { 1 } ^ { \\Lambda } ( \\lambda ) \\right) } { T _ { t } \\left( \\sigma _ { 1 } ^ { \\Lambda } ( 0 ) \\right) } = \\operatorname * { a r g m i n } _ { P \\in \\mathbb { R } _ { t } [ X ] , P ( 0 ) = 1 } \\operatorname * { s u p } _ { \\lambda \\in \\Lambda } \\lvert P ( \\lambda ) \\rvert , \\mathrm { w i t h } \\sigma _ { 1 } ^ { \\Lambda } ( \\lambda ) = \\frac { L + \\mu } { L - \\mu } - \\frac { 2 } { L - \\mu } \\lambda ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            236,
+            642,
+            761,
+            679
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "121 where $\\sigma _ { 1 } ^ { \\Lambda }$ maps the interval $[ \\mu , L ]$ to $[ - 1 , 1 ]$ . The optimization method associated with this minimax   \n122 polynomial is the Chebyshev semi-terative method [Flanders and Shortley, 1950, Golub and Varga,   \n123 1961] (described also in Appendix B.1). This method achieves the lower complexity bound for   \n124 smooth strongly convex quadratic minimization, see for instance [Nemirovsky, 1995, Chapter 12] or   \n125 [Nemirovsky, 1992, Nesterov, 2003].   \n126 The next proposition provides the main results in this subsection, which is key for obtaining Algo  \n127 rithm 2. It characterizes the even degree minimax polynomial in the setting of Section 3, that is,   \n128 129 when Cheb $\\Lambda$ is the union of 2 intervals of same size. In this case, the minihev polynomials, but composed with a degree-two polynomial $\\sigma _ { 2 } ^ { \\Lambda }$ solution is also based on. ",
+        "bbox": [
+            140,
+            683,
+            825,
+            753
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            762,
+            826,
+            819
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "130 Proposition 4.5. Let $\\Lambda = [ \\mu _ { 1 } , L _ { 1 } ] \\cup [ \\mu _ { 2 } , L _ { 2 } ]$ be an union of two intervals of the same size 131 $( L _ { 1 } - \\mu _ { 1 } = L _ { 2 } - \\mu _ { 2 } )$ and let m be as defined in Algorithm 2. Then the minimax polynomial (solution to (12)) is, for all 132 $t = 2 n$ , $n \\in \\mathbb { N } _ { 0 } ^ { + }$ , ",
+        "bbox": [
+            143,
+            820,
+            825,
+            864
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/43817578e46184d65a2b42dff65aeb025add60f4d56166212da5f6618c208767.jpg",
+        "text": "$$\n{ \\frac { T _ { n } \\left( \\sigma _ { 2 } ^ { \\Lambda } ( \\lambda ) \\right) } { T _ { n } \\left( \\sigma _ { 2 } ^ { \\Lambda } ( 0 ) \\right) } } = \\underset { P ( 0 ) = 1 } { \\operatorname { a r g m i n } } \\ \\underset { \\lambda \\in \\Lambda } { \\operatorname* { s u p } } | P ( \\lambda ) | , \\ w i t h \\ \\ \\sigma _ { 2 } ^ { \\Lambda } ( \\lambda ) = 2 \\left( { \\frac { 1 + m } { 2 \\sqrt { m } } } \\right) ^ { 2 } \\left( 1 - { \\frac { \\lambda } { L _ { 1 } } } \\right) \\left( 1 - { \\frac { \\lambda } { \\mu _ { 2 } } } \\right) - 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            866,
+            815,
+            915
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "134 The polynomial in Example 4.4 uses a linear link function $\\sigma _ { 1 } ^ { \\Lambda }$ to map $\\Lambda$ to $[ - 1 , 1 ]$ . In Proposition 4.5,   \n135 we see that a degree two link function $\\sigma _ { 2 } ^ { \\Lambda }$ can be used to find the minimax polynomial when $\\Lambda$ is the   \n136 union of two intervals. This section generalizes this approach and considers higher-order polynomials   \n137 for $\\sigma _ { K }$ . We start with the following parametrization, with an arbitrary polynomial $\\sigma _ { K }$ of degree $K$ , ",
+        "bbox": [
+            140,
+            116,
+            826,
+            172
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d3d97501cf74f291de12fda79baf353ba2308413571db204f444d7c68abf014d.jpg",
+        "text": "$$\nP _ { t } ( \\lambda ; \\sigma _ { K } ) \\triangleq \\frac { T _ { n } \\left( \\sigma _ { K } ( \\lambda ) \\right) } { T _ { n } \\left( \\sigma _ { K } ( 0 ) \\right) } , \\quad \\forall t = K n , n \\in \\mathbb { N } _ { 0 } ^ { + } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            174,
+            665,
+            208
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "138 As we will see in the next subsection, this parametrization allows considering cycles of step-sizes.   \n139 Our goal now is to find the $\\sigma _ { K }$ that obtains the fastest convergence rate possible. The next proposition   \n140 quantifies its impact on the asymptotic rate and its proof can be found in Appendix D.1.   \n141 Proposition 4.6. For a given $\\sigma _ { K }$ such that $\\operatorname* { s u p } _ { \\lambda \\in \\Lambda } | \\sigma _ { K } ( \\lambda ) | = 1$ , the asymptotic rate factor $\\tau ^ { \\sigma _ { K } }$ of   \n142 the method associated to the polynomial (14) is ",
+        "bbox": [
+            140,
+            214,
+            826,
+            258
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            261,
+            826,
+            289
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/450545e6ef83784988d63a3e02ca5e76b879bb68d8075d651bba600927c6f519.jpg",
+        "text": "$$\n1 - \\tau ^ { \\sigma \\kappa } = \\operatorname * { l i m } _ { t  \\infty } \\sqrt { \\smash [ b ] { \\operatorname { s u p } _ { \\lambda \\in \\Lambda } | P _ { t } ( \\lambda ; \\sigma _ { K } ) | } } = ( \\sigma _ { 0 } - \\sqrt { \\sigma _ { 0 } ^ { 2 } - 1 } ) ^ { \\frac { 1 } { \\kappa } } , \\quad w i t h \\ \\sigma _ { 0 } \\triangleq \\sigma _ { K } ( 0 ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            290,
+            766,
+            330
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "143 For a fixed $K$ , the asymptotic rate (15) is a decreasing function of $\\sigma _ { 0 }$ . This motivates the introduction of the “optimal” degree 144 $K$ polynomial $\\sigma _ { K } ^ { \\Lambda }$ as the one that solves ",
+        "bbox": [
+            140,
+            337,
+            828,
+            367
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bf45748ad26ca5613ce1d5f9652ade9d69cfe0e33e8bb3d6c8f7dd055cb5a22e.jpg",
+        "text": "$$\n\\sigma _ { K } ^ { \\Lambda } \\triangleq \\underset { \\sigma \\in \\mathbb { R } _ { K } [ X ] } { \\arg \\operatorname* { m a x } } \\sigma ( 0 ) \\quad \\mathrm { s . t . } ~ \\underset { \\lambda \\in \\Lambda } { \\operatorname* { s u p } } | \\sigma ( \\lambda ) | = 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            369,
+            640,
+            397
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Using the above definition, we recover the 145 $\\sigma _ { 1 } ^ { \\Lambda }$ and $\\sigma _ { 2 } ^ { \\Lambda }$ from Example 4.4 and Proposition 4.5. ",
+        "bbox": [
+            147,
+            400,
+            779,
+            417
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "146 Finding the polynomial. Finding an exact and explicit solution for the general $K$ and $\\Lambda$ case   \n147 148 equations. Here we describe an approximate approach. Let σΛK (x) = PKi=0 σixi . We propose to   \n149 ",
+        "bbox": [
+            140,
+            429,
+            825,
+            489
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4e971d6cf3c46235e39ee5b116c19cfc53c607df7c52d87e37b95ab78c2c3a17.jpg",
+        "text": "$$\n\\begin{array} { r } { \\underset { \\sigma _ { i } } { \\operatorname* { m a x } } \\sigma _ { 0 } \\quad \\mathrm { ~ s . t . ~ } - 1 \\leq \\sum _ { i = 0 } ^ { K } \\sigma _ { i } \\lambda _ { j } ^ { i } \\leq 1 , \\quad \\forall j = 1 , \\ldots , N . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            491,
+            694,
+            517
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To check the optimality, it suffices to verify that the polynomial 50 $\\sigma _ { K } ^ { \\Lambda }$ satisfies the equioscillation 151 property (Definition 4.3), as depicted in Figure 3. ",
+        "bbox": [
+            150,
+            520,
+            825,
+            550
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "152 Remark 4.7 (Relationship between optimal and minimax polynomials). For later reference, we note   \n153 that the optimal polynomial $\\sigma _ { K } ^ { \\Lambda }$ is equivalent to finding a minimax polynomial on $\\Lambda$ and to rescale it.   \n154 More precisely, $\\sigma _ { K } ^ { \\Lambda }$ is optimal if and only if $\\sigma _ { K } ^ { \\Lambda } / \\sigma _ { K } ^ { \\Lambda } ( 0 )$ is minimax. ",
+        "bbox": [
+            140,
+            551,
+            826,
+            597
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "155 4.3 Cyclical Heavy Ball and (Non-)asymptotic Rates of Convergence ",
+        "text_level": 1,
+        "bbox": [
+            151,
+            613,
+            660,
+            628
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We now describe the link between 156 $\\sigma _ { K } ^ { \\Lambda }$ and Algorithm 3. Using the recurrence for Chebyshev polynomials of the first kind in (14), we have 157 $\\forall t = K n$ , $n \\in \\mathbb { N } _ { 0 } ^ { + }$ , ",
+        "bbox": [
+            140,
+            638,
+            823,
+            670
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a6c201dda720a3bb6771260094bf2c33a74575ee06b5fbf46177c5a28afba237.jpg",
+        "text": "$$\n\\frac { T _ { n + 1 } ( \\sigma _ { K } ^ { \\Lambda } ( \\lambda ) ) } { T _ { n + 1 } ( \\sigma _ { K } ^ { \\Lambda } ( 0 ) ) } = 2 \\sigma _ { K } ^ { \\Lambda } ( \\lambda ) \\left[ \\frac { T _ { n } ( \\sigma _ { K } ^ { \\Lambda } ( \\lambda ) ) } { T _ { n } ( \\sigma _ { K } ^ { \\Lambda } ( 0 ) ) } \\right] \\underbrace { \\left[ \\frac { T _ { n } ( \\sigma _ { K } ^ { \\Lambda } ( 0 ) ) } { T _ { n + 1 } ( \\sigma _ { K } ^ { \\Lambda } ( 0 ) ) } \\right] } _ { = a _ { n } } - \\left[ \\frac { T _ { n - 1 } ( \\sigma _ { K } ^ { \\Lambda } ( \\lambda ) ) } { T _ { n - 1 } ( \\sigma _ { K } ^ { \\Lambda } ( 0 ) ) } \\right] \\underbrace { \\left[ \\frac { T _ { n - 1 } ( \\sigma _ { K } ^ { \\Lambda } ( 0 ) ) } { T _ { n + 1 } ( \\sigma _ { K } ^ { \\Lambda } ( 0 ) ) } \\right] } _ { = b _ { n } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            671,
+            818,
+            727
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "158 It still remains to find an algorithm associated with this polynomial. To obtain one in the form of   \n159 Algorithm 1, one can use the stationary behavior of the recurrence. From [Scieur and Pedregosa,   \n160 2020], the coefficients $a _ { n }$ and $b _ { n }$ converge as $n \\to \\infty$ to their fixed-points $a _ { \\infty }$ and $b _ { \\infty }$ . We therefore   \n161 consider here an asymptotic polynomial $\\big ( \\bar { P } _ { t } ( \\lambda ; \\sigma _ { K } ^ { \\Lambda } )$ , whose recurrence satisfies ",
+        "bbox": [
+            140,
+            728,
+            825,
+            785
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d9bf2c7b041da20a3e9b8c1a765d3f7f1e7434b92f6517b05b0fb2ea778634ca.jpg",
+        "text": "$$\n\\bar { P } _ { t } ( \\lambda ; \\sigma _ { K } ^ { \\Lambda } ) = 2 a _ { \\infty } \\sigma _ { K } ^ { \\Lambda } ( \\lambda ) \\bar { P } _ { t - K } ( \\lambda ; \\sigma _ { K } ^ { \\Lambda } ) - b _ { \\infty } \\bar { P } _ { t - 2 K } ( \\lambda ; \\sigma _ { K } ^ { \\Lambda } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            787,
+            699,
+            806
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "162 Similarly to $K = 1$ , where this limit recursion corresponds to PHB, this recursion corresponds to   \n163 an instance of Algorithm 3 (see Proposition 4.9 below), further motivating the cyclical heavy ball   \n164 algorithm.   \n165 The following theorem is the main result of this section and characterizes the convergence rate of   \n166 Algorithm 1 for arbitrary momentum and step-size sequences $\\{ h _ { i } \\} _ { i \\in [ [ 1 , K ] ] }$ . By optimizing over these   \n167 J Kparameters, we obtain a method associated to (18), whose rate is described in Proposition 4.9. All   \n168 proofs can be found in Appendix D.2.   \n169 Theorem 4.8. The worst-case rate of convergence of Algorithm $^ { l }$ on $\\mathcal { C } _ { \\Lambda }$ with an arbitrary momentum   \n170 $m$ and an arbitrary sequence of step-sizes $\\{ h _ { i } \\}$ is ",
+        "bbox": [
+            142,
+            808,
+            825,
+            849
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            854,
+            825,
+            912
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/9240a9958eaa348db8f2ba977a8a2708f567c10acd0a1745f092f5ee93f5b674.jpg",
+        "image_caption": [
+            "Figure 3: Examples of optimal polynomials $\\sigma _ { K } ^ { \\Lambda }$ from (16), all of them verifying the equioscillation property (Definition 4.3). The $\" \\star \"$ symbol highlights the degree of $\\sigma _ { K } ^ { \\Lambda }$ that achieves the best asymptotic rate $\\tau ^ { \\sigma _ { K } ^ { \\Lambda } }$ in (15) amongst all $K$ (see Section 4.4). (Left) When $\\Lambda$ is an unique interval, all 3 polynomials are equivalently optimal $\\begin{array} { r l r } { \\tau ^ { \\sigma _ { 1 } ^ { \\Lambda } } = } & { { } } & { = \\tau ^ { \\sigma _ { 3 } ^ { \\Lambda } } } \\end{array}$ . (Center) When $\\Lambda$ is the union of two intervals of the same size, the degree 2 polynomial is optimal $\\tau ^ { \\sigma _ { 2 } ^ { \\Lambda } } > \\tau ^ { \\sigma _ { 3 } ^ { \\Lambda } } > \\tau ^ { \\sigma _ { 1 } ^ { \\Lambda } }$ . This is expected given the result in Proposition 4.5. (Right) When $\\Lambda$ is the union of two unbalanced intervals, the degree 3 polynomial instead achieves the best asymptotic rate $\\tau ^ { \\sigma _ { 3 } ^ { \\Lambda } } > > > \\tau ^ { \\sigma _ { 1 } ^ { \\Lambda } }$ (see Section 4.4). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            111,
+            812,
+            246
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            404,
+            826,
+            434
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c3ceae7fb87f8ae56a446c07e3446534794676d0f61bc65b559cb63f5688aa94.jpg",
+        "text": "$$\n1 - \\tau = \\left\\{ \\begin{array} { l l } { \\sqrt { m } , } & { \\mathrm { i f ~ } \\sigma _ { \\operatorname* { s u p } } \\leq 1 } \\\\ { \\sqrt { m } \\left( \\sigma _ { \\operatorname* { s u p } } + \\sqrt { \\sigma _ { \\operatorname* { s u p } } ^ { 2 } - 1 } \\right) ^ { 1 / K } , } & { \\mathrm { i f ~ } \\sigma _ { \\operatorname* { s u p } } \\in \\left( 1 , \\displaystyle \\frac { 1 + m ^ { K } } { 2 \\left( \\sqrt { m } \\right) ^ { K } } \\right) } \\\\ { \\geq 1 \\left( n o c o n v e r g e n c e \\right) } & { \\mathrm { i f ~ } \\sigma _ { \\operatorname* { s u p } } \\geq \\displaystyle \\frac { 1 + m ^ { K } } { 2 \\left( \\sqrt { m } \\right) ^ { K } } } \\end{array} \\right. ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            441,
+            753,
+            534
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where 171 $\\sigma _ { \\operatorname* { s u p } } \\triangleq \\operatorname* { s u p } _ { \\lambda \\in \\Lambda } \\vert \\sigma ( \\lambda ; \\{ h _ { i } \\} , m ) \\vert$ , and $\\sigma ( \\lambda ; \\{ h _ { i } \\} , m )$ is the $K$ -degree polynomial ",
+        "bbox": [
+            142,
+            541,
+            709,
+            568
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/572cf4062cc4cb961b6ed25acb3caf39d62877873e8cf994e885c0912697bbfe.jpg",
+        "text": "$$\n\\sigma ( \\lambda ; \\{ h _ { i } \\} , m ) \\triangleq \\frac { 1 } { 2 } \\mathrm { T r } \\left( \\left[ \\begin{array} { c c c } { \\frac { 1 + m - h _ { K - 1 } \\lambda } { \\sqrt { m } } } & { - 1 } \\\\ { 1 } & { 0 } \\end{array} \\right] \\left[ \\begin{array} { c c c } { \\frac { 1 + m - h _ { K - 2 } \\lambda } { \\sqrt { m } } } & { - 1 } \\\\ { 1 } & { 0 } \\end{array} \\right] \\cdots \\left[ \\begin{array} { c c } { \\frac { 1 + m - h _ { 0 } \\lambda } { \\sqrt { m } } } & { - 1 } \\\\ { 1 } & { 0 } \\end{array} \\right] \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            575,
+            784,
+            613
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "172 Proposition 4.9. Let $\\sigma ( \\lambda ; \\{ h _ { i } \\} , m )$ be the polynomial defined by (20), and $\\sigma _ { K } ^ { \\Lambda }$ be the optimal link   \n173 function of degree $K$ defined by (16). If the momentum $m$ and the sequence of step-sizes $\\{ h _ { i } \\}$ satisfy ",
+        "bbox": [
+            140,
+            630,
+            825,
+            660
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/96f2561be1240ffecd1276d87c108e462331d75f9c81332a5f90a50735bdfb2d.jpg",
+        "text": "$$\n\\sigma ( \\lambda ; \\{ h _ { i } \\} , m ) = \\sigma _ { K } ^ { \\Lambda } ( \\lambda ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            415,
+            667,
+            583,
+            688
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "174 then 1) the parameters are optimal, in the sense that they minimize the asymptotic rate factor from   \n175 Theorem 4.8, 2) the optimal momentum parameter is ",
+        "bbox": [
+            137,
+            695,
+            830,
+            724
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6a2553c61f8a1e8ebfe0e0edf660872dc25b357b3c55e438cdb6a646ade30592.jpg",
+        "text": "$$\nm = \\left( \\sigma _ { 0 } - \\sqrt { \\sigma _ { 0 } ^ { 2 } - 1 } \\right) ^ { 2 / K } , \\quad w h e r e \\sigma _ { 0 } = \\sigma _ { K } ^ { \\Lambda } ( 0 ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            733,
+            668,
+            756
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "176 3) the iterates from Algo. 3 with parameters $\\{ h _ { i } \\}$ and $m$ form a polynomial with recurrence (18), and 4) Algorithm 3 achieves the worst-case rate 177 $r _ { t } ^ { A l g . \\ 3 }$ and the asymptotic rate factor $1 - \\tau ^ { A l g }$ . 3 ",
+        "bbox": [
+            140,
+            765,
+            828,
+            797
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a9a8e01981bb16c7276851aa9022b25b00229a6b39c50d905ed6e67e47bb3fb2.jpg",
+        "text": "$$\nr _ { t } ^ { A l g . ~ 3 } = O \\left( t \\left( \\sigma _ { 0 } - \\sqrt { \\sigma _ { 0 } ^ { 2 } - 1 } \\right) ^ { t / K } \\right) , \\qquad 1 - \\tau ^ { A l g . ~ 3 } = \\left( \\sigma _ { 0 } - \\sqrt { \\sigma _ { 0 } ^ { 2 } - 1 } \\right) ^ { 1 / K } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            230,
+            804,
+            766,
+            840
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "178 Solving the system (21) The system is constructed by identification of the coefficients in both   \n179 polynomials $\\bar { \\sigma } _ { K } ^ { \\Lambda }$ and $\\sigma ( \\lambda ; \\{ h _ { i } \\} , m )$ , which can be solved using a naive grid-search for instance. We   \n180 are not aware of any efficient algorithm to solve this system exactly, although it is possible to use   \n181 iterative methods such as steepest descent or Newton’s method. ",
+        "bbox": [
+            140,
+            854,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This section discusses the (asymptotic) optimality of Algorithm 3. In Section 4.2, the polynomial $P _ { t } ( \\cdot ; \\sigma _ { K } ^ { \\Lambda } )$ was written as a composition of Chebyshev polynomials with $\\sigma _ { K } ^ { \\Lambda }$ , defined in (16). The best the $K$ is chosen as follows: we solve (16) for several values of imizers of (15). However, following such steps does not $K$ , then pick the smallest arantee that the polyno $K$ aal $P _ { t , K } ^ { \\Lambda }$ is minimax, as it is not guaranteed to minimize the worst-case rate $\\operatorname* { s u p } _ { \\lambda \\in \\Lambda } \\left| P _ { t } ( \\lambda ) \\right|$ (see (11)). ",
+        "bbox": [
+            171,
+            117,
+            825,
+            190
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We give here an optimality certificate, linked to a generalized version of equioscillation. In short, if we can find $K$ non overlapping intervals (more formally, whose interiors are disjoint) $\\Lambda _ { i }$ in $\\Lambda$ such that $\\sigma _ { K } ^ { \\Lambda } ( \\Lambda _ { i } ) = [ - 1 , 1 ]$ then $P _ { t , K } ^ { \\tilde { \\Lambda } }$ is minimax for all $t = n K$ , $n \\in \\mathbb { N } _ { 0 } ^ { + }$ . The detailed result is provided by Theorem C.2. A direct consequence of this result is the asymptotic optimality of Algorithm 3, i.e., there exists no first order algorithm with a better asymptotic rate $1 - \\tau$ for the function class $\\mathcal { C } _ { \\Lambda }$ . ",
+        "bbox": [
+            171,
+            194,
+            825,
+            265
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "It is possible that such $\\sigma _ { K } ^ { \\Lambda }$ does not exist for a given $\\Lambda$ . A complete characterization of the set $\\Lambda$ for which there exists such $\\sigma _ { K } ^ { \\Lambda }$ is out of the scope of this paper. A partial answer is given in [Fischer, 2011] when $\\Lambda$ is the union of two intervals. However, the problem remains open in the general case. ",
+        "bbox": [
+            171,
+            270,
+            826,
+            315
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "96 5 Local Convergence for Non-Quadratic Functions ",
+        "text_level": 1,
+        "bbox": [
+            156,
+            335,
+            612,
+            353
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "197 When $f$ is twice-differentiable, it is possible to show local convergence rates when $x _ { 0 }$ is close   \n198 enough to $x _ { * }$ [Polyak, 1964]. We give here a similar result that applies to Algorithm 1 (see proof in   \n199 Appendix E). Those results are only local, as it is possible to find pathological counter-examples for   \n200 which even PHB does not converge globally, for some specific initialization [Lessard et al., 2016].   \n201 Theorem 5.1 (Local convergence). Let $f : \\mathbb { R } ^ { d } \\mapsto \\mathbb { R }$ be a (potentially non-quadratic) twice continu  \n202 ously differentiable function, $x _ { * }$ a local minimizer, and $H$ be the Hessian of $f$ at $x _ { * }$ with $S p ( H ) \\subseteq \\Lambda$   \n203 Let $x _ { t }$ denote the result of running Algorithm 1 with parameters $h _ { 1 } , h _ { 2 } , \\cdot \\cdot \\cdot , h _ { K } , m$ , and let $1 - \\tau$ be   \n204 the linear convergence rate on the quadratic objective (OPT). Then we have ",
+        "bbox": [
+            143,
+            367,
+            825,
+            424
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            428,
+            825,
+            486
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d9fa1877b1a190700c8af2cd0aae74cc4078c4ec5666b1c1d9e4ba5cceb2d624.jpg",
+        "text": "$$\n\\forall \\varepsilon > 0 , \\exists \\mathrm { o p e n ~ s e t } \\ : V _ { \\varepsilon } : x _ { 0 } , x _ { * } \\in V _ { \\varepsilon } \\implies \\| x _ { t } - x _ { * } \\| = O ( ( 1 - \\tau + \\varepsilon ) ^ { t } ) \\| x _ { 0 } - x _ { * } \\| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            204,
+            492,
+            766,
+            511
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "205 In short, when Algorithm 1 is guaranteed to converge at rate $1 - \\tau$ on (OPT), then the convergence   \n206 rate on a nonlinear functions can be arbitrary close to $1 - \\tau$ when $x _ { 0 }$ is sufficiently close to $x _ { * }$ . ",
+        "bbox": [
+            147,
+            525,
+            828,
+            554
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            163,
+            575,
+            312,
+            592
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section we present an empirical comparison of the cyclical heavy ball method for different length cycles across 4 different problems. We consider two different problems, quadratic and logistic regression, each applied on two datasets, the MNIST handwritten digits [Le Cun et al., 2010] and a synthetic dataset. The results of these experiments, together with a histogram of the Hessian’s eigenvalues are presented in Figure 4 (see caption for a discussion). ",
+        "bbox": [
+            173,
+            607,
+            825,
+            678
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Dataset description. The MNIST dataset consists of a data matrix $A$ with 60000 images of handwritten digits each one with $2 8 \\times 2 8 = 7 8 4$ pixels. The synthetic dataset is generated according to a spiked covariance model [Johnstone, 2001], which has been shown to be an accurate model of covariance matrices arising for instance in spectral clustering [Couillet and Benaych-Georges, 2016] and deep networks [Pennington and Worah, 2017, Granziol et al., 2020]. In this model, the data matrix $A = X Z$ is generated from a $m \\times n$ random Gaussian matrix $X$ and an $m \\times m$ deterministic matrix $Z$ . In our case, we take $n = 1 0 0 0 , m = 1 2 0 0$ and $Z$ is the identity where the first three entries are multiplied by 100 (this will lead to three outlier eigenvalues). We also generate an $n$ -dimensional target vector $b$ as $b = A x$ or $b = { \\mathrm { s i g n } } ( A x )$ for the quadratic and logistic problem respectively. ",
+        "bbox": [
+            173,
+            683,
+            825,
+            809
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Objective function For each dataset, we consider a quadratic and a logistic regression problem, leadin to 4 different problems. All pro ms are of the form $\\begin{array} { r } { \\operatorname* { m i n } _ { x \\in \\mathbb { R } ^ { p } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\ell ( \\breve { A } _ { i } ^ { \\top } x , b _ { i } ) \\stackrel { \\cdot } { + } \\lambda \\| x \\| ^ { 2 } } \\end{array}$ , $\\ell$ $A$ is the data matrix and $b$ are the target values. We set the regularization parameter to $\\lambda = 1 0 ^ { - 3 } \\Vert A \\Vert ^ { 2 }$ . For logistic regression, since guarantees only hold at a neighborhood of the solution (even for the 1-cycle algorithm), we initialize the first iterate as the result of 100 iteration of gradient descent. In the case of logistic regression, the Hessian eigenvalues are computed at the optimum. ",
+        "bbox": [
+            171,
+            814,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/1e22ce525180458a522848876892c69e42308b6733154fc2bd6a5e5629064183.jpg",
+        "image_caption": [
+            "Figure 4: Hessian Eigenvalue histogram (top row) and Benchmarks (bottom row). The top row shows the Hessian eigenvalue histogram at optimum for the 4 problems consider, together with the interval boundaries $\\mu _ { 1 } < L _ { 1 } < \\mu _ { 2 } < L _ { 2 }$ for the two-interval split of the eigenvalue support described in Section 3. In all cases, there’s a non-zero gap radius $R$ . This is shown in the bottom row, where we compare the suboptimality in terms of gradient norm as a function of the number of iterations. As predicted by the theory, the non-zero gap radius translates into a faster convergence of the cyclical approach, compared to PHB in all cases. The improvement is observed on both quadratic and logistic regression problems, even through the theory for the latter is limited to local convergence. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            87,
+            823,
+            296
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "229 7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            145,
+            458,
+            299,
+            474
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This work is motivated by two recent observations from the optimization practice of machine learning. First, cyclical step-sizes have been shown to enjoy excellent empirical convergence [Loshchilov and Hutter, 2017, Smith, 2017]. Second, spectral gaps are pervasive in the Hessian spectrum of deep learning models [Sagun et al., 2017, Papyan, 2018, Ghorbani et al., 2019, Papyan, 2019]. Based on the simpler context of quadratic convex minimization, we develop a convergence-rate analysis and optimal parameters for the heavy ball method with cyclical step-sizes. This analysis highlights the regimes under which cyclical step-sizes have faster rates than classical accelerated methods. Finally, we illustrate these findings through numerical benchmarks. ",
+        "bbox": [
+            171,
+            489,
+            825,
+            601
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Main Limitations. In Section 3 we gave explicit formulas for the optimal parameters in the case of the 2-cycle heavy ball algorithm. These formulas depend not only on extremal eigenvalues—as is usual for accelerated methods—but also on the spectral gap $R$ . The gap can sometimes be computed after computed the top eigenvalues (e.g. top-2 eigenvalue for MNIST). However, in general, there is no guarantee on how many eigenvalues are needed to estimate it. Moreover, global convergence result rely heavily on the quadratic assumption. ",
+        "bbox": [
+            163,
+            616,
+            825,
+            699
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "244 Another limitation regards long cycles. For cycles longer than 2, we have only given an implicit   \n245 formula to set the optimal parameters (Proposition 4.9). This involves solving a set of non-linear   \n246 equations whose complexity increases with the cycle length. That being said, cyclical step-sizes   \n247 might significantly enhance convergence speeds both in terms of worst-case rates and empirically,   \n248 and this work advocates that new tuning practices involving different cycle lengths might be relevant. ",
+        "bbox": [
+            143,
+            705,
+            825,
+            775
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Broader Impact. This work is mostly theoretical, and as such we believe it does not present direct societal consequences. However, the methods described in this paper can be used to train machine learning models which could themselves have societal consequences. For example, the deployment of machine learning models in decision-making has been shown to suffer from gender and racial bias and to amplify existing inequalities, see for instance [Hutchinson and Mitchell, 2019, Barocas et al., 2017, Obermeyer et al., 2019]. ",
+        "bbox": [
+            173,
+            789,
+            825,
+            872
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            90,
+            267,
+            106
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "256 Solon Barocas, Moritz Hardt, and Arvind Narayanan. Fairness in machine learning. Nips tutorial, 2017.   \n258 Dimitri P. Bertsekas. Nonlinear programming. Journal of the Operational Research Society, 1997.   \n59 Romain Couillet and Florent Benaych-Georges. Kernel spectral clustering of large dimensional data. Electronic Journal of Statistics, 2016. Bernd Fischer. Polynomial based iteration methods for symmetric linear systems. SIAM, 2011. Donald A. Flanders and George Shortley. Numerical determination of fundamental modes. Journal of Applied Physics, 1950. Behrooz Ghorbani, Shankar Krishnan, and Ying Xiao. An investigation into neural net optimization via hessian eigenvalue density. In International Conference on Machine Learning (ICML), 2019. Gene H. Golub and Richard S. Varga. Chebyshev semi-iterative methods, successive overrelaxation iterative methods, and second order Richardson iterative methods. Numerische Mathematik, 1961. Diego Granziol, Xingchen Wan, Samuel Albanie, and Stephen Roberts. Explaining the Adaptive Generalisation Gap. arXiv preprint arXiv:2011.08181, 2020.   \n70 Magnus R. Hestenes and Eduard Stiefel. Methods of conjugate gradients for solving linear systems, volume 49. NBS Washington, DC, 1952. Ben Hutchinson and Margaret Mitchell. 50 years of test (un) fairness: Lessons for machine learning. In Proceedings of the Conference on Fairness, Accountability, and Transparency, 2019.   \n274 Iain M. Johnstone. On the distribution of the largest eigenvalue in principal components analysis. Annals of statistics, 2001.   \n76 Yann Le Cun, Corinna Cortes, and Chris Burges. MNIST handwritten digit database. ATT Labs [Online], 2010. Laurent Lessard, Benjamin Recht, and Andrew Packard. Analysis and design of optimization algorithms via integral quadratic constraints. SIAM Journal on Optimization, 2016. Ilya Loshchilov and Frank Hutter. SGDR: stochastic gradient descent with warm restarts. In International Conference on Learning Representations (ICLR), 2017.   \n282 Arkadi S. Nemirovsky. Information-based complexity of linear operator equations. Journal of Complexity, 1992.   \n284 Arkadi S. Nemirovsky. Information-based complexity of convex programming. Lecture Notes, 1995.   \n285 Yurii Nesterov. Introductory Lectures on Convex Optimization. Springer, 2003.   \n286 Ziad Obermeyer, Brian Powers, Christine Vogeli, and Sendhil Mullainathan. Dissecting racial bias in an algorithm used to manage the health of populations. Science, 2019. Vardan Papyan. The full spectrum of deepnet hessians at scale: Dynamics with SGD training and sample size. arXiv preprint arXiv:1811.07062, 2018. Vardan Papyan. Measurements of Three-Level Hierarchical Structure in the Outliers in the Spectrum of Deepnet Hessians. In International Conference on Machine Learning (ICML), 2019.   \n92 Jeffrey Pennington and Pratik Worah. Nonlinear random matrix theory for deep learning. In Advances on Neural Information Processing Systems (NIPS), 2017. Boris T. Polyak. Some methods of speeding up the convergence of iteration methods. USSR computational mathematics and mathematical physics, 1964. Levent Sagun, Utku Evci, V. Ugur Guney, Yann Dauphin, and Leon Bottou. Empirical analysis of the Hessian of over-parametrized neural networks. arXiv preprint arXiv:1706.04454, 2017.   \n98 Damien Scieur and Fabian Pedregosa. Universal Asymptotic Optimality of Polyak Momentum. In International Conference on Machine Learning (ICML), 2020. Leslie N. Smith. Cyclical learning rates for training neural networks. In 2017 IEEE Winter Conference on Applications of Computer Vision (WACV). IEEE, 2017.   \n02 Ilya Sutskever, James Martens, George Dahl, and Geoffrey Hinton. On the importance of initialization and momentum in deep learning. In International Conference on Machine Learning (ICML), 2013. ",
+        "bbox": [
+            151,
+            101,
+            830,
+            902
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            116,
+            339,
+            130
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] The Introduction (Section 1) details where all results can be found.   \n(b) Did you describe the limitations of your work? [Yes] There is a paragraph \"Main Limitations\" in the Conclusion (Section 7).   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] As stated in the \"Broader Impact\" section, this work is mostly theoretical and as such we believe it doesn’t present a direct societal impact.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            238,
+            135,
+            825,
+            281
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            215,
+            285,
+            493,
+            300
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] In each and every statement we make.   \n(b) Did you include complete proofs of all theoretical results? [Yes] All proofs are available in the supplementary material. ",
+        "bbox": [
+            238,
+            304,
+            825,
+            363
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            214,
+            367,
+            393,
+            381
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] URLs are provided in the supplementary material.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] In the experiments section (Section 6).   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] .   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] , in Appendix F ",
+        "bbox": [
+            238,
+            386,
+            825,
+            518
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            214,
+            522,
+            823,
+            536
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] In the \"Dataset description\" paragraph of the Experiments section 6.   \n(b) Did you mention the license of the assets? [N/A]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] There is no new asset. We used MNIST existing Dataset as well as a synthetic dataset whose construction is described in papers cited in the Experiments section 6.   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
+        "bbox": [
+            238,
+            541,
+            825,
+            689
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            214,
+            694,
+            705,
+            708
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            238,
+            712,
+            825,
+            801
+        ],
+        "page_idx": 10
+    }
+]

parse/train/B1l6y0VFPr/B1l6y0VFPr.md

@@ -0,0 +1,687 @@
+# IDENTITY CRISIS: MEMORIZATION AND GENERALIZATION UNDER EXTREME OVERPARAMETERIZATION
+
+Chiyuan Zhang & Samy Bengio Google Research, Brain Team Mountain View, CA 94043, USA {chiyuan,bengio}@google.com
+
+Moritz Hardt University of California, Berkeley Berkeley, CA 94720, USA hardt@berkeley.edu
+
+Michael C. Mozer Google Research, Brain Team Mountain View, CA 94043, USA mcmozer@google.com
+
+Yoram Singer Princeton University Princeton, NJ 08544, USA y.s@princeton.edu
+
+# ABSTRACT
+
+We study the interplay between memorization and generalization of overparameterized networks in the extreme case of a single training example and an identitymapping task. We examine fully-connected and convolutional networks (FCN and CNN), both linear and nonlinear, initialized randomly and then trained to minimize the reconstruction error. The trained networks stereotypically take one of two forms: the constant function (memorization) and the identity function (generalization). We formally characterize generalization in single-layer FCNs and CNNs. We show empirically that different architectures exhibit strikingly different inductive biases. For example, CNNs of up to 10 layers are able to generalize from a single example, whereas FCNs cannot learn the identity function reliably from 60k examples. Deeper CNNs often fail, but nonetheless do astonishing work to memorize the training output: because CNN biases are location invariant, the model must progressively grow an output pattern from the image boundaries via the coordination of many layers. Our work helps to quantify and visualize the sensitivity of inductive biases to architectural choices such as depth, kernel width, and number of channels.
+
+# 1 INTRODUCTION
+
+The remarkable empirical success of deep neural networks is often attributed to the availability of large data sets for training. However, sample size does not provide a comprehensive rationale since complex models often outperform simple ones on a given data set, even when the model size exceeds the number of training examples.
+
+What form of inductive bias leads to better generalization performance from highly overparameterized models? Numerous theoretical and empirical studies of inductive bias in deep learning have been conducted in recent years (Dziugaite & Roy, 2016; Kawaguchi et al., 2017; Bartlett et al., 2017; Neyshabur et al., 2017; Liang et al., 2017; Neyshabur et al., 2018; Arora et al., 2018; Zhou et al., 2019) but these postmortem analyses do not identify the root source of the bias.
+
+One cult belief among researchers is that gradient-based optimization methods provide an implicit bias toward simple solutions (Neyshabur et al., 2014; Soudry et al., 2018; Shah et al., 2018; Arora et al., 2019). However, when a network is sufficiently large (e.g., the number of hidden units in each layer is polynomial in the input dimension and the number of training examples), then under some mild assumptions, gradient methods are guaranteed to fit the training set perfectly (Allen-Zhu et al., 2018b; Du et al., 2018a;b; Zou et al., 2018). These results do not distinguish a model trained on a data distribution with strong statistical regularities from one trained on the same inputs but with randomly shuffled labels. Although the former model might achieve good generalization, the latter can only memorize the training labels. Consequently, these analyses do not tell the whole story on the question of inductive bias.
+
+![](images/fc06e529e49efb4ad77beac30d644b0988a4f1a8cfcf3b178d820be9ac9104aa.jpg)  
+Figure 1: Predictions of three architectures trained on the identity mapping task with 60k MNIST examples. The red dashed line separates 3 training examples from the test examples.
+
+Another line of research characterizes sufficient conditions on the input and label distribution that guarantee generalization from a trained network. These conditions range from linear separability (Brutzkus et al., 2018) to compact structures (Li & Liang, 2018). While very promising, this direction has thus far identified only structures that can be solved by linear or nearest neighbor classifiers over the original input space. The fact that in many applications deep neural networks significantly outperform these simpler models reveals a gap in our understanding of deep neural networks.
+
+As a formal understanding of inductive bias in deep networks has been elusive, we conduct a novel exploration in a highly restrictive setting that admits visualization and quantification of inductive bias, allowing us to compare variations in architecture, optimization procedure, initialization scheme, and hyperparameters. The particular task we investigate is learning an identity mapping in a regression setting. The identity mapping is interesting for four reasons. First, it imposes a structural regularity between the input and output, the type of regularity that could in principle lead to systematic generalization (He et al., 2016; Hardt & Ma, 2017). Second, it requires that every input feature is transmitted to the output and thus provides a sensitive indicator of whether a model succeeds in passing activations (and gradients) between inputs and outputs. Third, conditional image generation is a popular task in the literature (e.g., Mirza & Osindero, 2014; Ledig et al., 2017); an identity mapping is the simplest form of such a generative process. Fourth, and perhaps most importantly, it admits detailed analysis and visualization of model behaviors and hidden representations.
+
+Consider networks trained on the identity task with 60k MNIST digits. Although only digit images are presented during the training, one might expect the strong regularity of the task to lead to good generalization to images other than digits. Figure 1 compares three different architectures. The top row shows various input patterns, and the next three rows are outputs from a 20-layer convolutional net (CNN), a 10-layer fully connected net (FCN) with rectified-linear unit (ReLU) activation functions, and a 1-layer FCN. The 1-layer FCN amounts to a convex optimization problem with infinitely many solutions, however gradient decent converges to a unique closed-form solution. All nets perform well on the training set (first three columns) and transfer well to novel digits and digit blends (columns 4–6). Yet, outside of the hull of hand-printed digits, only the CNN discovers a reasonably good approximation to the identity function.
+
+Figure 1 reflects architecture-specific inductive bias that persists even with 60k training examples. Despite this persistence, a model’s intrinsic bias is more likely to be revealed with a smaller training set. In this paper, we push this argument to the limit by studying learning with a single training example. Although models are free to reveal their natural proclivities in this maximally overparameterized regime, our initial intuition was that a single example would be uninteresting as models would be algebraically equivalent to the constant function (e.g., via biases on output units). Further, it seemed inconceivable that inductive biases would be sufficiently strong to learn a mapping close to the identity. Unexpectedly, our experiments show that model behavior is subtle and architecture dependent. In a broad set of experiments, we highlight model characteristics—including depth, initialization, and hyperparameters—that determine where a model lands on the continuum between memorization (learning a constant function) and generalization (learning the identity function). The simplicity of the training scenario permits rich characterization of inductive biases.
+
+![](images/7f95d9984e35102ecaf1f46e21f9d9070407f298d035ffaf6af1398d578ddcfe.jpg)  
+Figure 2: Visualization of the outputs of fully connected networks trained on a single example. The first row shows the single training example (7) and a set of evaluation images consisting of a linear combination of two digits, random digits from MNIST test set, random images from Fashion MNIST, and some algorithmically generated image patterns. Each row below indicates an architecture and the output from that architecture for a given input.
+
+# 2 RELATED WORK
+
+The consequences of overparameterized models in deep learning have been extensively studied in recently years, on the optimization landscape and convergence of SGD (Allen-Zhu et al., 2018b; Du et al., 2018a;b; Bassily et al., 2018; Zou et al., 2018; Oymak & Soltanolkotabi, 2018), as well as the generalization guarantees under stronger structural assumptions of the data (Li & Liang, 2018; Brutzkus et al., 2018; Allen-Zhu et al., 2018a). Another line of related work is the study of the implicit regularization effects of SGD on training overparameterized models (Neyshabur et al., 2014; Zhang et al., 2017; Soudry et al., 2018; Shah et al., 2018; Arora et al., 2019).
+
+The traits of memorization in learning are also explicitly studied from various perspectives such as prioritizing learning of simple patterns (Arpit et al., 2017) or perfect interpolation of the training set (Belkin et al., 2018; Feldman, 2019). More recently, coincidentally with the writing of this paper, Radhakrishnan et al. (2018) reported on the effects of the downsampling operator in convolutional auto-encoders on image memorization. Their empirical framework is similar to ours, fitting CNNs to the autoregression problem with few training examples. We focus on investigating the general inductive bias in the extreme overparameterization case, and study a broader range of network types without enforcing a bottleneck in the architectures.
+
+# 3 EXPERIMENTS
+
+We explore a progression of models: linear convex models, linear non-convex models (with multiple linear layers), fully-connected multilayered architectures with nonlinearities, and finally the case of greatest practical importance, fully convolutional networks. In all architectures we study, we ensure that there is a simple realization of the identity function (see Appendix B). We train networks by minimizing the mean squared error using standard gradient descent.
+
+# 3.1 FULLY CONNECTED NETWORKS
+
+Figure 2 shows examples of predictions from multi-layer fully connected networks. Infinitely many solutions exist for all models under this extreme over-parameterization, and the figure shows that all the models fit the training example perfectly. However, on new test examples, contrasting behaviors are observed between shallow and deep networks. In particular, deeper models bias toward predicting a constant output, whereas shallower networks tend to predict random white noises on unseen inputs. The random predictions can be characterized as follows (proof in Appendix C).
+
+Theorem 1. A one-layer fully connected network, when trained with gradient descent on a single training example $\hat { x }$ , converges to a solution that makes the following prediction on a test example $x$ :
+
+$$
+f ( x ) = \Pi _ { \parallel } ( x ) + \mathbf { R } \Pi _ { \perp } ( x ) ,
+$$
+
+![](images/508fdbb34aa6fd7839074ce691345d3d8b0c5a84139c42651a093cae62bcc51a.jpg)  
+Figure 3: Visualization of predictions from CNNs trained on a single example. The first row shows the single training example (7) and a set of test inputs. Each row below shows the output of a CNN whose depth is indicated to the left of the row. The hidden layers of the CNN consist of $5 \times 5$ convolution filters organized as 128 channels.
+
+where $x = \Pi _ { \parallel } ( x ) + \Pi _ { \perp } ( x )$ decomposes $x$ into orthogonal components that are parallel and perpendicular to the training example $\hat { x }$ , respective. R is a random matrix from the network initialization, independent of the training data.
+
+For test examples similar to the training example—i.e., where $\Pi _ { \parallel } ( x )$ dominates $\Pi _ { \perp } ( x )$ —the outputs resemble the training output; on the other hand, for test examples that are not highly correlated to the training example, $\bar { \Pi } _ { \perp } ( \bar { x } )$ dominates and the outputs looks like white noise due to the random projection by $\mathbf { R }$ . The behavior can be empirically verified from the second row in Figure 2. Specifically, the first test example is a mixture of the training and an unseen test example, and the corresponding output is a mixture of white noise and the training output. For the remaining test examples, the outputs appear random.
+
+Although Theorem 1 characterizes only the 1-layer linear case, the empirical results in Figure 2 suggest that shallow (2 layer) networks tend to have this inductive bias. However, this inductive bias does not miraculously obtain good generalization: the trained model fails to learn either the identity or the constant function. Specifically, it predicts well in the vicinity (measured by correlations) of the training example $\hat { x }$ , but further away its predictions are random. In particular, when the test example $x$ is orthogonal to $\hat { x }$ , the prediction is completely random. In deeper (6 layer) nets, the deviations from the identity function take on a quite different characteristic form.
+
+Interestingly, deeper linear networks behave more like deeper ReLU networks, with a strong bias towards a constant function that maps any input to the single training output. A multilayer linear network with no hidden-layer bottleneck has essentially the same representational power as a 1-layer linear network, but gradient descent produces different learning dynamics that alter the inductive biases. See Appendix $\mathrm { E }$ for more results and analysis on FCNs.
+
+![](images/19f8f600de478ee721e977113e2289cd664ef326f1269db5bef137161ce62845.jpg)  
+Figure 4: Predictions of CNNs on test examples at different angles to the training image. The horizontal axis shows the train-test correlation, while the vertical axis indicate the number of hidden layers for the CNNs being evaluated. The heatmap shows the similarity (measured in correlation) between the model prediction and the reference function (the constant or the identity function).
+
+# 3.2 CONVOLUTIONAL NETWORKS
+
+We next study the inductive bias of convolutional neural networks with ReLU activation functions. Figure 3 shows predictions on various test patterns obtained by training CNNs of varying depths. Compared to FCNs, CNNs have strong structural constraints that limit the receptive field of each neuron to a spatially local neighborhood and the weights are tied and being used across the spatial array. These two constraints match the structure of the identity target function. (See Appendix B.3 for an example of constructing the identity function with CNNs.) Similar to the fully connected case, for one-layer CNN, we can bound the error as follows (proof in Appendix D).
+
+Theorem 2. A one-layer convolutional neural network can learn the identity map from a single training example with the mean squared error over all output pixels bounded as
+
+$$
+M S E \leq \tilde { \mathcal { O } } \left( \frac { m ( m / C - r ) } { C } \right)
+$$
+
+where m is the number of network parameters, $C$ is the number of channels in the image, and $r \leq m / C$ is the rank of the subspace formed by the span of the local input patches.
+
+The error grows with $m$ , the number of parameters in the network. For example, learning CNNs with larger receptive field sizes will be harder. Even though the bound seems to decrease with more (input and output) channels in the image, note that the number of channels $C$ also contributes to the number of parameters $( m = K _ { H } K _ { W } C ^ { 2 } )$ , so there is a trade-off. Unlike typical generalization bound that decays with number of i.i.d. training examples, we have only one training example here, and the key quantity that reduces the bound is the rank $r$ of the subspace formed by the local image patches. The size of the training image implicitly affects bounds as larger image generates more image patches. Note the rank $r$ also heavily depends on the contents of the training image. For example, simply padding the image with zeros on all boundaries will not reduce the error bound. With enough linearly independent image patches, the subspace becomes full rank $r = m / c$ , and learning of the global identity map is guaranteed.
+
+The theorem guarantees only the one-layer case. Empirically—as shown in Figure 3—CNNs with depth up-to-5 layers learn a fairly accurate approximation to the identity function, with the exception of a few artifacts at the boundaries. For a quantitative evaluation, we measure the performance by calculating the correlation (See Appendix J for the results in MSE.) to two reference functions: the identity function and the constant function that maps every input to the training point $\hat { x }$ . To examine how a model’s response varies with similarity to the training image, we generate test images having correlation $\rho \in [ 0 , 1 ]$ to the training image by: (1) sampling an image with random pixels, (2) adding $\alpha \hat { x }$ to the image, picking $\alpha$ such that the correlation with $\hat { x }$ is $\rho$ ; (3) renormalizing the image to be of the same norm as $\hat { x }$ . For $\rho = 0$ , the test images are orthogonal to $\hat { x }$ , whereas for $\rho = 1$ , the test images equal $\hat { x }$ . The results for CNNs of different depths are shown in Figure 4. The quantitative findings are consistent with the visualizations: shallow CNNs are able to learn the identity function from only one training example; very deep CNNs bias towards the constant function; and CNNs of intermediate depth correlate well with neither the identity nor the constant function. However, unlike FCNs that produce white-noise-like predictions, from Figure 3 CNNs of intermediate depth behave like edge detectors.
+
+![](images/5302d643a21f7e634e0609e79e49dcd62cbdcd8976d11c7072f5c23626856ad9.jpg)  
+Figure 5: Illustration of the collapse of predictive power as function of layer’s depth. Error rate is measured by using the representations computed at each layer to a simple averaging based classifier on the MNIST test set. The error rate at each layer is plotted for a number of trained CNNs of different depth. The thick red line shows the curve of an untrained 20-layer CNN for reference.
+
+To evaluate how much information is lost in the intermediate layers, we use the following simple criterion to assess the representation in each layer. We feed each image in the MNIST dataset through a network trained on our single example. We collect the representations at a given layer and perform a simple similarity-weighted classification. For each example $x _ { i }$ from the MNIST test set, we predict its class as a weighted average of the (one hot) label vector of each example $x _ { j }$ from the MNIST training set, where the weight is the inner-product of $x _ { i }$ and $x _ { j }$ .
+
+This metric does not quantify how much information is preserved as the image representation propagates through the layers, because the representation could still be maintained yet not captured by the simple correlation-weighted classifier. It nonetheless provides a simple metric for exploring the (identity) mapping: using the input image as the baseline, if a layer represents the identity function, then the representation at that layer would obtain a similar error rate when using the input representation; on the other hand, if a layer degenerates into the constant function, then the corresponding representation would have error rate close to random guessing of the label. The results are plotted in Figure 5. The error curve for a randomly initialized 20-layer CNN is shown as reference: at random initialization, the smoothing effect renders the representations beyond the sixth layer unuseful for the averaging-based classifier. After training, the concave nonmonotonicity in the curves indicates loss and then recovery of the information present in the input. Trained networks try to recover the washed out intermediate layer representations as means to link the input and the output layer. However, if the depth is too large, the network tries to infer input-output relations using partial information, resulting in models that behave like edge detectors. Finally, for the case of 20 layers, the curve shows that the bottom few layers do get small improvements in error rate comparing to random initialization, but the big gap between the input and output layer drives the network to learn the constant function instead. On a first sight, this deems to underscore a vanishing gradient problem, but Figure 1 reveals that given a sufficient number of training examples, a 20-layer CNN can still learn the identity map. See also Appendix G for further discussions on vanishing gradients. Since CNNs preserve spatial structure, we can also visualize information loss in the intermediate layers. The visualization results, described in Appendix F, are consistent with the aforementioned observations.
+
+# 3.3 ROBUSTNESS TO CHANGES IN INPUT SCALE
+
+Whether a relatively shallow CNN learns the identity function from a single training example or a relatively deep CNN learns the constant function, both outcomes reflect an inductive bias because the training objective never explicitly mandates the model to learn one structure or another. The spatial structure of CNNs enables additional analyses of the encoding induced by the learned function. In particular, we examined how changing the size of the input by scaling the dimensions of the spatial map affects the model predictions. Figure 6 depicts the predictions of a 5-layer CNN trained on a $2 8 \times 2 8$ image and tested on $7 \times 7$ and $1 1 2 \times 1 1 2$ images. Although the learned identity map generally holds up against a larger-than-trained input, the identity map is disturbed on smaller-thantrained inputs. See Appendix H.1 for a comprehensive description of the results. Note that CNNs are capable of encoding the identity function for arbitrary input and filter sizes (Appendix B.3).
+
+Figure 7 shows the predictions on the rescaled input patterns of Figure 6 by a 20-layer CNN that has learned the constant function on a $2 8 \times 2 8$ image. The learned constant map holds up over a smaller range of input sizes than the learned identity map in a 5-layer CNN. It is nonetheless interesting to see smooth changes as the input size increases to reveal the network’s own notion of “7”, clearly defined with respect to the corners of the input map (Figure 21 in Appendix H reveals interesting details on how the patterns progressively grow from image boundaries in intermediate layers). We performed additional experiments to directly feed test images to the upper subnet, which reveals more about the generative process by which the net synthesizes a constant output (Appendix H).
+
+![](images/50361bfd49385b0af2ce09088bb39a944a38e055b3b95ecc28ef9f4f2ec8faea.jpg)  
+Figure 6: Visualization of a 5-layer CNN on test images of different sizes. The two subfigures show the results on $7 \times 7$ inputs and $1 1 2 \times 1 1 2$ inputs, respectively.
+
+![](images/905ef711245755d979e3d471ea22ff1370cdb8b26e1dd57269d6aa57423323a6.jpg)  
+Figure 7: Visualization of the predictions of a 20-layer CNN on test images of different sizes (indicated by the number on each row). The input patterns are the same as in Figure 6 (constructed in different resolutions), which are not shown for brevity.
+
+# 3.4 VARYING OTHER FACTORS DURING TRAINING
+
+We studied a variety of common hyperparameters, including image dimensions, convolutional filter dimensions, number of kernels, weight initialization scheme, and the choice of gradient-based optimizer. With highly overparameterized networks, training converges to zero error for a wide range of hyperparameters. Within the models that all perform optimally on the training set, we now address how the particular hyperparameter settings affect the inductive bias. We briefly present some of the most interesting observations here; please refer to Appendix I for the full results and analyses.
+
+Size of training image. Figure 8 shows, for varying training-image size, the mean correlation to the constant and the identity function at different depths within the network. The training examples are resized versions of the same image. The bias toward the constant function at a given depth increases with smaller training images. This finding makes sense considering that smaller images provide fewer pixel-to-pixel mapping constraints, which is aligned with Theorem 2.
+
+Size of convolution filters. Figure 9 illustrates the inductive bias with convolution filter size varying from $5 \times 5$ to $5 7 \times 5 7$ . Predictions become blurrier as the filter size grows. With extremely large filters that cover the entire input array, CNNs exhibit a strong bias towards the constant function.
+
+![](images/5f77ce886b838d746477baef4725f52f6ec50f5c752fc71fa1e3e0b9c270e876.jpg)  
+Figure 8: Comparing bias towards constant and identity when trained with different image sizes. The xaxis is the depth of the CNNs, while the y-axis is the mean correlation (average of each row from the heatmaps like in Figure 4). Each curve corresponds to training with a different image size.
+
+![](images/372bfa2839f350185a592b490a68e013ed03a448f4317aa49086febfcd773194.jpg)  
+Figure 9: Visualizing the predictions from 5-layer CNNs with various filter sizes. The first row shows the single training example (7) and a set of test inputs. Each row below shows the output of a CNN whose filter size is indicated by the number on the left. See Appendix I for more results.
+
+![](images/4f05637849de81fe28505f93f1aebbdb55d416efabaa20d872bbba7936fd890f.jpg)  
+Figure 10: Visualization of the predictions from CNNs for various number of hidden channels. The first row shows a single training example (7) and a set of test inputs. Each row below shows the output of CNNs differing in the number of channels. This number is indicated on the left of each row; for intermediate layers, the number specifies both the input and output channel count.
+
+Because the training inputs are of size $2 8 \times 2 8$ , a $2 9 \times 2 9$ filter size allows each neuron to see at least half of the spatial domain of the previous layer, assuming large boundary padding of the inputs. With $5 7 \times 5 7$ filters centered at any location within the image, each neuron sees the entire previous layer. This is also consistent with Theorem 2, in which the error bound deteriorates as the filter sizes increases. Note even with a large filter, CNNs do not perform the same elementary computation as FCNs because the (shared) convolution filter is repeatedly applied throughout the spatial domain.
+
+Number of channels. Appendix B.3 shows that in principle two channels suffice to encode the identity function for gray-scale inputs via a straightforward construction. However, in practice the outcome of training may be quite different depending on the channel count, as shown in Figure 10(a). On the one hand, the aggressively overparameterized network with 1024 channels $\mathrm { \sim } 2 5 \mathrm { M }$ parameters per middle convolution layer) does not seem to suffer from overfitting. On the other hand, the results with three channels often lose content in the image center. The problem is not underfitting as the network reconstructs the training image (first column) correctly.
+
+A potential reason why 3-channel CNNs do not learn the identity map well might be poor initialization when there are very few channels (Frankle & Carbin, 2019). This issue is demonstrated in Figure 29 of Appendix I. Our study of training with different initialization schemes indeed confirms that the random initial conditions have a big impact on the inductive bias (Appendix I).
+
+Figure 10(b) shows the case for 20-layer CNNs. Surprisingly, having an order of magnitude more feature channels in every layer does not seem to help much at making the information flow through layers, as the network still learns to ignore the inputs and construct a constant output.
+
+# 4 CONCLUSIONS
+
+We presented an systematic study of the extreme case of overparameterization when learning from a single example. We investigated the interplay between memorization and generalization in deep neural networks. By restricting the learning task to the identity function, we sidestepped issues such as the underlying optimal Bayes error of the problem and the approximation error of the hypothesis classes. This choice also facilitated rich visualization and intuitive interpretation of the trained models. Under this setup, we investigated gradient-based learning procedures with explicit memorization-generalization characterization. Our results indicate that different architectures exhibit vastly different inductive bias towards memorization and generalization.
+
+For future work, we plan to extend the study to other domains and neural network architectures, like natural language processing and recurrent neural networks, and aim for more qualitative relationship between the inductive bias and various architecture configurations.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Kunal Talwar, Hanie Sedghi, and Rong Ge for helpful discussions.
+
+# REFERENCES
+
+Zeyuan Allen-Zhu, Yuanzhi Li, and Yingyu Liang. Learning and generalization in overparameterized neural networks, going beyond two layers. CoRR, 1811.04918, 2018a.
+
+Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. A convergence theory for deep learning via OverParameterization. CoRR, arXiv:1811.03962, 2018b.
+
+Sanjeev Arora, Rong Ge, Behnam Neyshabur, and Yi Zhang. Stronger generalization bounds for deep nets via a compression approach. CoRR, arXiv:1802.05296, 2018.
+
+Sanjeev Arora, Nadav Cohen, Wei Hu, and Yuping Luo. Implicit regularization in deep matrix factorization. arXiv preprint arXiv:1905.13655, 2019.
+
+Devansh Arpit, Stanisław Jastrzebski, Nicolas Ballas, David Krueger, Emmanuel Bengio, Maxinder S Kanwal, Tegan Maharaj, Asja Fischer, Aaron Courville, Yoshua Bengio, et al. A closer look at memorization in deep networks. In International Conference on Machine Learning, 2017.
+
+Peter L Bartlett, Dylan J Foster, and Matus J Telgarsky. Spectrally-normalized margin bounds for neural networks. In Advances in Neural Information Processing Systems, pp. 6240–6249, 2017.
+
+Raef Bassily, Mikhail Belkin, and Siyuan Ma. On exponential convergence of SGD in non-convex over-parametrized learning. CoRR, arXiv:1811.02564, 2018.
+
+Mikhail Belkin, Daniel Hsu, and Partha Mitra. Overfitting or perfect fitting? risk bounds for classification and regression rules that interpolate. In Advances in Neural Information Processing Systems, 2018.
+
+Alon Brutzkus, Amir Globerson, Eran Malach, and Shai Shalev-Shwartz. SGD learns overparameterized networks that provably generalize on linearly separable data. In ICLR, 2018.
+
+Simon S Du, Jason D Lee, Haochuan Li, Liwei Wang, and Xiyu Zhai. Gradient descent finds global minima of deep neural networks. CoRR, arXiv:1811.03804, 2018a.
+
+Simon S Du, Xiyu Zhai, Barnabas Poczos, and Aarti Singh. Gradient descent provably optimizes over-parameterized neural networks. CoRR, arXiv:1810.02054, 2018b.
+
+John Duchi, Elad Hazan, and Yoram Singer. Adaptive subgradient methods for online learning and stochastic optimization. Journal of Machine Learning Research, 12(Jul):2121–2159, 2011.
+
+Gintare Karolina Dziugaite and Daniel M Roy. Computing nonvacuous generalization bounds for deep (stochastic) neural networks with many more parameters than training data. In UAI, 2016.
+
+Vitaly Feldman. Does learning require memorization? a short tale about a long tail. arXiv preprint arXiv:1906.05271, 2019.
+
+Jonathan Frankle and Michael Carbin. The lottery ticket hypothesis: Finding sparse, trainable neural networks. In ICLR, 2019.
+
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pp. 249–256, 2010.
+
+Alex Graves. Generating sequences with recurrent neural networks. arXiv preprint arXiv:1308.0850, 2013.
+
+Moritz Hardt and Tengyu Ma. Identity matters in deep learning. 2017.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In Proceedings of the IEEE international conference on computer vision, pp. 1026–1034, 2015.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In European conference on computer vision, pp. 630–645. Springer, 2016.
+
+Kenji Kawaguchi, Leslie Pack Kaelbling, and Yoshua Bengio. Generalization in deep learning. CoRR, arXiv:1710.05468, 2017.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Christian Ledig, Lucas Theis, Ferenc Huszar, Jose Caballero, Andrew Cunningham, Alejandro ´ Acosta, Andrew Aitken, Alykhan Tejani, Johannes Totz, Zehan Wang, et al. Photo-realistic single image super-resolution using a generative adversarial network. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 4681–4690, 2017.
+
+Yuanzhi Li and Yingyu Liang. Learning overparameterized neural networks via stochastic gradient descent on structured data. In NIPS, 2018.
+
+Tengyuan Liang, Tomaso Poggio, Alexander Rakhlin, and James Stokes. Fisher-rao metric, geometry, and complexity of neural networks. CoRR, arXiv:1711.01530, 2017.
+
+Mehdi Mirza and Simon Osindero. Conditional generative adversarial nets. arXiv preprint arXiv:1411.1784, 2014.
+
+Behnam Neyshabur, Ryota Tomioka, and Nathan Srebro. In search of the real inductive bias: On the role of implicit regularization in deep learning. CoRR, arXiv:1412.6614, 2014.
+
+Behnam Neyshabur, Srinadh Bhojanapalli, David McAllester, and Nati Srebro. Exploring generalization in deep learning. In Advances in Neural Information Processing Systems, pp. 5947–5956, 2017.
+
+Behnam Neyshabur, Srinadh Bhojanapalli, and Nathan Srebro. A PAC-Bayesian approach to Spectrally-Normalized margin bounds for neural networks. In ICLR, 2018.
+
+Samet Oymak and Mahdi Soltanolkotabi. Overparameterized nonlinear learning: Gradient descent takes the shortest path? CoRR, arXiv:1812.10004, 2018.
+
+Adityanarayanan Radhakrishnan, Mikhail Belkin, and Caroline Uhler. Downsampling leads to image memorization in convolutional autoencoders. CoRR, arXiv:1810.10333, 2018.
+
+Andrew M Saxe, James L McClelland, and Surya Ganguli. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. In International Conference on Learning Representations, 2014.
+
+Vatsal Shah, Anastasios Kyrillidis, and Sujay Sanghavi. Minimum norm solutions do not always generalize well for over-parameterized problems. CoRR, arXiv:1811.07055, 2018.
+
+Daniel Soudry, Elad Hoffer, Mor Shpigel Nacson, Suriya Gunasekar, and Nathan Srebro. The implicit bias of gradient descent on separable data. Journal of Machine Learning Research, 19(70), 2018.
+
+Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, et al. Google’s neural machine translation system: Bridging the gap between human and machine translation. arXiv preprint arXiv:1609.08144, 2016.
+
+Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. In International Conference on Learning Representations, 2017.
+
+Wenda Zhou, Victor Veitch, Morgane Austern, Ryan P Adams, and Peter Orbanz. Non-vacuous generalization bounds at the ImageNet scale: a PAC-Bayesian compression approach. In ICLR, 2019.
+
+Difan Zou, Yuan Cao, Dongruo Zhou, and Quanquan Gu. Stochastic gradient descent optimizes over-parameterized deep ReLU networks. CoRR, arXiv:1811.08888, 2018.
+
+# A EXPERIMENT DETAILS AND HYPER-PARAMETERS
+
+We specify the experiment setups and the hyper-parameters here. Unless otherwise specified in each study (e.g. when we explicitly vary the number of convolution channels), all the hyper-parameters are set according to the default values here.
+
+The main study is done with the MNIST dataset. It consists of grayscale images of hand written digits of size $2 8 \times 2 8$ . For training, we randomly sample one digit from the training set (a digit $^ { \bullet } 7 ^ { \bullet }$ ) with a fixed random seed. For testing, we use random images from the test set of MNIST and Fashion-MNIST, as well as algorithmically generated structured patterns and random images. The training and test images are all normalized by mapping the pixel values in $\{ 0 , 1 , \ldots , 2 5 5 \}$ to $[ 0 , 1 ]$ , and then standardize with the mean 0.1307 and standard deviation 0.3081 originally calculated on the (full) MNIST training set.
+
+The models are trained by minimizing the mean squared error (MSE) loss with a vanilla SGD (base learning rate 0.01 and momentum 0.9). The learning rate is scheduled as stagewise constant that decays with a factor of 0.2 at the $30 \%$ , $60 \%$ and $80 \%$ of the total training steps (2,000,000). No weight decay is applied during training.
+
+For neural network architectures, the rectified linear unit (ReLU) activation is used for both fully connected networks (FCNs) and convolutional networks (CNNs). The input and output dimensions are decided by the data. The hidden dimensions for the FCNs are 2,048 by default. The CNNs use $5 \times 5$ kernels with stride 1 and padding 2, so that the geometry does not change after each convolution layer.
+
+# B REPRESENTATION OF THE IDENTITY FUNCTION USING DEEP NETWORKS
+
+In this section, we provide explicit constructions on how common types of neural networks can represent the identity function. Those constructions are only proof for that the models in our study have the capacity to represent the target function. There are many different ways to construct the identity map for each network architecture, but we try to provide the most straightforward and explicit constructions. However, during our experiments, even when the SGD learns (approximately) the identity function, there is no evidence suggesting that it is encoding the functions in similar ways as described here. We put some mild constraints (e.g. no “bottleneck” in the hidden dimensions) to allow more straightforward realization of the identity function, but this by no means asserts that networks violating those constraints cannot encode the identity function.
+
+# B.1 LINEAR MODELS
+
+For a one-layer linear network $f ( x ) = W x$ , where $W \in \mathbb { R } ^ { d \times d }$ , setting $W$ to the identity matrix will realize the identity function. For a multi-layer linear network $\begin{array} { r } { f ( x ) { \stackrel { - } { = } } ( \prod _ { \ell } W _ { \ell } ) x } \end{array}$ , we need to require that all the hidden dimensions are not smaller than the input dimension. In this case, a simple concrete construction is to set each $W _ { \ell }$ to an identity matrix.
+
+# B.2 MULTI-LAYER RELU NETWORKS
+
+The ReLU activation function $\sigma ( \cdot ) = \operatorname* { m a x } ( 0 , \cdot )$ discards all the negative values. There are many ways one can encode the negative values and recover it after ReLU. We provide a simple approach that uses hidden dimensions twice the input dimension. Consider a ReLU network with one hidden layer $f ( x ) = W _ { 2 } \sigma ( W _ { 1 } x )$ , where $W _ { 2 } \in \mathbf { \tilde { \mathbb { R } } } ^ { d \times 2 d }$ , $W _ { 1 } \in \mathbb { R } ^ { 2 d \times d }$ . The idea is to store the positive and negative part of $x$ separately, and then re-construct. This can be achieved by setting
+
+$$
+W _ { 1 } = \left( { \small { \frac { I _ { d } } { - I _ { d } } } } \right) , \quad W _ { 2 } = ( I _ { d } \quad - I _ { d } )
+$$
+
+where $I _ { d }$ is the $d$ -dimensional identity matrix. For the case of more than two layers, we can use the bottom layer to split the positive and negative part, and the top layer to merge them back. All the intermediate layers can be set to $2 d$ -dimensional identity matrix. Since the bottom layer encode all the responsives in non-negative values, the ReLU in the middle layers will pass through.
+
+# B.3 CONVOLUTIONAL NETWORKS
+
+In particular, we consider 2D convolutional networks for data with the structure of multi-channel images. A mini-batch of data is usually formatted as a four-dimensional tensor of the shape $B \times C \times H \times W$ , where $B$ is the batch size, $C$ the number of channels (e.g. RGB or feature channels for intermediate layer representations), $H$ and $W$ are image height and width, respectively. A convolutional layer (ignoring the bias term) is parameterized with another four-dimensional tensor of the shape $\bar { C } ^ { \bullet } \times \bar { C } ^ { \bullet } K _ { H } \times K _ { W }$ , where $\bar { C }$ is the number of output feature channels, $K _ { H }$ and $K _ { W }$ are convolutional kernel height and width, respectively. The convolutional kernel is applied at local $K _ { H } \times K _ { W }$ patches of the input tensor, with optional padding and striding.
+
+For one convolution layer to represent the identity function, we can use only the center slice of the kernel tensor and set all the other values to zero. Note it is very rare to use even numbers as kernel size, in which case the “center” of the kernel tensor is not well defined. When the kernel size is odd, we can set
+
+$$
+W _ { \bar { c } c h w } = \left\{ \begin{array} { l l } { 1 } & { \bar { c } = c , \ h = \lfloor K _ { H } / 2 \rfloor , \ w = \lfloor K _ { W } / 2 \rfloor } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+By using only the center of the kernel, we essentially simulate a $1 \times 1$ convolution, and encode a local identity function for each (multi-channel) pixel.
+
+For multi-layer convolutional networks with ReLU activation functions, the same idea as in multilayer fully-connected networks can be applied. Specifically, we ask for twice as many channels as the input channels for the hidden layers. At the bottom layer, separately the positive and negative part of the inputs, and reconstruct them at the top layer.
+
+# C PROOF OF THEOREM 1
+
+Consider the 1-layer linear model $f _ { W } ( x ) = W x$ , where $\boldsymbol { x } \in \mathbb { R } ^ { d }$ and $W \in \mathbb { R } ^ { d \times d }$ . Let $\hat { x }$ be the single training example. The training objective is to minimize the empirical risk $\hat { R } = 1 / 2 \lVert f _ { W } ( x ) - x \rVert _ { 2 } ^ { 2 }$ . The optimization problem is convex and well understood. Due to overparameterization, the solution of the empirical risk minimization is not unique. However, given randomly initialized weights $W ^ { 0 }$ , gradient descent obtains a unique global minimizer.
+
+The gradient of the empirical risk is
+
+$$
+\frac { \partial \hat { R } } { \partial W } = ( W - I ) \hat { x } \hat { x } ^ { \top }
+$$
+
+Gradient descent with step sizes $\eta _ { t }$ and initialization weights $W ^ { 0 }$ updates weights as
+
+$$
+\begin{array} { r l } { \displaystyle } & { \boldsymbol { W } ^ { T } = \boldsymbol { W } ^ { 0 } - \sum _ { t = 1 } ^ { T } \eta _ { t } \big ( \boldsymbol { W } ^ { t - 1 } - I ) \hat { \boldsymbol { x } } \hat { \boldsymbol { x } } ^ { \top } } \\ { : = \boldsymbol { W } ^ { 0 } + u _ { T } \hat { \boldsymbol { x } } ^ { \top } } \end{array}
+$$
+
+where $u _ { T } \in \mathbb { R } ^ { d }$ is a vector decided via the accumulation in the optimization trajectory. Because of the form of the gradient, it is easy to see the solution found by gradient descent will always have such parameterization structure. Moreover, under this parameterization, a unique minimizer exists that solves the equation
+
+$$
+\hat { x } = f _ { W } ( \hat { x } ) = W ^ { 0 } \hat { x } + u \hat { x } ^ { \top } \hat { x }
+$$
+
+via
+
+$$
+\hat { u } = \frac { ( I - W ^ { 0 } ) \hat { x } } { \Vert \hat { x } \Vert _ { 2 } ^ { 2 } }
+$$
+
+Therefore, the global minimizer can be written as
+
+$$
+\begin{array} { r l } { \hat { f } _ { W } ( x ) = ( W ^ { 0 } + \hat { u } \hat { x } ^ { \top } ) x } \\ { = } & { { } + W ^ { 0 } \left( x - \frac { \hat { x } ^ { \top } x } { \Vert \hat { x } \Vert _ { 2 } ^ { 2 } } \cdot \hat { x } \right) } \end{array}
+$$
+
+For the one-layer network case, the optimization problem is convex. Under standard conditions in convex optimization, gradient descent will converge to the global minimizer shown above.
+
+We can easily verify that the red term is exactly the projection of $x$ onto the training example $\hat { x }$ , while the blue term is the residual projection onto the orthogonal subspace.
+
+# D PROOF OF THEOREM 2
+
+Lemma 1. Consider the linear model $f ( x ) = W ^ { \top } x$ , where $\boldsymbol { x } \in \mathbb { R } ^ { D }$ , $W \in \mathbb { R } ^ { D \times d }$ . Let the training set be $\left\{ ( x _ { 1 } , y _ { 1 } ) , \dotsc ( x _ { N } , y _ { N } ) \right\}$ . Assume the model is overparameterized $\prime N \leq d )$ and there is no sample redundancy (rank of the data matrix is $N$ ). Fitting $f ( \cdot )$ by optimizing the square loss with gradient descent on the training set converges to $\hat { f } ( x ) = \hat { W } ^ { \top } x$ , where
+
+$$
+\hat { W } = W ^ { 0 } + X ^ { \top } ( X X ^ { \top } ) ^ { - 1 } ( Y - X W ^ { 0 } ) .
+$$
+
+$W ^ { 0 } \in \mathbb { R } ^ { D \times d }$ is the random initialization of weights, $\boldsymbol { X } = \left( x _ { 1 } , \ldots , x _ { N } \right) ^ { \top } \in \mathbb { R } ^ { N \times D }$ and $Y =$ $\left( y _ { 1 } , \ldots , y _ { N } \right) ^ { \top } \in \mathbb { R } ^ { N \times d }$ are the matrices formed by the inputs and outputs, respectively.
+
+Lemma 2. With the same notation of Lemma $^ { l }$ , assume the model is underparameterized $N > d ,$ ) and there is no feature redundancy (rank of the data matrix is $d ,$ ). Fitting $f ( \cdot )$ by optimizing the square loss with gradient descent on the training set converges to $\hat { f } ( x ) = \hat { W } ^ { \top } x$ where
+
+$$
+\hat { W } = \left( \boldsymbol { X } ^ { \top } \boldsymbol { X } \right) ^ { - 1 } \boldsymbol { X } ^ { \top } \boldsymbol { Y } ,
+$$
+
+Proof of Theorem 2. Let $\hat { x } \in \mathbb { R } ^ { H \times W \times C }$ be the single training image of size $H \times W$ , and $C$ color channels. Let $K _ { H } \times K _ { W }$ be the convolution receptive field size, so the weights of convolution filters can be parameterized via a 4-dimensional tensor $\Theta \in \mathbb { R } ^ { K _ { H } \times K _ { W } \times C \times C }$ . Let $\Xi$ be the collection of 2D coordinates of the local patches from the input image that the convolutional filter is applied to, and $\mathsf { P } _ { i j } ( \hat { x } ) \in \mathbb { R } ^ { K _ { H } \times K _ { W } \times C }$ be the local patch of $\hat { x }$ centered at the coordinate $( i , j ) \in \Xi$ . Note for our case the input and output are of the same shape, so the convolution stride is one, and $| \Xi | = H \times W$ .
+
+The empirical risk of fitting $\hat { x }$ with a one-layer convolution model can be written as
+
+$$
+\hat { R } = \frac { 1 } { 2 } \sum _ { ( i , j ) \in \Xi } \sum _ { k = 1 } ^ { C } \left( \langle \Theta _ { : : : k } , \mathsf { P } _ { i j } ( \hat { x } ) \rangle - \hat { x } _ { i j k } \right) ^ { 2 } ,
+$$
+
+where $\Theta _ { : : : k } \in \mathbb { R } ^ { K _ { H } \times K _ { W } \times C }$ is the subset of convolution weights corresponding to the $k$ -th output channel, and $\hat { x } _ { i j k }$ is the pixel value at coordinate $( i , j )$ and channel $k$ .
+
+CASE 1 — overparameterization: $| \Xi | \le K _ { H } \times K _ { W } \times C \times C$ . Assumme the patches are linearly independent1, with slight abuse of notatons, we represent the empirical risk in matrix form with the
+
+following matrices2:
+
+$$
+\begin{array} { r l } & { W : = ( \Theta _ { : : : 1 } , \dots , \Theta _ { : : : C } ) \in \mathbb R ^ { ( K _ { H } K _ { W } C ) \times C } , } \\ & { X : = ( \dots , \mathsf { P } _ { i j } ( \hat { x } ) , \dots ) ^ { \top } \in \mathbb R ^ { | \Xi | \times ( K _ { H } K _ { W } C ) } , } \\ & { Y : = \left( \begin{array} { c c c } { \vdots } & { \vdots } & { \vdots } \\ { \hat { x } _ { i j 1 } } & { \dots } & { \hat { x } _ { i j C } } \\ { \vdots } & { \vdots } & { \vdots } \end{array} \right) \in \mathbb R ^ { | \Xi | \times C } , } \end{array}
+$$
+
+and apply Lemma 1 to obtain $\hat { W }$ . For a test example $x$ , the prediction of the $( i , j )$ -th (multi-channel) pixel is
+
+$$
+\begin{array} { r l } & { \hat { W } ^ { \top } { \mathsf P } _ { i j } ( x ) = \left( W ^ { 0 } + X ^ { \top } \left( X X ^ { \top } \right) ^ { - 1 } ( Y - X W ^ { 0 } ) \right) ^ { \top } { \mathsf P } _ { i j } ( x ) } \\ & { \qquad = W ^ { 0 \top } \left( I - X ^ { \top } \left( X X ^ { \top } \right) ^ { - 1 } X \right) { \mathsf P } _ { i j } ( x ) + Y ^ { \top } ( X X ^ { \top } ) ^ { - 1 } X { \mathsf P } _ { i j } ( x ) } \end{array}
+$$
+
+Note we are learning the identity map, the $( i , j )$ -th row of the learning target matrix $Y$ is the (multi-channel) pixel at the $( i , j )$ -th coordinate of $\hat { x }$ . This is exactly the center of the $( i , j )$ -th patch $\mathsf P _ { i j } ( \hat { x } )$ , i.e. the $( i , j )$ -th row of the matrix $X$ . As a result, there is a linear projection matrix $\boldsymbol { \Lambda } \in \mathrm { \bar { \mathbb { R } } } ^ { ( K _ { H } K _ { W } C ) \times C }$ that maps $X$ to $Y \colon X \Lambda = Y$ . So
+
+$$
+\begin{array} { r } { \hat { W } ^ { \top } P _ { i j } ( x ) = W ^ { 0 \top } \Pi _ { X } ^ { \bot } \mathsf { P } _ { i j } ( x ) + \Lambda ^ { \top } \Pi _ { X } ^ { \| } \mathsf { P } _ { i j } ( x ) } \end{array}
+$$
+
+where $\Pi _ { X } ^ { \parallel } = X ^ { \top } ( X X ^ { \top } ) ^ { - 1 } X$ is the linear operator that projects a vector into the subspace spanned by the rows of $X$ , and $\Pi _ { X } ^ { \bot } = I - \Pi _ { X } ^ { \| }$ is the operator projecting into the orthogonal subspace.
+
+To compute the prediction errors, it is suffice to look at the errors at each $( i , j )$ -th (multi-channel) pixel separately:
+
+$$
+\begin{array} { r l } & { \mathrm { E r r o r } _ { i j } ^ { 2 } = \Vert \hat { f } ( x ) _ { i j } - \Lambda ^ { \top } \mathsf { P } _ { i j } ( x ) \Vert ^ { 2 } } \\ & { \qquad = \left. W ^ { 0 ^ { \top } } \Pi _ { X } ^ { \bot } \mathsf { P } _ { i j } ( x ) + \Lambda ^ { \top } \Pi _ { X } ^ { \| } \mathsf { P } _ { i j } ( x ) - \Lambda ^ { \top } \mathsf { P } _ { i j } ( x ) \right. ^ { 2 } } \\ & { \qquad = \left. ( W ^ { 0 } - \Lambda ) ^ { \top } \Pi _ { X } ^ { \bot } \mathsf { P } _ { i j } ( x ) \right. ^ { 2 } } \\ & { \qquad \le \left( \left( \left. W ^ { 0 ^ { \top } } \Pi _ { X } ^ { \bot } \right. + \left. \Lambda ^ { \top } \Pi _ { X } ^ { \bot } \right. \right) \left. \mathsf { P } _ { i j } ( x ) \right. \right) ^ { 2 } . } \end{array}
+$$
+
+Assume the absolute value of pixel values are bounded by $B$ , then
+
+$$
+\| \mathsf { P } _ { i j } ( x ) \| \leq B \sqrt { K _ { H } K _ { W } C } .
+$$
+
+The first two terms can be bounded according to the rank r of the subspace projection $\Pi _ { X } ^ { \parallel }$ . Let the nullity of the projection be ${ \mathfrak { n } } = K _ { H } K _ { W } C - { \mathfrak { r } }$ . Note the projection matrix can be decomposed as
+
+$$
+\Pi _ { X } ^ { \perp } = \sum _ { k = 1 } ^ { \mathfrak { n } } v _ { k } v _ { k } ^ { \top }
+$$
+
+where $\{ v _ { k } \} _ { k }$ is a orthonormal basis for the projection subspace.
+
+$$
+\begin{array} { r l } { \| \Lambda ^ { \top } \Pi _ { X } ^ { \perp } \| = \bigg \| \Lambda ^ { \top } \displaystyle \sum _ { k = 1 } ^ { n } v _ { k } v _ { k } ^ { \top } \bigg \| } & { } \\ { = \sqrt { \displaystyle \sum _ { k = 1 } ^ { n } \| \Lambda ^ { \top } v _ { k } \| ^ { 2 } } } & { } \\ { \leq \sqrt { \mathsf { n } \| \Lambda \| ^ { 2 } } } & { } \\ { = \sqrt { \mathsf { n } C } . } \end{array}
+$$
+
+(sub-multiplicativity of the Frobenius norm)
+
+Similarly,
+
+$$
+\left\| W ^ { 0 ^ { \top } } \Pi _ { X } ^ { \perp } \right\| = \sqrt { \sum _ { k = 1 } ^ { \mathfrak n } \left\| W ^ { 0 ^ { \top } } v _ { k } \right\| ^ { 2 } }
+$$
+
+Assume the entries of $W ^ { 0 }$ are initialized as i.i.d. Gaussians ${ \mathcal { N } } ( 0 , \sigma ^ { 2 } )$ . Since $\{ v _ { k } \} _ { k }$ are orthonormal vectors, for each $k$ , let $\rho _ { k } = \| \boldsymbol { W ^ { 0 \top } v _ { k } } \| ^ { 2 } / \sigma ^ { 2 }$ , then $\rho _ { k }$ is distributed according to $\hat { \chi } ^ { 2 }$ distribution with $C$ degree of freedom. For any $1 < \zeta \leq M$ , where $M$ is a constant chosen a priori, and for each $k = 1 , \ldots , \mathfrak { n }$ ,
+
+(Markov’s inequality)
+
+$$
+\begin{array} { r l } & { \mathsf { P } ( \rho _ { k } \ge \zeta C ) \le e ^ { - \zeta t C } \mathbb { E } [ e ^ { t \rho _ { k } } ] , } \\ & { \quad \quad \quad = e ^ { - \zeta t C } ( 1 - 2 t ) ^ { - C / 2 } , \quad \forall t < ^ { 1 / 2 } , } \\ & { \quad \quad \quad = \exp ( - C ( \zeta t + ^ { 1 } / 2 \log ( 1 - 2 t ) ) ) , } \\ & { \quad \quad \quad \le \exp ( - C / 2 ( \zeta - 1 + \log ( 1 / \zeta ) ) ) , } \\ & { \quad \quad \quad \le \exp ( - C / 2 ( \zeta - 1 + \log ( 1 / M ) ) ) . } \end{array}
+$$
+
+By union bound,
+
+$$
+\begin{array} { r } { \mathsf { P } \left( \exists k \in \{ 1 , \dots , \mathfrak { n } \} : \rho _ { k } \ge \zeta C \right) \le \mathsf { n e x p } \left( - C / 2 ( \zeta - 1 + \log ( 1 / M ) ) \right) . } \end{array}
+$$
+
+Let $\delta$ equals the right hand side, and solve for $\zeta$ , we get for any $\delta \ge \delta _ { 0 } > 0$ , with probability at least $1 - \delta$
+
+$$
+\forall k : \rho _ { k } < 2 \log \left( \frac { \mathfrak { n } } { \delta } \right) + C ( 1 + \log M )
+$$
+
+where $\delta _ { 0 } \geq 2 \mathfrak { n } / ( C ( M - \log M - 1 ) )$ is chosen to satisfy $\zeta \leq M$ . We can also choose $\delta _ { 0 }$ first and set $M$ accordingly.
+
+Putting everyting together, with probability at least $1 - \delta$ , the mean squared error (averaged over all output pixels)
+
+$$
+\begin{array} { l } { \displaystyle \mathrm { E r r o r } ^ { 2 } = \frac { 1 } { | \Xi | C } \sum _ { i j \in \Xi } \mathrm { E r r o r } _ { i j } ^ { 2 } , } \\ { \displaystyle \mathrm { \quad } \le \left( \left( \sqrt { \mathfrak { n } \sigma ^ { 2 } \left( 2 \log \left( \frac { \mathfrak { n } } { \delta } \right) + C ( 1 + \log M ) \right) } + \sqrt { \mathfrak { n } C } \right) B \sqrt { K _ { H } K _ { W } C } \right) ^ { 2 } \Biggl / c , } \\ { \displaystyle \mathrm { \quad } = { \cal O } \left( \frac { K _ { H } K _ { W } C ^ { 2 } \mathfrak { n } ( 1 + 1 / C \log ( \mathfrak { n } / \delta ) ) } { C } \right) . } \end{array}
+$$
+
+Note $K _ { H } K _ { W } C ^ { 2 }$ is the number of parameters in the convolution net.
+
+CASE 2 — underparameterization: $| \Xi | > K _ { H } \times K _ { W } \times C \times C$ . Using the same notation above, assuming no redundant features, we apply Lemma 2 to get the prediction of the $( i , j )$ -th (multichannel) pixel of a test example $x$ as
+
+$$
+\begin{array} { r l } & { \hat { W } ^ { \top } \mathsf { P } _ { i j } ( x ) = Y ^ { \top } X ( X ^ { \top } X ) ^ { - 1 } \mathsf { P } _ { i j } ( x ) } \\ & { \quad \quad = \Lambda ^ { \top } X ^ { \top } X ( X ^ { \top } X ) ^ { - 1 } \mathsf { P } _ { i j } ( x ) } \\ & { \quad \quad = \Lambda ^ { \top } \mathsf { P } _ { i j } ( x ) } \end{array}
+$$
+
+Recall the definition of $\Lambda$ , which maps a patch to the corresponding (multi-channel) pixel. In this case, the prediction is exact, and the error is zero. Since in this case $\mathfrak { n }$ , this case can be merged with equation 12. □
+
+Proof of Lemma $^ { l }$ . The proof is an extension of Theorem 1 to the case of more than one training examples. Using the notation in the Lemma, the training objective can be written as the matrix form
+
+$$
+\hat { R } = \frac { 1 } { 2 } \left\| X W - Y \right\| _ { 2 } ^ { 2 } ,
+$$
+
+and the gradient as
+
+$$
+\frac { \partial \hat { R } } { \partial W } = X ^ { \top } ( X W - Y ) .
+$$
+
+Since the model is overparameterized, and there is no unique minimizer to the empirical risk. However, gradient descent converges to a unique solution. Note the gradient descent step is:
+
+$$
+\begin{array} { r c l } { { W ^ { t } } } & { { = } } & { { W ^ { t - 1 } - \eta _ { t } \left. \frac { \partial \hat { R } } { \partial W } \right| _ { W = W _ { t - 1 } } } } \\ { { } } & { { = } } & { { W ^ { t - 1 } - \eta _ { t } X ^ { \top } ( X W ^ { t - 1 } - Y ) } } \\ { { } } & { { = } } & { { W ^ { 0 } - { \displaystyle \sum _ { \tau = 1 } ^ { t - 1 } } \eta _ { \tau } X ^ { \top } ( X W ^ { \tau } - Y ) } } \\ { { } } & { { : = } } & { { W ^ { 0 } + X ^ { \top } U ^ { t } , } } \end{array}
+$$
+
+where $\begin{array} { r } { U ^ { t } = - \sum _ { \tau = 1 } ^ { t - 1 } \eta _ { \tau } ( X W ^ { \tau } - Y ) \in \mathbb { R } ^ { N \times d } } \end{array}$ parameterizes the solution at iteration $t$ . Since $X X ^ { \top }$ is invertible in this case. A unique solution exists under this parameterization, which can be obtained by solving
+
+$$
+\begin{array} { r l } & { X ( W ^ { 0 } + X ^ { \top } \hat { { \boldsymbol { U } } } ) = Y } \\ { \Rightarrow } & { \hat { { \boldsymbol { U } } } = \left( X X ^ { \top } \right) ^ { - 1 } ( Y - X W ^ { 0 } ) . } \end{array}
+$$
+
+Plug this into the parameterization, we get
+
+$$
+\hat { W } = W ^ { 0 } + X ^ { \top } \left( X X ^ { \top } \right) ^ { - 1 } ( Y - X W ^ { 0 } ) .
+$$
+
+Proof of Lemma 2. Using the same notation as in the proof of Lemma 1, since the model is underparameterized, a unique minimizer of the empirical risk exists. Directly solving for the optimality condition $\partial \hat { R } / \partial W = 0$ , we get
+
+$$
+\hat { W } = \left( X ^ { \top } X \right) ^ { - 1 } X ^ { \top } Y .
+$$
+
+# E FULL RESULTS OF FULLY CONNECTED MULTI-LAYER NETWORKS
+
+In this section, we present the detailed results on fully connected networks that are omitted from Section 3.1 due to space limit.
+
+# E.1 FULLY CONNECTED LINEAR NETWORKS
+
+Figure 11 shows the results on multi-layer linear networks with various number of hidden layers and hidden units. The depth of the architecture has a stronger effect on the inductive bias than the width. For example, the network with one hidden layer of dimension 2048 has 3.2M parameters, more than the $2 . 5 \mathrm { M }$ parameters of the network with three hidden layers of dimension 784. But the latter behaves less like the convex case.
+
+# E.2 TWO-LAYER FULLY CONNECTED RELU NETWORKS
+
+Li & Liang (2018) offer a theoretical characterization of learning in a two-layer ReLU neural network. They show that when the data consists of well separated clusters (i.e., the cluster diameters are much smaller than the distances between each cluster pair), training an overparameterized twolayer ReLU network will generalize well. To simplify the analysis, they study a special case where the weights in the top layer are randomly initialized and fixed; only the bottom layer weights are learned.
+
+![](images/659faa8c2c6afc1e3ba6de5a30c28d5f5029e82f722b89d923742287a67ae93c.jpg)  
+Figure 11: Visualization of predictions from trained multi-layer linear networks. The first row shows the input images for evaluation, including the single training image “7” at the beginning of the row. The remaining rows shows the prediction from a trained linear network with 1, 3, and 5 hidden layers, respectively.
+
+![](images/ead5de59d1859c12d1120fbb25abd1de2ef97cc5669d8878651bb79441694872.jpg)  
+Figure 12: Visualization of predictions from two-layer ReLU networks. The first row shows the input images for evaluation, including the single training image “7” at the beginning of the row. The remaining rows shows the predictions from trained models with hidden dimension 2,048 and 16,384, repectively.
+
+We study the problem of learning a two-layer ReLU network under our identity-mapping task. Figure 12 compares the cases of learning the bottom layer only and learning both layers (left and right panels, respectively). The two cases demonstrate different inductive biases for predictions on unseen test images. When only the first layer is trained, the tendency is toward speckled noise, but when both layers are trained, the tendency is toward a constant output (the image used for training). Our observation does not contradict the theoretical results of Li & Liang (2018), which assume a well separated and clustered data distribution.
+
+For bottom-layer only training, we can analyze it in a similar way to the one-layer networks. Let us denote
+
+$$
+f _ { W } ( x ) = \langle \alpha , \operatorname { R e L U } ( z ) \rangle , \quad z = W x
+$$
+
+where $W \in \mathbb { R } ^ { m \times d }$ is the learnable weight matrix, and $\alpha \in \mathbb { R } ^ { m }$ is randomly initialized and fixed. Although the trained weights no longer have a closed-form solution, the solution found by gradient descent is always parameterized as
+
+$$
+W ^ { T } = W ^ { 0 } + u ^ { T } \hat { x } ^ { \top }
+$$
+
+where $\hat { x }$ is the training example, and $\boldsymbol { u } ^ { T } \in \mathbb { R } ^ { m }$ summarizes the efforts of gradient descent up to time $T$ . In particular, the gradient of the empirical risk $\hat { R }$ with respect to each row $W _ { : r }$ of the learnable weight is
+
+$$
+\frac { \partial \hat { R } } { \partial W _ { : r } } = \frac { \partial \hat { R } } { \partial z _ { r } } \frac { \partial z _ { r } } { \partial W _ { : r } } = \frac { \partial \hat { R } } { \partial z _ { r } } \hat { x } ^ { \top }
+$$
+
+Putting it together, the full gradient is
+
+$$
+\frac { \partial \hat { R } } { \partial W } = \frac { \partial \hat { R } } { \partial z } \cdot \hat { x } ^ { \top }
+$$
+
+Since the gradient lives in the span of the training example $\hat { x }$ , the solution found by gradient descent is always parameterized as equation 16.
+
+![](images/1bdf3f433e740db389585f471a9e3f08290c550ce8f33d077a3736551bedef62.jpg)  
+Figure 13: Visualization of predictions from multi-layer ReLU networks. The first row shows the input images for evaluation, including the single training image “7” at the beginning of the row. The remaining rows show the predictions from trained multi-layer ReLU FCNs with 1, 3, and 9 hidden layers.
+
+![](images/197a470915fe3621b346407f875d6eca26fd066e19779ace2b40996581d56387.jpg)  
+Figure 14: Quantitative evaluation of the learned model on randomly generated test samples at various angles (correlation) to the training image. The horizontal axis shows the train-test correlation, while the vertical axis indicate the number of hidden layers for the FCNs being evaluated. The heatmap shows the similarity (measured in correlation) between the model prediction and the reference function (the constant function or the identity function). (a) shows the results for FCNs with the ReLU activation function; (b) shows the results for linear FCNs.
+
+The same arguments applies to multi-layer neural networks. The prediction on any test example that is orthogonal to $\hat { x }$ will depend only on randomly initialized $W ^ { 0 }$ and upper layer weights. When only the bottom layer is trained, the upper layer weights will also be independent from the data, therefore the prediction is completely random. However, when all the layers are jointly trained, the arguments no longer apply. The empirical results presented in the main text that multi-layer networks bias towards the constant function verify this.
+
+In particular, if the test example is orthogonal to $\hat { x }$ (i.e., $\hat { x } ^ { \top } x = 0 )$ ), the prediction depends solely on the randomly initialized values in $W ^ { 0 }$ and therefore can be characterized by the distribution used for parameter initialization.
+
+However, when both layers are trained, the upper layer weights are also tuned to make the prediction fit the training output. In particular, the learned weights in the upper layer depend on $W ^ { 0 }$ . Therefore, the randomness arguments shown above no longer apply even for test examples orthogonal to $\hat { x }$ . As the empirical results show, the behavior is indeed different.
+
+# E.3 NONLINEAR MULTI-LAYER FULLY CONNECTED NETWORKS
+
+In this section, we consider the general case of multilayer fully connected networks (FCNs) with ReLU activation functions. Figure 13 visualizes predictions from trained ReLU FCNs with 1, 3, or 9 hidden layers. The deepest network encodes the constant map with high confidence; the shallowest network shows behavior similar to that of a one-layer linear net. Quantitative evaluations of the behaviors are shown in Figure 14, computed in the same way as Figure 4 for CNNs (see Section 3.2). The results for linear FCNs are also shown for comparison. The linear and ReLU FCNs behave similarly when measuring the correlation to the identity function: neither of them performs well for test images that are nearly orthogonal to $\hat { x }$ . For the correlation to the constant function, ReLU FCNs overfit sooner than linear FCNs when the depth increases. This is consistent with our previous visual inspections: for shallow models, the networks learn neither the constant nor the identity function, as the predictions on nearly orthogonal examples are random.
+
+![](images/7c6a7d9e023ce3888fda648236392ffe1d20f6c481781a7b631b72617e45b33a.jpg)  
+Figure 15: Visualization the intermediate layers of CNNs with different number of layers. The first column shows a randomly initialized 20-layer CNN (random shallower CNNs look similar to the truncation of this). The rest of the columns show the trained CNNs with various number of layers.
+
+# F VISUALIZATION OF THE INTERMEDIATE LAYER REPRESENTATIONS FOR CNNS
+
+Unlike the FCNs, CNNs preserve the spatial relation between neurons in the hidden layers, so we can easily visualize the intermediate layers as images in comparison to the inputs and outputs, to gain more insights on how the networks are computing the functions layer-by-layer. In Figure 15, we visualize the intermediate layer representations on some test patterns for CNNs with different depths. In particular, for each example, the outputs from a convolutional layer in an intermediate layer is a three dimensional tensor of shape (#channel, height, width). To get a compact visualization for multiple channels in each layer, we compute flatten the 3D tensor to a matrix of shape (#channel, height $\times$ width), compute SVD and visualize the top singular vector as a one-channel image.
+
+In the first column, we visualize a 20-layer CNN at random initialization3. As expected, the randomly initialized convolutional layers gradually smooth out the input images. The shape of the input images are (visually) wiped out after around 8 layers of (random) convolution. On the right of the figure, we show several trained CNNs with increasing depths. For a 7-layer CNN, the holistic structure of inputs are still visible all the way to the top at random initialization. After training, the network approximately renders an identity function at the output, and the intermediate activations also become less blurry. Next we show a 14-layer CNN, which fails to learn the identity function.
+
+![](images/f341b65ecabdd62852773be98169a978b203562cc1e0e20014e809b836a9a549.jpg)  
+Figure 16: Visualizing the intermediate layers of a trained 7-layer CNN. The three subfigures show for each layer: 1) the top singular vector across the channels; 2) the channel that maximally correlate with the input image; 2) a random channel, respectively.
+
+![](images/542e736de500dbb8448d4d82823711bca838400a0510103c745e752f56c707ea.jpg)  
+Figure 17: Visualizing the intermediate layers of a trained 14-layer CNN. The three subfigures show for each layer: 1) the top singular vector across the channels; 2) the channel that maximally correlate with the input image; 2) a random channel, respectively.
+
+However, it manages to recover meaningful information in the higher layer activations that were (visually) lost in the random initialization. On the other hand, in the last column, the network is so deep that it fails to make connection from the input to the output. Instead, the network start from scratch and constructs the digit $\bullet \mathbf { 7 } ^ { \bullet }$ from empty and predict everything as ‘7’. However, note that around layer-8, we see the activations depict slightly more clear structures than the randomly initialized network. This suggests that some efforts have been made during the learning, as opposed to the case that the bottom layers not being learned due to complete gradient vanishing. Please refer to Appendix G for further details related to potential gradient vanishing problems.
+
+Two alternative visualizations to the intermediate multi-channel representations are provided that show the channel that is maximally correlated with the input image, and a random channel (channel 0). Figure 16, Figure 17 and Figure 18 illustrate a 7-layer CNN, a 14-layer CNN and a 20-layer CNN, respectively.
+
+# G MEASURING THE CHANGE IN WEIGHTS OF LAYERS POST TRAINING
+
+In this section, we study the connection between the inductive bias of learning the constant function and the potential gradient vanishing problem. Instead of measuring the norm of gradient during training, we use a simple proxy that directly compute the distance of the weight tensor before and after training. In particular, for each weight tensor $W ^ { 0 }$ at initialization and $W ^ { \star }$ after training, we compute the relative $\ell _ { 2 }$ distance as
+
+$$
+d ( W ^ { 0 } , W ^ { \star } ) : = \frac { \| W ^ { 0 } - W ^ { \star } \| } { \| W ^ { 0 } \| }
+$$
+
+The results for CNNs with various depths are plotted in Figure 19(a). As a general pattern, we do see that as the network architecture gets deeper, the distances at lower layers do become smaller. But they are still non-zero, which is consistent with the visualization in Figure 15 showing that even for the 20-layer CNN, where the output layer fits to the constant function, the lower layers does get enough updates to allow them to be visually distinguished from the random initialization.
+
+![](images/6d105fb966f6f410c36bc8beb86e0ef43bf9e88cd411d85a8cac3a73b9010f52.jpg)  
+Figure 18: Visualizing the intermediate layers of a trained 20-layer CNN. The three subfigures show for each layer: 1) the top singular vector across the channels; 2) the channel that maximally correlate with the input image; 2) a random channel, respectively.
+
+![](images/90b1c242498cd2b341b66f0dc6a72ebdb2dcebc0b733ae8443e6b7db4e060628.jpg)  
+Figure 19: The relative $\ell _ { 2 }$ distance of the weight tensors before and after training at each layer. The curves compare models at different depth. Most of the networks have significantly larger distances on the top-most layer. To see a better resolution at the bottom layers, we cut off the top layer in the figures by manually restricting the y axis.
+
+![](images/d607d6dd5f6411f7fdbdf5ef3fc4522b583dd73ad26212a43c233299d3042a05.jpg)  
+Figure 20: Visualization of a 5-layer CNN on test images of different sizes. Every two rows show the inputs and model predictions. The numbers on the left indicate the input image size (both width and height).
+
+In Figure 19(b) and (c), we show the same plots for linear FCNs and FCNs with ReLU activation, respectively. We see that especially for ReLU FCN with 11 hidden layers, the distances for the weight tensors at the lower 5 layers are near zero. However, recall from Figure 13 in Section E.3, the ReLU FCNs start to bias towards the constant function with only three hidden layers, which are by no means suffering from vanishing gradients as the plots here demonstrate.
+
+# H FULL RESULTS ON ROBUSTNESS OF INDUCTIVE BIASES
+
+# H.1 TESTING ON DIFFERENT INPUT IMAGE SIZES
+
+Figure 6 in Section 3.3 visualize the predictions of a 5-layer CNN on inputs of the extreme sizes of $7 \times 7$ and $1 1 2 \times 1 1 2$ . Here in Figure 20 we provide more results on some intermediate image sizes.
+
+We also test a trained 20-layer CNN on various input sizes (Figure 7). It is interesting that the constant map also holds robustly across a wide range of input sizes. Especially from the prediction on large input images, we found hints on how the CNNs actually compute the constant function. It seems the artifacts from the outside boundary due to convolution padding are used as cues to “grow” a pattern inwards to generate the digit “7”. The visualizations of the intermediate layer representatiosn in Figure 21 is consistent with our hypothesis: the CNN take the last several layers to realize this construction. This is very clever, because in CNNs, the same filter and bias is applied to all the spatial locations. Without relying on the artifacts on the boundaries, it would be very challenging to get a sense of the spatial location in order to construct an image with a holistic structure (e.g. the digit “7” in our case).
+
+# H.2 THE UPPER SUBNETWORKS
+
+In the visualization of intermediate layers (Figure 15), the intermediate layers actually represent the “lower” subnetwork from the inputs. Here we investigate the “upper” subnetwork. Thanks again to the spatial structure of CNNs, we can skip the lower layers and feed the test patterns directly to the intermediate layers and still get interpretable visualizations4. Figure 22 shows the results for the top-one layer from CNNs with various depths. A clear distinction can be found at 15-layer CNN, which according to Figure 3 is where the networks start to bias away from edge detector and towards the constant function.
+
+![](images/f339233266ce3782df4fe6463bea1e19b98e4de54c7dc904a9d7e7ce73b6ea49.jpg)  
+Figure 21: Visualization of intermediate representations when testing on images of different sizes from training images for a 20-layer trained CNN. The CNN is trained on a $2 8 \times 2 8$ image of the digit $\mathbf { \bar { \Psi } } ^ { 6 6 } 7 ^ { 9 }$ .
+
+![](images/bc0fdb734d0eed43a166af2b9d5f09d52d565ce65ff76880005febf162e7f633.jpg)  
+Figure 22: Visualizing only the final layer in trained networks. The first row are the input images, which are directly fed into the final layer of trained networks (skipping the bottom layers). The remaining rows shows the predictions from the top layers of CNNs, with the numbers on the left indicating their (original) depth.
+
+![](images/957dcef0bad66e4996f4e45e2adcedd7511cd7f8a675cf7dc31eb96517fa9ea1.jpg)  
+Figure 23: Visualizing only the top two layers in trained networks. The first row are the input images, which are directly fed into the top two layer of trained networks (skipping the bottom layers). The remaining rows shows the predictions from the top two layers of CNNs, with the numbers on the left indicating their (original) depth. More specifically, each of the two top layers occupies one row. The colorful rows are the visualizations (as the top singular vector across channels) of the outputs of the second to the last layer from each network. The grayscale rows are the outputs of the final layer from each network.
+
+The predictions from the final two layers of each network are visualized in Figure 23. Figure 24 focuses on the 20-layer CNN that learns the constant map, and visualize the upper 3 layers, 6 layers and 10 layers, respectively. In particular, the last visualization shows that the 20-layer CNN is already starting to construct the digit “7” from nowhere when using only the upper half of the model.
+
+![](images/f3f0cb6d2461426b5a508e1e03c3748a12725847df8a820502ace4081f5f5af6.jpg)  
+Figure 24: Visualzing the top 3 layers, 6 layers and 10 layers of a 20-layer CNN. Visualizations formatted in the same way as Figure 23.
+
+![](images/02f09bafeb0d38cf5cd4390662c220e4c56d83e9143b19f57acdd2d741a55138.jpg)  
+Figure 25: Inductive bias of a 5-layer CNN with varying convolutional filter size. The heatmap is arranged similarly as Figure 4, except that the rows correspond to CNNs filter sizes.
+
+# I FULL RESULTS ON VARYING DIFFERENT FACTORS WHEN TRAINING CNNS
+
+The study on the effects of various hyperparameters on the inductive bias of trained CNNs is briefly presented in Section 3.4. The full results are shown here.
+
+Convolution filter sizes Figure 25 and Figure 26 illustrate the inductive bias with varying convolution filter size from $5 \times 5$ to $5 7 \times 5 7$ . The visualization shows that the predictions become more and more blurry as the filter sizes grow. The heatmaps, especially the correlation to the identity function, are not as helpful in this case as the correlation metric is not very good at distinguishing images with different levels of blurry. With extremely large filter sizes that cover the whole inputs, the CNNs start to bias towards the constant function. Note our training inputs are of size $2 8 \times 2 8$ , so $2 9 \times 2 9$ filter size allows all the neurons to see no less than half of the spatial domain from the previous layer. $5 7 \times 5 7$ filters centered at any location within the image will be able to see the whole previous layer. On the other hand, the repeated application of the same convolution filter through out the spatial domain is still used (with very large boundary paddings in the inputs). So the CNNs are not trivially doing the same computation as FCNs.
+
+Convolution channel depths Figure 10 in Section 3.4 shows that the learned identity function is severely corrupted when only 3 channels are used in the 5-layer CNNs. Figure 27 presents the correlation to the constant and the identity function when different numbers of convolution channels are used. The heatmap is consistent with the visualizations, showing that the 5-layer CNN fails to approximate the identity function when only three channels are used in each convolution layer.
+
+Furthermore, Figure 28 visualize the predictions of trained 3-channel CNNs with various depths. The 3-channel CNNs beyond 8 layers fail to converge during training. The 5-layer and the 7-layer CNNs implement functions biased towards edge-detecting or countour-finding. But the 6-layer and the 8-layer CNNs demonstrate very different biases. The potential reason is that with only a few channels, the random initialization does not have enough randomness to smooth out “unlucky” bad cases. Therefore, the networks have higher chance to converge to various corner cases. Figure 29 and Figure 30 compare the random initialization with the converged network for a 3-channel CNN and a 128-channel CNN. From the visualizations of the intermediate layers, the 128-channel CNN already behave more smoothly than the 3-channel CNN at initialization.
+
+![](images/01e4ce8ad15c7bb2d1224c09f1980e5a688fe133b21c09aa265d5e8581d89118.jpg)  
+Figure 26: Visualizing the predictions from CNNs with various filter sizes. The first row is the inputs, including the single training image “7”. The remaining rows are predictions, with the numbers on the left showing the corresponding filter sizes.
+
+![](images/71295b1b1aea971e3502703ff8997c69c14da45601610fc808cdabb781538755.jpg)  
+Figure 27: Correlation to the constant and the identity function for different convolution channels in a 5-layer CNN.
+
+![](images/697513d93bc13e659876de226ccafc62e337a286cb9427ceca5915d283cdfba0.jpg)  
+Figure 28: Visualization predictions from CNNs with 3 convolution channels and with various number of layers (numbers on the left). The first row is the inputs, and the remaining rows illustrate the network predictions.
+
+![](images/08e1555e18df6c2753c1ccf82e4ab43648fa3ba1ac45ba9b2015d7911a671895.jpg)  
+Figure 29: Visualizing the randomly initialized models to compare two 5-layer CNNs with 3 convolution channels per layer and 128 convolution channels per layer, respectively. The subfigures visualize the predictions of intermediate layers of the two network at random initialization. The multi-channel intermediate layers are visualized as the top singular vectors.
+
+![](images/851aeada21b6b8fc515da3d230adb1cb6f9bf7a4e25cdb802b6b3c29603a7976.jpg)  
+Figure 30: Comparing two 5-layer CNNs with 3 convolution channels per layer and 128 convolution channels per layer, respectively. Layout is similar to Figure 29.
+
+Initialization schemes In transfer learning, it is well known that initializing with pre-trained network weights could affect the inductive bias of trained models. On the other hand, different random initialization schemes are mainly proposed to help with optimization by maintaining information flow or norms of representations at different depths. It turns out that they also strongly affect the inductive bias of the trained models. Figure 31 visualizes the different inductive biases. Let $f _ { i }$ , $f _ { o }$ be the fan in and fan out of the layer being initialized. We tested the following commonly used initialization schemes: 1) default: $\bar { \mathcal { N } } ( 0 , \sigma ^ { 2 } = 1 / ( f _ { i } f _ { o } ) ) ; 2$ ) Xavier (a.k.a. Glorot) init (Glorot & Bengio, 2010): $\mathcal { N } ( 0 , \sigma ^ { 2 } = 2 / ( f _ { i } + f _ { o } ) )$ ; 3) Kaiming init (He et al., 2015): $\mathcal { N } ( 0 , \sigma ^ { 2 } = 2 / f _ { i } )$ ; 4) Orthogonal init (Saxe et al., 2014). Variations with uniform distributions instead of Gaussian distributions are also evaluated for Xavier and Kaiming inits. All initialization schemes bias toward the identity function for shallow networks. But Kaiming init produces heavy artifacts on test predictions. For the bias towards the constant function in deep networks, Xavier init behaves similarly to the default init scheme, though more layers are needed to learn a visually good identity function. On the other hand, the corresponding results from the Kaiming init is less interpretable.
+
+Optimizers First order stochastic optimizers are dorminately used in deep learning due to the huge model sizes and dataset sizes. To improve convergence speed, various adaptive methods are introduced (Duchi et al., 2011; Kingma & Ba, 2014; Graves, 2013). It is known that those methods lead to worse generalization performances in some applications (e.g. Wu et al. (2016)). But they are extremely popular in practice due to the superior convergence speed and easier hyper-parameter tuning than the vanilla SGD. We compare several popular optimizers in our framework, and confirm that different optimizers find different global minimizers, and those minimizers show drastically different inductive biases on test examples, as shown in Figure 32. Some optimizer requires a smaller base learning rate to avoid parameter exploding during training. In particular, we use base learning rate 0.001 for Adagrad and Adamax, 0.0001 for Adam and RMSprop. For comparison, we also include results from SGD with those corresponding base learning rates.
+
+![](images/19b15cc067410e800a8f85a516d3a1788d61beedb8a90618ae0e88f5e6d14caa.jpg)  
+Figure 31: Visualization of the predictions from CNNs trained with the D) default, Xn) Xavier normal, Xu) Xavier uniform, Kn) Kaiming normal, Ku) Kaiming uniform, and Or) orthogonal initialization schemes. The first row shows the inputs, and the remaining rows shows the predictions from each trained networks.
+
+![](images/a8539e8c8d81c3d8f93d9294a9f0f58d35e9e4704e4b311471ba71cba9d8d9c3.jpg)  
+Figure 32: Visualization of the predictions from CNNs trained with different optimizers. The first row shows the inputs, and the remaining rows shows the predictions from SGD (lr 0.01), SGD (lr 0.001), SGD (lr 0.0001), Adagrad, RMSprop, Aam, and Adamax respectively.
+
+# J CORRELATION VS MSE
+
+Figure 33, Figure 34 and Figure 35 can be compared to their corresponding figures in the main text. The figures here are plotted with the MSE metric between the prediction and the groundtruth, while the figures in the main text uses the correlation metric. Each corresponding pair of plots are overall consistent. But the correlation plots show the patterns more clearly and has a fixed value range of [0, 1] that is easier to interpret.
+
+![](images/5956fc22d34cecf233e95399d4e4ec8618fcc9a4d15a84c50fb32dd12cc4f2f9.jpg)  
+Figure 33: Quantitative evaluation of linear FCNs. The same as Figure 14(a), except MSE is plotted here instead of correlation.
+
+![](images/b5861cee52a4ed7c9b0cfcb4960195eb138a665ab69e15f7611e97fda4f432d0.jpg)  
+Figure 34: Quantitative evaluation of ReLU FCNs. The same as Figure 14(b), except MSE is plotted here instead of correlation.
+
+![](images/17422661cd39e889e007db9a581037a521fbf1c9880d9f888a2a105d9bf9dc96.jpg)  
+Figure 35: Quantitative evaluation of CNNs. The same as Figure 4, except MSE is plotted here instead of correlation.
+
+![](images/a03bae7a519e7c4fdd01773c3b07ad88b9f9e1552e1a85f6fa0b134262360d2f.jpg)  
+mse to identity function
+
+![](images/7ff820390e9e976449745f8002dc48fba0ec401b3c715c1b52239fca59b8b71f.jpg)  
+Figure 36: Visualization of predictions from CNNs trained on a single example. The same training example is used as in the main text, but two extra runs of training and evaluation are listed to show the robustness of our main observations to the randomness in the experiments.
+
+![](images/21998f75a53baf74b4115083b1863db78e6d847c444e7ffb6ac5ef17eca80a67.jpg)  
+Figure 37: Visualization of predictions from CNNs trained on a single example. Two different (randomly chosen) training images (a) digit 3, (b) digit 0, are shown to compare robustness of our main observations to different training images.
+
+# K ROBUSTNESS OF OBSERVATIONS TO RANDOM SEEDS
+
+We evaluate the robustness of our main observations to randomness from the experiments in this section. In Figure 36, two different runs of training and evaluation are compared side by side, and the results are very consistent. In Figure 37 we further perturb the random seeds for data loading, so that different (single) training images are loaded for training. We can see that the main observations hold for both cases: CNNs learn the identity map, edge detector and the constant map as the depth increases.
+
+![](images/3a5895677a106a77e48a6f24bc0cbce510e9c82b81a4362361a18c3595f504d3.jpg)  
+Figure 38: Visualization of the predictions from networks trained with 2 examples. The first two columns are training examples, and the remaining columns are unseen test cases. H0—H9 shows fully connected networks with the corresponding number of hidden layers. C3—C24 shows CNNs with the corresponding number of layers.
+
+# L LEARNING WITH MULTIPLE EXAMPLES
+
+We focus on learning from a single example in this paper because the two extreme inductive biases of the identity map and the constant map can be precisely defined and tested against. When training with multiple examples, the constant map is no longer a viable solution to the optimization problem. Nevertheless, we can still qualitatively evaluate the inductive bias via visualization of predictions. Figure 38 shows the results on various networks trained with two examples. In particular, for networks that are known to overfit to the constant map when trained with one example (e.g. fully connected networks with 9 hidden layers, and 20-layer convolutional networks), similar overfitting behaviors are observed. In this case, a proper definition of a constant map no longer holds, as the network perfectly reconstruct each individual digit from the training set. On the test set, a notion of memorization can be recognized as always predicting a mixture of the training samples. Note sometimes the prediction is biased more towards one of the training samples depending on the input patterns.
+
+Figure 39 shows the results with three examples, with similar observations. Figure 40 shows the situation when training with the full MNIST training set (60k examples). In this case, even the deepest convolutional networks we tried successfully learn the identity function. Fully connected networks learn the identity function on the manifold of digit inputs, but still cannot reconstruct meaningful results on other test patterns.
+
+![](images/389b8ef3b6b8935b312b999c5b9abcdd594e7d3e7f13005adb3504629c7cc356.jpg)  
+Figure 39: Visualization of the predictions from networks trained with 3 examples. The first three columns are training examples, and the remaining columns are unseen test cases. H0—H9 shows fully connected networks with the corresponding number of hidden layers. C3—C24 shows CNNs with the corresponding number of layers.
+
+![](images/ad6a36f12ddb6cfde31d617b1d268f374d612661bc35708390ec4146b197ba7f.jpg)  
+Figure 40: Visualization of the predictions from networks trained with 60k examples. The first three columns are (3 out of 60k) training examples, and the remaining columns are unseen test cases. H0—H9 show fully connected networks (ReLU activation) with the corresponding number of hidden layers. LinH1—LinH9 show linear fully connected networks (without activation) with the corresponding number of hidden layers. And C3—C24 show CNNs with the corresponding number of layers.
+
+Table 1: Number of parameters of different neural network architectures used in this paper. The number of parameters are calculated for $2 8 \times 2 8$ grayscale input / output sizes.   
+
+<table><tr><td>FCN num layers</td><td>1</td><td>2</td><td>4</td><td>6</td><td>8</td><td>10</td></tr><tr><td>hidden dim = 784</td><td>614,656</td><td>1,229,312</td><td>2,458.624</td><td>3,687,936</td><td>4,917,248</td><td>6,146,560</td></tr><tr><td>hidden dim = 2048</td><td></td><td>3,211,264</td><td>11,599,872</td><td>19,988,480</td><td>28,377,088</td><td>36,765,696</td></tr><tr><td>CNN num layers</td><td>1</td><td>3</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td>5 × 5 filter,3channels</td><td>25</td><td>375</td><td>825</td><td>1,050</td><td>1,275</td><td>1,500</td></tr><tr><td>5 × 5 filter,128 channels</td><td>25</td><td>416.000</td><td>1,235,200</td><td>1,644,800</td><td>2,054,400</td><td>2,464,000</td></tr><tr><td>5 × 5 filter,1024 channels</td><td></td><td></td><td>78,694,400</td><td></td><td>131,123,200</td><td></td></tr><tr><td>7 × 7 filter,3 channels</td><td>49</td><td>735</td><td>1,617</td><td>2.058</td><td>2,499</td><td>2,940</td></tr><tr><td>7 × 7 filter,128 channels 7 × 7 filter,1024 channels</td><td>49</td><td>815,360</td><td>2,420,992 154,241,024</td><td>3,223,808</td><td>4,026,624</td><td>4,829,440</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>257,001,472</td><td></td></tr><tr><td>CNN num layers</td><td>9</td><td>10</td><td>11</td><td>12</td><td>13</td><td>14</td></tr><tr><td>5 × 5 filter,3channels</td><td>1,725</td><td>1,950</td><td>2,175</td><td>2,400</td><td>2.625</td><td>2,850</td></tr><tr><td>5 × 5 filter,128 channels</td><td>2,873,600</td><td>3,283,200</td><td>3,692,800</td><td>4,102,400</td><td>4,512,000</td><td>4,921,600</td></tr><tr><td>5 × 5 filter,1024 channels</td><td></td><td></td><td></td><td>262,195,200</td><td></td><td>314,624,000</td></tr><tr><td>7 × 7 filter,3channels</td><td>3,381</td><td>3,822</td><td>4,263</td><td>4,704</td><td>5,145</td><td>5,586</td></tr><tr><td>7 × 7 filter,128 channels 7 × 7 filter,1024 channels</td><td>5,632,256</td><td>6,435,072</td><td>7,237,888</td><td>8,040,704</td><td>8,843,520</td><td>9,646,336</td></tr><tr><td></td><td></td><td></td><td></td><td>513,902,592</td><td></td><td>616,663,040</td></tr><tr><td>CNN num layers</td><td>15</td><td>16</td><td>17</td><td>19</td><td>20</td><td>24</td></tr><tr><td>5 × 5 filter,3channels</td><td>3,075</td><td>3,300</td><td>3,525</td><td>3,975</td><td>4,200</td><td>5,100</td></tr><tr><td>5 × 5 flter,128 channels</td><td>5,331,200</td><td>5,740,800</td><td>6,150,400</td><td>6,969,600</td><td>7,379,200</td><td>9,017,600</td></tr><tr><td>5 × 5 filter,1024 channels</td><td></td><td></td><td></td><td></td><td>471,910,400</td><td></td></tr><tr><td>7 × 7 filter,3channels</td><td>6,027</td><td>6.468</td><td>6,909</td><td>7,791</td><td>8.232</td><td>9,996</td></tr><tr><td>7 × 7 flter,128 channels 7 × 7 filter,1024 channels</td><td>10,449,152</td><td>11,251,968</td><td>12,054,784</td><td>13,660,416</td><td>14,463,232</td><td>17,674,496</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>924,944,384</td><td></td></tr><tr><td>CNN filter size</td><td>9×9</td><td>13×13</td><td>17× 17</td><td>25×25</td><td>29×29</td><td>57×57</td></tr><tr><td>5 layers,128 channels</td><td>4,002.048</td><td>8,349,952</td><td>14,278,912</td><td>30,880.000</td><td>41,552,128</td><td>160,526,592</td></tr></table>
+
+# M PARAMETER COUNTS FOR NEURAL NETWORKS USED IN THIS PAPER
+
+For convenience of cross comparing the results of different architectures with similar number of parameters, we list in Table 1 the architectures used in this paper and their associated number of parameters.
+
+# N RESIDUAL NETWORKS
+
+In this section, we evaluate training FCNs with residual connections. In particular, an identity skip connection is added for every two fully connected layers. In other words, the networks are built with two-layer blocks that computes $x \mapsto { \\\dot { x } } + \operatorname { R e L U } ( W _ { 2 } \mathrm { R e L U } ( W _ { 1 } x ) )$ , except that no ReLU is applied at the output layer. Adding skip connection after ReLU ensures the input gets passed directly to the output layer. Because the residual structure requires the same input-output shape for every block, we use 784 hidden dimensions.
+
+Comparing Figure 41 with the vanilla FCNs in Figure 2, the identity skip connection strongly biased the FCNs towards learning the identity map. On the other hand, we can still observe that the prediction noises become stronger with larger number of hidden layers, suggesting that learning the identity function is non-trivial even with the explicit identity skip connections built into the architectures.
+
+# O EXPERIMENTS ON CIFAR-10 IMAGES
+
+In this section, we show experiments on some colored images from the CIFAR-10 dataset $( 3 2 \times 3 2$ RGB images). Figure 42 and Figure 43 show the results from two different randomly sampled training images, respectively. The network architectures used here are nearly identical to the ones used to train on MNIST digits in the main text of the paper: the CNNs are with 128 channels and $5 \times 5$ filters. The FCNs are slightly modified to have larger hidden dimensions $3 0 7 2 = 3 2 { \times } 3 2 { \times } 3$ , to avoid explicitly enforcing bottlenecks in the hidden representations. The training loss and optimization algorithms are the same as in the MNIST case.
+
+![](images/36c72d4e3ecad1a9549d340324d9a0086972acd8f5df90861d21b3a12eca2a44.jpg)  
+Figure 41: Visualization of predictions from multi-layer ReLU networks with residual connections. The first row shows the input images for evaluation, including the single training image “7” at the beginning of the row. The remaining rows show the predictions from trained multi-layer ReLU FCNs with residual connections, with the numbers on the left indicating the number of hidden layers.
+
+The main observations are consistent the results on the MNIST digits. In particular, shallow FCNs produce random noisy predictions on unseen evaluation inputs, but deep FCNs bias towards memorization and hallucinate the training image on all test outputs. For CNNs, shallow networks are capable of learning the identity function, but deep networks learn the constant function instead.
+
+![](images/28600abd66569558541fa47ac63ff17d9550dbc4b6bd5711d9af98a07aebfc29.jpg)  
+Figure 42: Visualization of predictions from various networks trained on a single CIFAR-10 image (dog). The first row shows the input images, where the first one is the training image and the rest are images for evaluation. The evaluation images consist of unseen images from the CIFAR-10 test set and artificially generated grayscale patterns. The grayscale patterns are duplicated into three channels before feeding into the networks. The remaining rows show the predictions from trained FCNs and CNNs. For FCNs, the numbers indicate the number of hidden layers. For example, $H 3$ means a 3-hidden-layer FCN. For CNNs, the numbers indicate the number of (convolutional) layers. For example, C16 means a 16-layer CNN.
+
+![](images/318ddd80230a0d1f5070040c20dd2ddfd242613de33e2b8bd98c7fa577a70f7b.jpg)  
+Figure 43: Visualization of predictions from various networks trained on a single CIFAR-10 image (air plane). The first row shows the input images, where the first one is the training image and the rest are images for evaluation. The evaluation images consist of unseen images from the CIFAR-10 test set and artificially generated grayscale patterns. The grayscale patterns are duplicated into three channels before feeding into the networks. The remaining rows show the predictions from trained FCNs and CNNs. For FCNs, the numbers indicate the number of hidden layers. For example, $H 3$ means a 3-hidden-layer FCN. For CNNs, the numbers indicate the number of (convolutional) layers. For example, C16 means a 16-layer CNN.

parse/train/Hyg1G2AqtQ/Hyg1G2AqtQ.md

@@ -0,0 +1,503 @@
+# VARIANCE REDUCTION FOR REINFORCEMENT LEARNING IN INPUT-DRIVEN ENVIRONMENTS
+
+Hongzi Mao, Shaileshh Bojja Venkatakrishnan, Malte Schwarzkopf, Mohammad Alizadeh   
+MIT Computer Science and Artificial Intelligence Laboratory   
+{hongzi,bjjvnkt,malte,alizadeh}@csail.mit.edu
+
+# ABSTRACT
+
+We consider reinforcement learning in input-driven environments, where an exogenous, stochastic input process affects the dynamics of the system. Input processes arise in many applications, including queuing systems, robotics control with disturbances, and object tracking. Since the state dynamics and rewards depend on the input process, the state alone provides limited information for the expected future returns. Therefore, policy gradient methods with standard state-dependent baselines suffer high variance during training. We derive a bias-free, input-dependent baseline to reduce this variance, and analytically show its benefits over state-dependent baselines. We then propose a meta-learning approach to overcome the complexity of learning a baseline that depends on a long sequence of inputs. Our experimental results show that across environments from queuing systems, computer networks, and MuJoCo robotic locomotion, input-dependent baselines consistently improve training stability and result in better eventual policies.
+
+# 1 INTRODUCTION
+
+Deep reinforcement learning (RL) has emerged as a powerful approach for sequential decision-making problems, achieving impressive results in domains such as game playing (Mnih et al., 2015; Silver et al., 2017) and robotics (Levine et al., 2016; Schulman et al., 2015a; Lillicrap et al., 2015). This paper concerns RL in input-driven environments. Informally, input-driven environments have dynamics that are partially dictated by an exogenous, stochastic input process. Queuing systems (Kleinrock, 1976; Kelly, 2011) are an example; their dynamics are governed by not only the decisions made within the system (e.g., scheduling, load balancing) but also the arrival process that brings work (e.g., jobs, customers, packets) into the system. Input-driven environments also arise naturally in many other domains: network control and optimization (Winstein & Balakrishnan, 2013; Mao et al., 2017), robotics control with stochastic disturbances (Pinto et al., 2017), locomotion in environments with complex terrains and obstacles (Heess et al., 2017), vehicular traffic control (Belletti et al., 2018; Wu et al., 2017), tracking moving targets, and more (see Figure 1).
+
+We focus on model-free policy gradient RL algorithms (Williams, 1992; Mnih et al., 2016; Schulman et al., 2015a), which have been widely adopted and benchmarked for a variety of RL tasks (Duan et al., 2016; Wu & Tian, 2017). A key challenge for these methods is the high variance in the gradient estimates, as such variance increases sample complexity and can impede effective learning (Schulman et al., 2015b; Mnih et al., 2016). A standard approach to reduce variance is to subtract a “baseline” from the total reward (or “return”) to estimate the policy gradient (Weaver & Tao, 2001). The most common choice of a baseline is the value function — the expected return starting from the state.
+
+Our main insight is that a state-dependent baseline — such as the value function — is a poor choice in input-driven environments, whose state dynamics and rewards are partially dictated by the input process. In such environments, comparing the return to the value function baseline may provide limited information about the quality of actions. The return obtained after taking a good action may be poor (lower than the baseline) if the input sequence following the action drives the system to unfavorable states; similarly, a bad action might end up with a high return with an advantageous input sequence. Intuitively, a good baseline for estimating the policy gradient should take the specific instance of the input process — the sequence of input values — into account. We call such a baseline an input-dependent baseline; it is a function of both the state and the entire future input sequence.
+
+We formally define input-driven Markov decision processes, and we prove that an input-dependent baseline does not introduce bias in standard policy gradient algorithms such as Advantage Actor
+
+![](images/534c88038d0ad0972447d05769c0444ea38f50b165e99642567a7e7630a5efb6.jpg)  
+Figure 1: Input-driven environments: (a) load-balancing heterogeneous servers (Harchol-Balter & Vesilo, 2010) with stochastic job arrival as the input process; (b) adaptive bitrate video streaming (Mao et al., 2017) with stochastic network bandwidth as the input process; (c) Walker2d in wind with a stochastic force (wind) applied to the walker as the input process; (d) HalfCheetah on floating tiles with the stochastic process that controls the buoyancy of the tiles as the input process; (e) 7-DoF arm tracking moving target with the stochastic target position as the input process. Environments (c)–(e) use the MuJoCo physics simulator (Todorov et al., 2012).
+
+Critic (A2C) (Mnih et al., 2016) and Trust Region Policy Optimization (TRPO) (Schulman et al., 2015a), provided that the input process is independent of the states and actions. We derive the optimal input-independent baseline and a simpler one to work with in practice; this takes the form of a conditional value function — the expected return given the state and the future input sequence.
+
+Input-dependent baselines are harder to learn than their state-dependent counterparts; they are highdimensional functions of the sequence of input values. To learn input-dependent baselines efficiently, we propose a simple approach based on meta-learning (Finn et al., 2017; Vilalta & Drissi, 2002). The idea is to learn a “meta baseline” that can be specialized to a baseline for a specific input instantiation using a small number of training episodes with that input. This approach applies to applications in which an input sequence can be repeated during training, e.g., applications that use simulations or experiments with previously-collected input traces for training (McGough et al., 2017).
+
+We compare our input-dependent baseline to the standard value function baseline for the five tasks illustrated in Figure 1. These tasks are derived from queuing systems (load balancing heterogeneous servers (Harchol-Balter & Vesilo, 2010)), computer networks (bitrate adaptation for video streaming (Mao et al., 2017)), and variants of standard continuous control RL benchmarks in the MuJoCo physics simulator (Todorov et al., 2012). We adapted three widely-used MuJoCo benchmarks (Duan et al., 2016; Clavera et al., 2018a; Heess et al., 2017) to add a stochastic input element that makes these tasks significantly more challenging. For example, we replaced the static target in a 7-DoF robotic arm target-reaching task with a randomly-moving target that the robot aims to track over time. Our results show that input-dependent baselines consistently provide improved training stability and better eventual policies. Input-dependent baselines are applicable to a variety of policy gradient methods, including A2C, TRPO, PPO, robust adversarial RL methods such as RARL (Pinto et al., 2017), and meta-policy optimization such as MB-MPO (Clavera et al., 2018b). Video demonstrations of our experiments are available at https://sites.google.com/view/input-dependent-baseline/.
+
+# 2 PRELIMINARIES
+
+Notation. We consider a discrete-time Markov decision process (MDP), defined by $( S , \mathcal { A } , \mathcal { P } , \rho _ { 0 } , r , \gamma )$ , where $S \subseteq \mathbb { R } ^ { n }$ is a set of $n$ -dimensional states, ${ \mathcal { A } } \subseteq \mathbb { R } ^ { m }$ is a set of $m$ -dimensional actions, $\mathcal { P } : \mathcal { S } \times \mathcal { A } \times \mathcal { S }  [ 0 , 1 ]$ is the state transition probability distribution, $\rho _ { 0 } : { \cal S }  [ 0 , 1 ]$ is the distribution over initial states, $\overset { \cdot } { r : S } \times \mathcal { A }  \mathbb { R }$ is the reward function, and $\gamma \in ( 0 , 1 )$ is the discount factor. We denote a stochastic policy as $\pi : S \times A \to [ 0 , 1 ]$ , which aims to optimize the expected return $\begin{array} { r } { \eta ( \pi ) = \mathbb { E } _ { \tau } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r ( \dot { s _ { t } } , a _ { t } ) \right] } \end{array}$ , where ${ \boldsymbol \tau } = ( s _ { 0 } , a _ { 0 } , \ldots )$ is the trajectory following $s _ { 0 } \sim \rho _ { 0 }$ $a _ { t } \sim \pi ( a _ { t } | s _ { t } )$ , $s _ { t + 1 } \sim \mathcal { P } ( s _ { t + 1 } | s _ { t } , a _ { t } )$ . We use $\begin{array} { r } { V _ { \pi } ( s _ { t } ) = \mathbb { E } _ { a _ { t } , s _ { t + 1 } , a _ { t + 1 } , \dots } \left[ \sum _ { l = 0 } ^ { \infty } \gamma ^ { l } r ( s _ { t + l } , a _ { r + l } ) | s _ { t } \right] } \end{array}$ to define the value function, and Q $\tau ( s _ { t } , a _ { t } ) = \mathbb { E } _ { s _ { t + 1 } , a _ { t + 1 } , \dots } \left[ \sum _ { l = 0 } ^ { \infty } \gamma ^ { l } r ( s _ { t + l } , a _ { r + l } ) | s _ { t } , a _ { t } \right]$ to define the state-action value function. For any sequence $( x _ { 0 } , x _ { 1 } , \ldots )$ , we use $_ { \textbf { \em x } }$ to denote the entire sequence and $x _ { i : j }$ to denote $( x _ { i } , x _ { i + 1 } , . . . , x _ { j } )$ .
+
+Policy gradient methods. Policy gradient methods estimate the gradient of expected return with respect to the policy parameters (Sutton et al., 2000; Kakade, 2002; Gu et al., 2017). To train a policy $\pi _ { \theta }$ parameterized by $\theta$ , the Policy Gradient Theorem (Sutton et al., 2000) states that
+
+![](images/0c66c87ebc6be2821061428a362dae49e7bc4d9ccbeb66ccffc8dd524597f371.jpg)  
+Figure 2: Load balancing over two servers. (a) Job sizes follow a Pareto distribution and jobs arrive as a Poisson process; the RL agent observes the queue lengths and picks a server for an incoming job. (b) The input-dependent baseline (blue) results in a $5 0 \times$ lower policy gradient variance (left) and a $33 \%$ higher test reward (right) than the standard, state-dependent baseline (green). (c) The probability heatmap of picking server 1 shows that using the input-dependent baseline (left) yields a more precise policy than using the state-dependent baseline (right).
+
+$$
+\begin{array} { r } { \nabla _ { \theta } \eta ( \pi _ { \theta } ) = \mathbb { E } _ { s \sim \rho _ { \pi } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a | s ) Q _ { \pi _ { \theta } } ( s , a ) \right] , } \end{array}
+$$
+
+where $\begin{array} { r } { \rho _ { \pi } ( s ) = \sum _ { t = 0 } ^ { \infty } \left[ \gamma ^ { t } \operatorname* { P r } ( s _ { t } = s ) \right] } \end{array}$ denotes the discounted state visitation frequency. Practical algorithms often use the undiscounted state visitation frequency (i.e., $\gamma = 1$ in $\rho _ { \pi }$ ), which can make the estimation slightly biased (Thomas, 2014).
+
+Estimating the policy gradient using Monte Carlo estimation for the $Q$ function suffers from high variance (Mnih et al., 2016). To reduce variance, an appropriately chosen baseline $b ( s _ { t } )$ can be subtracted from the Q-estimate without introducing bias (Greensmith et al., 2004). The policy gradient estimation with a baseline in Equation (1) becomes $\mathbb { E } _ { \rho _ { \pi } , \pi _ { \theta } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a | s ) \left( Q _ { \pi _ { \theta } } ( s , a ) - b ( s ) \right) \right]$ While an optimal baseline exists (Greensmith et al., 2004; Wu et al., 2018), it is hard to estimate and often replaced by the value function $b ( s _ { t } ) = V _ { \pi } ( s _ { t } )$ (Sutton & Barto, 2017; Mnih et al., 2016).
+
+# 3 MOTIVATING EXAMPLE
+
+We use a simple load balancing example to illustrate the variance introduced by an exogenous input process. As shown in Figure 2a, jobs arrive over time and a load balancing agent sends them to one of two servers. The jobs arrive according to a Poisson process, and the job sizes follow a Pareto distribution. The two servers process jobs from their queues at identical rates. On each job arrival, the load balancer observes state $s _ { t } = ( q _ { 1 } , q _ { 2 } )$ , denoting the queue length at the two servers. It then takes an action $a _ { t } \in \{ 1 , 2 \}$ , sending the job to one of the servers. The goal of the load balancer is to minimize the average job completion time. The reward corresponding to this goal is $\boldsymbol { r } _ { t } = - \boldsymbol { \tau } \times \dot { \boldsymbol { j } }$ where $\tau$ is the time elapsed since the last action and $j$ is total number of enqueued jobs.
+
+In this example, the optimal policy is to send the job to the server with the shortest queue (Daley, 1987). However, we find that a standard policy gradient algorithm, A2C (Mnih et al., 2016), trained using a value function baseline struggles to learn this policy. The reason is that the stochastic sequence of job arrivals creates huge variance in the reward signal, making it difficult to distinguish between good and bad actions. Consider, for example, an action at the state shown in Figure 2a. If the arrival sequence following this action consists of a burst of large jobs (e.g., input sequence 1 in Figure 2a), the queues will build up, and the return will be poor compared to the value function baseline (average return from the state). On the other hand, a light stream of jobs (e.g., input sequence 2 in Figure 2a) will lead to short queues and a better-than-average return. Importantly, this difference in return has little to do with the action; it is a consequence of the random job arrival process.
+
+We train two A2C agents (Mnih et al., 2016), one with the standard value function baseline and the other with an input-dependent baseline tailored for each specific instantiation of the job arrival process (details of this baseline in $\ S 4 _ { , }$ ). Since the the input-dependent baseline takes each input sequence into account explicitly, it reduces the variance of the policy gradient estimation much more effectively (Figure 2b, left). As a result, even in this simple example, only the policy learned with the input-dependent baseline comes close to the optimal (Figure 2b, right). Figure 2c visualizes the policies learned using the two baselines. The optimal policy (pick-shortest-queue) corresponds to a clear divide between the chosen servers at the diagonal.
+
+In fact, the variance of the standard baseline can be arbitrarily worse than an input-dependent baseline: we refer the reader to Appendix A for an analytical example on a 1D grid world.
+
+# 4 REDUCING VARIANCE FOR INPUT-DRIVEN MDPS
+
+We now formally define input-driven MDPs and derive variance-reducing baselines for policy gradient methods in environments with input processes.
+
+Definition 1. An input-driven MDP is defined by $( S , \mathcal { A } , \mathcal { Z } , \mathcal { P } _ { s } , \mathcal { P } _ { z } , \rho _ { 0 } ^ { s } , \rho _ { 0 } ^ { z } , r , \gamma )$ , where $\mathcal { Z } \subseteq \mathbb { R } ^ { k }$ is a set of $k$ -dimensional input values, $\mathcal { P } _ { s } ( s _ { t + 1 } | s _ { t } , a _ { t } , z _ { t } )$ is the transition kernel of the states, $\mathcal { P } _ { z } \left( z _ { t + 1 } | z _ { 0 : t } \right)$ is the transition kernel of the input process, $\rho _ { 0 } ^ { z } ( z _ { 0 } )$ is the distribution of the initial input, $r ( s _ { t } , a _ { t } , z _ { t } )$ is the reward function, and $\begin{array} { r } { S , A , \rho _ { 0 } ^ { s } , } \end{array}$ , $\gamma$ follow the standard definition in $\ S 2$ .
+
+An input-driven MDP adds an input process, $\boldsymbol { z } = ( z _ { 0 } , z _ { 1 } , \cdot \cdot \cdot )$ , to a standard MDP. In this setting, the next state $s _ { t + 1 }$ depends on $\left( { { s _ { t } } , { a _ { t } } , { z _ { t } } } \right)$ . We seek to learn policies that maximize cumulative expected rewards. We focus on two cases, corresponding to the graphical models shown in Figure 3:
+
+Case 1: $z _ { t }$ is a Markov process, and $\omega _ { t } = ( s _ { t } , z _ { t } )$ is observed at time $t$ . The action $a _ { t }$ can hence depend on both $s _ { t }$ and $z _ { t }$ .
+
+Case 2: $z _ { t }$ is a general process (not necessarily Markov), and $\omega _ { t } = s _ { t }$ is observed at time $t$ . The action $a _ { t }$ hence depends only on $s _ { t }$ .
+
+In Appendix B, we prove that case 1 corresponds to a fully-observable MDP. This is evident from the graphical model in Figure 3a by considering $\omega _ { t } = ( s _ { t } , z _ { t } )$ to be the ‘state’ of the MDP at time $t$ . Case 2, on the other hand, corresponds to a partially-observed MDP (POMDP) if we define the state to contain both $s _ { t }$ and $z _ { 0 : t }$ , but leave $z _ { 0 : t }$ unobserved at time $t$ (see Appendix B for details).
+
+# 4.1 VARIANCE REDUCTION
+
+![](images/b349ac0e08173aa9c007e9e7a8205cce32ea28fd05a203fa588324c9d0977b82.jpg)  
+Figure 3: Graphical model of input-driven MDPs.
+
+In input-driven MDPs, the standard input-agnostic baseline is ineffective at reducing variance, as shown by our motivating example (§3). We propose to use an input-dependent baseline of the form $b ( \omega _ { t } , z _ { t : \infty } )$ — a function of both the observation at time $t$ and the input sequence from $t$ onwards. An input-dependent baseline uses information that is not available to the policy. Specifically, the input sequence $z _ { t : \infty }$ cannot be used when taking an action at time $t$ , because $z _ { t + 1 : \infty }$ has not yet occurred at time $t$ . However, in many applications, the input sequence is known at training time. In some cases, we know the entire input sequence upfront, e.g., when training in a simulator. In other situations, we can record the input sequence on the fly during training. Then, after a training episode, we can use the recorded values, including those that occurred after time $t$ , to compute the baseline for each step $t$ .
+
+We now analyze input-dependent baselines. Our main result is that input-dependent baselines are bias-free. We also derive the optimal input-dependent baseline for variance reduction. All the results hold for both cases in Figure 3. We first state two useful lemmas required for our analysis. The first lemma shows that under the input-driven MDP definition, the input sequence $z _ { t : \infty }$ is conditionally independent of the action $a _ { t }$ given the observation $\omega _ { t }$ , while the second lemma states the policy gradient theorem for input-driven MDPs.
+
+Lemma 1. $\operatorname* { P r } ( z _ { t : \infty } , a _ { t } | \omega _ { t } ) = \operatorname* { P r } ( z _ { t : \infty } | \omega _ { t } ) \pi _ { \theta } ( a _ { t } | \omega _ { t } )$ , i.e., $z _ { t : \infty } - \omega _ { t } - a _ { t }$ forms a Markov chain.   
+Proof. See Appendix C.
+
+Lemma 2. For an input-driven MDP, the policy gradient theorem can be rewritten as
+
+$$
+\begin{array} { r } { \nabla _ { \theta } \eta \left( \pi _ { \theta } \right) = \mathbb { E } _ { ( \omega , z ) \sim \rho _ { \pi } } \Big [ \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) Q ( \omega , a , z ) \Big ] , } \end{array}
+$$
+
+where $\begin{array} { r l r } { \rho _ { \pi } ( \omega , z ) } & { { } = } & { \sum _ { t = 0 } ^ { \infty } \left[ \gamma ^ { t } \operatorname* { P r } ( \omega _ { t } = \omega , z _ { t : \infty } = z ) \right] } \end{array}$ denotes the discounted visitation frequency of the observation $\begin{array} { r } { \mathbb { E } \left[ \sum _ { l = 0 } ^ { \infty } \gamma ^ { l } r _ { t + l } \middle | \omega _ { t } = \omega , a _ { t } = a , z _ { t : \infty } = z \right] } \end{array}$ $\omega$ . and input sequence $_ z$ , and $\begin{array} { r l } { Q ( \omega , a , z ) } & { { } = } \end{array}$
+
+Proof. See Appendix D.
+
+Equation (2) generalizes the standard Policy Gradient Theorem in Equation (1). $\rho _ { \pi } ( \omega , z )$ can be thought of as a joint distribution over observations and input sequences. $Q ( \omega , a , z )$ is a “state-actioninput” value function, i.e., the expected return when taking action $a$ after observing $\omega$ , with input sequence $_ z$ from that step onwards. The key ingredient in the proof of Lemma 2 is the conditional independence of the input process $z _ { t : \infty }$ and the action $a _ { t }$ given the observation $\omega _ { t }$ (Lemma 1).
+
+Theorem 1. An input-dependent baseline does not bias the policy gradient.
+
+Proof. Using Lemma 2, we need to show: $\begin{array} { r } { \mathbb { E } _ { ( \omega , z ) \sim \rho _ { \pi } , a \sim \pi _ { \theta } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) b ( \omega , z ) \right] = 0 } \end{array}$ . We have:
+
+$$
+\begin{array} { r l } { \mathbb { E } _ { ( \omega , z ) \sim \rho _ { \pi } } [ \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) b ( \omega , z ) ] = \displaystyle \sum _ { \omega } \sum _ { z } \sum _ { a } \rho _ { \pi } ( \omega , z ) \pi _ { \theta } ( a | \omega ) \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) b ( \omega , z ) } & { } \\ { = \displaystyle \sum _ { \omega } \sum _ { z } \rho _ { \pi } ( \omega , z ) b ( \omega , z ) \sum _ { a } \pi _ { \theta } ( a | \omega ) \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) . } & { } \end{array}
+$$
+
+Since $\begin{array} { r } { \sum _ { a } \pi _ { \theta } ( a | \omega ) \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) = \sum _ { a } \nabla _ { \theta } \pi _ { \theta } ( a | \omega ) = \nabla _ { \theta } \sum _ { a } \pi _ { \theta } ( a | \omega ) = 0 } \end{array}$ , the theorem follows.
+
+Input-dependent baselines are also bias-free for policy optimization methods such as TRPO (Schulman et al., 2015a), as we show in Appendix F. Next, we derive the optimal input-dependent baseline for variance reduction. As the gradient estimates are vectors, we use the trace of the covariance matrix as the minimization objective (Greensmith et al., 2004).
+
+Theorem 2. The input-dependent baseline that minimizes variance in policy gradient is given by
+
+$$
+\begin{array} { r } { b ^ { * } ( \omega , z ) = \frac { \mathbb { E } _ { a \sim \pi _ { \theta } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) Q ( \omega , a , z ) \right] } { \mathbb { E } _ { a \sim \pi _ { \theta } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) \right] } . } \end{array}
+$$
+
+Proof. See Appendix E.
+
+Operationally, for observation $\omega _ { t }$ at each step $t$ , the input-dependent baseline takes the form $b ( \omega _ { t } , z _ { t : \infty } )$ . In practice, we use a simpler alternative to Equation (4): $b ( \omega _ { t } , z _ { t : \infty } ) ~ =$ $\mathbb { E } _ { a _ { t } \sim \pi _ { \theta } } \left[ Q ( \omega _ { t } , a _ { t } , z _ { t : \infty } ) \right]$ . This can be thought of as a value function $V ( \omega _ { t } , z _ { t : \infty } )$ that provides the expected return given observation $\omega _ { t }$ and input sequence $z _ { t : \infty }$ from that step onwards. We discuss how to estimate input-dependent baselines efficiently in $\ S 5$ .
+
+Remark. Input-dependent baselines are generally applicable to reducing variance for policy gradient methods in input-driven environments. In this paper, we apply input-dependent baselines to A2C (§6.2), TRPO $( \ S 6 . 1 )$ and PPO (Appendix L). Our technique is complementary and orthogonal to adversarial RL (e.g., RARL (Pinto et al., 2017)) and meta-policy adaptation (e.g., MB-MPO (Clavera et al., 2018b)) for environments with external disturbances. Adversarial RL improves policy robustness by co-training an “adversary” to generate a worst-case disturbance process. Meta-policy optimization aims for fast policy adaptation to handle model discrepancy between training and testing. By contrast, input-dependent baselines improve policy optimization itself in the presence of stochastic input processes. Our work primarily focuses on learning a single policy in input-driven environments, without policy adaptation. However, input-dependent baselines can be used as a general method to improve the policy optimization step in adversarial RL and meta-policy adaptation methods. For example, in Appendix M, we empirically show that if an adversary generates high-variance noise, RARL with a standard state-based baseline cannot train good controllers, but the input-dependent baseline helps improve the policy’s performance. Similarly, input-dependent baselines can improve meta-policy optimization in environments with stochastic disturbances, as we show in Appendix N.
+
+# 5 LEARNING INPUT-DEPENDENT BASELINES EFFICIENTLY
+
+Input-dependent baselines are functions of the sequence of input values. A natural approach to train such baselines is to use models that operate on sequences (e.g., LSTMs (Gers et al., 1999)). However, learning a sequential mapping in a high-dimensional space can be expensive (Bahdanau et al., 2014). We considered an LSTM approach, but ruled it out when initial experiments showed that it fails to provide significant policy improvement over the standard baseline in our environments (Appendix G).
+
+Fortunately, we can learn the baseline much more efficiently in applications where we can repeat the same input sequence multiple times during training. Input-repeatability is feasible in many applications: it is straightforward when using simulators for training, and also feasible when training a real system with previously-collected input traces outside simulation. For example, training a robot in the presence of exogenous forces might apply a set of time-series traces of these forces repeatedly to the physical robot. We now present two approaches that exploit input-repeatability to learn input-dependent baselines efficiently.
+
+Multi-value-network approach. A straightforward way to learn $b ( \omega _ { t } , z _ { t : \infty } )$ for different input instantiations $_ z$ is to train one value network to each particular instantiation of the input process. Specifically, in the training process, we first generate $N$ input sequences $\{ z _ { 1 } , z _ { 2 } , \cdots , z _ { N } \}$ and restrict training only to those $N$ sequences. To learn a separate baseline function for each input sequence, we use $N$ value networks with independent parameters $\theta _ { V _ { 1 } } , \theta _ { V _ { 2 } } , \cdot \cdot \cdot , \theta _ { V _ { N } }$ , and single policy network with parameter $\theta$ . During training, we randomly sample an input sequence $z _ { i }$ , execute a rollout based on $z _ { i }$ with the current policy $\pi _ { \theta }$ , and use the (state, action, reward) data to train the value network parameter $\theta _ { V _ { i } }$ and the policy network parameter $\theta$ (details in Appendix I).
+
+Meta-learning approach. The multi-value-network approach does not scale if the task requires training over a large number of input instantiations to generalize. The number of inputs needed is environment-specific, and can depend on a variety of factors, such as the time horizon of the problem, the distribution of the input process, the relative magnitude of the variance due to the input process compared to other sources of randomness (e.g., actions). Ideally, we would like an approach that enables learning across many different input sequences. We present a method based on meta-learning to train with an unbounded number of input sequences. The idea is to use all (potentially infinitely many) input sequences to learn a “meta value network” model. Then, for each specific input sequence, we first customize the meta value network using a few example rollouts with that input sequence. We then compute the actual baseline values for training the policy network parameters, using the customized value network for the specific input sequence. Our implementation uses Model-Agnostic Meta-Learning (MAML) (Finn et al., 2017).
+
+Require: $\alpha , \beta$ : meta value network step size hyperparameters
+
+1: Initialize policy network parameters $\theta$ and meta-value-network parameters $\theta _ { V }$   
+2: while not done do   
+3: Generate a new input sequence $_ z$   
+4: Sample $k$ rollouts $\mathcal { T } _ { 1 } , \mathcal { T } _ { 2 } , . . . , \mathcal { T } _ { k }$ using policy $\pi _ { \theta }$ and input sequence $_ z$   
+5: Adapt $\theta _ { V }$ with the first $k / 2$ rollouts: $\theta _ { V } ^ { 1 } = \theta _ { V } - \alpha \nabla _ { \theta _ { V } } \mathcal { L } _ { T _ { 1 : k / 2 } } [ V _ { \theta _ { V } } ]$   
+6: Estimate baseline value $V _ { \theta _ { V } ^ { 1 } } ( \omega _ { t } )$ for $s _ { t } \sim \mathcal T _ { k / 2 : k }$ using adapted $\theta _ { V } ^ { 1 }$   
+7: Adapt $\theta _ { V }$ with the second $k / 2$ rollouts: $\theta _ { V } ^ { 2 } = \theta _ { V } - \alpha \nabla _ { \theta _ { V } } \mathcal { L } _ { T _ { k / 2 : k } } \left[ V _ { \theta _ { V } } \right]$   
+8: Estimate baseline value $V _ { \theta _ { V } ^ { 2 } } \left( \omega _ { t } \right)$ for $s _ { t } \sim \mathcal { T } _ { 1 : k / 2 }$ using adapted $\theta _ { V } ^ { 2 }$   
+9: Update policy with Equation (2) using the values from line (6) and (8) as baseline   
+10: Update meta value network: $\theta _ { V } \gets \bar { \theta _ { V } } - \beta \nabla _ { \theta _ { V } } \mathcal { L } _ { k / 2 : k } \left[ V _ { \theta _ { V } ^ { 1 } } \right] - \beta \nabla _ { \theta _ { V } } \mathcal { L } _ { 1 : k / 2 } \left[ V _ { \theta _ { V } ^ { 2 } } \right]$
+
+The pseudocode in Algorithm 1 depicts the training algorithm. We follow the notation of MAML, in the value function. We perform rollouts $V _ { \theta _ { V } } ( \cdot )$ on a rollout  with the sam $\tau$ $\begin{array} { r } { \mathcal { L } _ { \mathcal { T } } \left[ V _ { \theta _ { V } } \right] = \sum _ { \omega _ { t } , r _ { t } \sim \mathcal { T } } \| V _ { \theta _ { V } } ( \omega _ { t } ) - } \end{array}$   
+$\scriptstyle \sum _ { t ^ { \prime } = t } ^ { T } \gamma ^ { t ^ { \prime } - t } r _ { t } \parallel ^ { 2 }$ $k$ $_ z$ $k / 2$ $_ { z }$   
+apply the customized value network on the states of the other $k / 2$ rollouts to compute the baseline   
+for those rollouts (line 6); similarly, we swap the two groups of rollouts and repeat the same process   
+(lines 7 and 8). We use different rollouts to adapt the meta value network and compute the baseline to   
+avoid introducing extra bias to the baseline. Finally, we use the baseline values computed for each   
+rollout to update the policy network parameters (line 9), and we apply the MAML (Finn et al., 2017)   
+gradient step to update the meta value network model (line 10).
+
+# 6 EXPERIMENTS
+
+Our experiments demonstrate that input-dependent baselines provide consistent performance gains across multiple continuous-action MuJoCo simulated robotic locomotions and discrete-action environments in queuing systems and network control. We conduct experiments for both policy gradient methods and policy optimization methods (see Appendix K for details). The videos for our experiments are available at https://sites.google.com/view/input-dependent-baseline/.
+
+![](images/fdece8f11ba9937f11dad1dffe4e8b7bad8112bbb9099d19f59372082315dbbd.jpg)  
+Figure 4: In continuous-action MuJoCo environments, TRPO (Schulman et al., 2015a) with input-dependent baselines achieve $2 5 \% - 3 \times$ better testing reward than with a standard state-dependent baseline. Learning curves are on 100 testing episodes with unseen input sequences; shaded area spans one standard deviation.
+
+# 6.1 SIMULATED ROBOTIC LOCOMOTION
+
+We use the MuJoCo physics engine (Todorov et al., 2012) in OpenAI Gym (Brockman et al., 2016) to evaluate input-dependent baselines for robotic control tasks with external disturbance. We extend the standard Walker2d, HalfCheetah and 7-DoF robotic arm environments, adding a different external input to each (Figure 1).
+
+Walker2d with random wind (Figure 1c). We train a 2D walker with varying wind, which randomly drags the walker backward or forward with different force at each step. The wind vector changes randomly, i.e., the wind forms a random input process. We add a force sensor to the state to enable the agent to quickly adapt. The goal is for the walker to walk forward while keeping balance.
+
+HalfCheetah on floating tiles with random buoyancy (Figure 1d). A half-cheetah runs over a series of tiles floating on water (Clavera et al., 2018a). Each tile has different damping and friction properties, which moves the half-cheetah up and down and changes its dynamics. This random buoyancy is the external input process; the cheetah needs to learn running forward over varying tiles.
+
+7-DoF arm tracking moving target (Figure 1e). We train a simulated robot arm to track a randomly moving target (a red ball). The robotic arm has seven degrees of freedom and the target is doing a random walk, which forms the external input process. The reward is the negative squared distance between the robot hand (blue square) and the target.
+
+The Walker2d and 7-DoF arm environments correspond to the fully observable MDP case in Figure 3, i.e. the agent observes the input $z _ { t }$ at time $t$ . The HalfCheetah environment is a POMDP, as the agent does not observe the buoyancy of the tiles. In Appendix H, we show results for the POMDP version of the Walker2d environment.
+
+Results. We build 10-value networks and a meta-baseline using MAML, both on top of the OpenAI’s TRPO implementation (Dhariwal et al., 2017). Figure 4 shows the performance comparison among different baselines with 100 unseen testing input sequences at each training checkpoint. These learning curves show that TRPO with a state-dependent baseline performs worst in all environments. With the input-dependent baseline, by contrast, performance in unseen testing environments improves by up to $3 \times$ , as the agent learns a policy robust against disturbances. For example, it learns to lean into headwind and quickly place its leg forward to counter the headwind; it learns to apply different force on tiles with different buoyancy to avoid falling over; and it learns to co-adjust multiple joints to keep track of the moving object. The meta-baseline eventually outperforms 10-value networks as it effectively learns from a large number of input processes and hence generalizes better.
+
+The input-dependent baseline technique applies generally on top of policy optimization methods. In Appendix L, we show a similar comparison with PPO (Schulman et al., 2017). Also, in Appendix M we show that adversarial RL (e.g., RARL (Pinto et al., 2017)) alone is not adequate to solve the high variance problem, and the input-dependent baseline helps improve the policy performance (Figure 9).
+
+# 6.2 DISCRETE-ACTION ENVIRONMENTS
+
+Our discrete-action environments arise from widely-studied problems in computer systems research: load balancing and bitrate adaptation.1 As these problems often lack closed-form optimal solutions (Grandl et al., 2016; Yin et al., 2015), hand-tuned heuristics abound. Recent work suggests that model-free reinforcement learning can achieve better performance than such human-engineered heuristics (Mao et al., 2016; Evans & Gao, 2016; Mao et al., 2017; Mirhoseini et al., 2017). We consider a load balancing environment (similar to the example in $\ S 3$ ) and a bitrate adaptation environment in video streaming (Yin et al., 2015). The detailed setup of these environments is in Appendix J.
+
+![](images/86a3fc137a01af36693a1470bcb2cbc489df5fd97ee3015b20a9edc454e88408.jpg)  
+Figure 5: In environments with discrete action spaces, A2C (Mnih et al., 2016) with input-dependent baselines outperforms the best heuristic and achieves $2 5 \mathrm { - } 3 3 \%$ better testing reward than vanilla A2C (Mnih et al., 2016). Learning curves are on 100 test episodes with unseen input sequences; shaded area spans one standard deviation.
+
+Results. We extend OpenAI’s A2C implementation (Dhariwal et al., 2017) for our baselines. The learning curves in Figure 5 illustrate that directly applying A2C with a standard value network as the baseline results in unstable test reward and underperforms the traditional heuristic in both environments. Our input-dependent baselines reduce the variance and improve test reward by $2 5 \mathrm { - } 3 3 \%$ , outperforming the heuristic. The meta-baseline performs the best in all environments.
+
+# 7 RELATED WORK
+
+Policy gradient methods compute unbiased gradient estimates, but can experience a large variance (Sutton & Barto, 2017; Weaver & Tao, 2001). Reducing variance for policy-based methods using a baseline has been shown to be effective (Williams, 1992; Sutton & Barto, 2017; Weaver & Tao, 2001; Greensmith et al., 2004; Mnih et al., 2016). Much of this work focuses on variance reduction in a general MDP setting, rather than variance reduction for MDPs with specific stochastic structures. Wu et al. (2018)’s techniques for MDPs with multi-variate independent actions are closest to our work. Their state-action-dependent baseline improves training efficiency and model performance on high-dimensional control tasks by explicitly factoring out, for each action, the effect due to other actions. By contrast, our work exploits the structure of state transitions instead of stochastic policy.
+
+Recent work has also investigated the bias-variance tradeoff in policy gradient methods. Schulman et al. (2015b) replace the Monte Carlo return with a $\lambda$ -weighted return estimation (similar to $\mathrm { T D } ( \lambda )$ with value function bootstrap (Tesauro, 1995)), improving performance in high-dimensional control tasks. Other recent approaches use more general control variates to construct variants of policy gradient algorithms. Tucker et al. (2018) compare the recent work, both analytically on a linearquadratic-Gaussian task and empirically on complex robotic control tasks. Analysis of control variates for policy gradient methods is a well-studied topic, and extending such analyses (e.g., Greensmith et al. (2004)) to the input-driven MDP setting could be interesting future work.
+
+In other contexts, prior work has proposed new RL training methodologies for environments with disturbances. Clavera et al. (2018b) adapts the policy to different pattern of disturbance by training the RL agent using meta-learning. RARL (Pinto et al., 2017) improves policy robustness by co-training an adversary to generate a worst-case noise process. Our work is orthogonal and complementary to these work, as we seek to improve policy optimization itself in the presence of inputs like disturbances.
+
+# 8 CONCLUSION
+
+We introduced input-driven Markov Decision Processes in which stochastic input processes influence state dynamics and rewards. In this setting, we demonstrated that an input-dependent baseline can significantly reduce variance for policy gradient methods, improving training stability and the quality of learned policies. Our work provides an important ingredient for using RL successfully in a variety of domains, including queuing networks and computer systems, where an input workload is a fundamental aspect of the system, as well as domains where the input process is more implicit, like robotics control with disturbances or random obstacles.
+
+We showed that meta-learning provides an efficient way to learn input-dependent baselines for applications where input sequences can be repeated during training. Investigating efficient architectures for input-dependent baselines for cases where the input process cannot be repeated in training is an interesting direction for future work.
+
+Acknowledgements. We thank Ignasi Clavera for sharing the HalfCheetah environment, Jonas Rothfuss for the comments on meta-policy optimization and the anonymous ICLR reviewers for their feedback. This work was funded in part by NSF grants CNS-1751009, CNS-1617702, a Google Faculty Research Award, an AWS Machine Learning Research Award, a Cisco Research Center Award, an Alfred P. Sloan Research Fellowship and the sponsors of MIT Data Systems and AI Lab.
+
+# REFERENCES
+
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014.
+
+Francois Belletti, Daniel Haziza, Gabriel Gomes, and Alexandre M. Bayen. Expert level control of ramp metering based on multi-task deep reinforcement learning. IEEE Transactions on Intelligent Transportation Systems, 19(4):1198–1207, 2018.
+
+Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. OpenAI Gym. https://gym.openai.com/docs/, 2016.
+
+Trishul Chilimbi, Yutaka Suzue, Johnson Apacible, and Karthik Kalyanaraman. Project adam: Building an efficient and scalable deep learning training system. In Proceedings of the $I I ^ { t h }$ USENIX Symposium on Operating Systems Design and Implementation (OSDI), pp. 571–582, October 2014.
+
+Ignasi Clavera, Anusha Nagabandi, Ronald S Fearing, Pieter Abbeel, Sergey Levine, and Chelsea Finn. Learning to adapt: Meta-learning for model-based control. arXiv preprint arXiv:1803.11347, 2018a.
+
+Ignasi Clavera, Jonas Rothfuss, John Schulman, Yasuhiro Fujita, Tamim Asfour, and Pieter Abbeel. Model-based reinforcement learning via meta-policy optimization. arXiv preprint arXiv:1809.05214, 2018b.
+
+D.J. Daley. Certain optimality properties of the first-come first-served discipline for G/G/s queues. Stochastic Processes and their Applications, 25:301–308, 1987.
+
+DASH Industry Form. Reference Client 2.4.0. http://mediapm.edgesuite.net/dash/ public/nightly/samples/dash-if-reference-player/index.html, 2016.
+
+Prafulla Dhariwal, Christopher Hesse, Oleg Klimov, Alex Nichol, Matthias Plappert, Alec Radford, John Schulman, Szymon Sidor, and Yuhuai Wu. OpenAI Baselines. https://github.com/ openai/baselines, 2017.
+
+Yan Duan, Xi Chen, Rein Houthooft, John Schulman, and Pieter Abbeel. Benchmarking deep reinforcement learning for continuous control. In International Conference on Machine Learning, pp. 1329–1338, 2016.
+
+Richard Evans and Jim Gao. DeepMind AI Reduces Google Data Centre Cooling Bill by $40 \%$ . https://deepmind.com/blog/ deepmind-ai-reduces-google-data-centre-cooling-bill-40/, 2016.
+
+Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In International Conference on Machine Learning, pp. 1126–1135, 2017.
+
+Felix A. Gers, Jürgen Schmidhuber, and Fred Cummins. Learning to forget: Continual prediction with lstm. 1999.
+
+Robert Grandl, Srikanth Kandula, Sriram Rao, Aditya Akella, and Janardhan Kulkarni. Graphene: Packing and dependency-aware scheduling for data-parallel clusters. In Proceedings of the $I 2 ^ { t h }$ USENIX Symposium on Operating Systems Design and Implementation (OSDI), pp. 81–97, 2016.
+
+Evan Greensmith, Peter L Bartlett, and Jonathan Baxter. Variance reduction techniques for gradient estimates in reinforcement learning. Journal of Machine Learning Research, 5(Nov):1471–1530, 2004.
+
+Shixiang Gu, Timothy P. Lillicrap, Richard E Turner, Zoubin Ghahramani, Bernhard Schölkopf, and Sergey Levine. Interpolated policy gradient: Merging on-policy and off-policy gradient estimation for deep reinforcement learning. In Advances in Neural Information Processing Systems, pp. 3849–3858, 2017.
+
+Mor Harchol-Balter and Rein Vesilo. To balance or unbalance load in size-interval task allocation. Probability in the Engineering and Informational Sciences, 24(2):219–244, April 2010.
+
+James Harrison, Animesh Garg, Boris Ivanovic, Yuke Zhu, Silvio Savarese, Li Fei-Fei, and Marco Pavone. Adapt: zero-shot adaptive policy transfer for stochastic dynamical systems. arXiv preprint arXiv:1707.04674, 2017.
+
+Nicolas Heess, Srinivasan Sriram, Jay Lemmon, Josh Merel, Greg Wayne, Yuval Tassa, Tom Erez, Ziyu Wang, Ali Eslami, Martin Riedmiller, et al. Emergence of locomotion behaviours in rich environments. arXiv preprint arXiv:1707.02286, 2017.
+
+Sham M. Kakade. A natural policy gradient. In Advances in Neural Information Processing Systems, pp. 1531–1538, 2002.
+
+Frank P. Kelly. Reversibility and stochastic networks. Cambridge University Press, 2011.
+
+Leonard Kleinrock. Queueing systems, volume 2: Computer applications, volume 66. Wiley, New York, 1976.
+
+Sergey Levine, Chelsea Finn, Trevor Darrell, and Pieter Abbeel. End-to-end training of deep visuomotor policies. Journal of Machine Learning Research, 17(1):1334–1373, January 2016.
+
+Timothy P. Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. arXiv preprint arXiv:1509.02971, 2015.
+
+Hongzi Mao, Mohammad Alizadeh, Ishai Menache, and Srikanth Kandula. Resource management with deep reinforcement learning. In Proceedings of the $1 5 ^ { t h }$ ACM Workshop on Hot Topics in Networks (HotNets), November 2016.
+
+Hongzi Mao, Ravi Netravali, and Mohammad Alizadeh. Neural adaptive video streaming with pensieve. In Proceedings of the ACM SIGCOMM 2017 Conference, 2017.
+
+Stephen McGough, Noura Al Moubayed, and Matthew Forshaw. Using machine learning in tracedriven energy-aware simulations of high-throughput computing systems. In Proceedings of the ${ { \boldsymbol { \& { \mathit { \varepsilon } } } } ^ { g t h } }$ ACM/SPEC on International Conference on Performance Engineering (ICPE), pp. 55–60. ACM, 2017.
+
+Azalia Mirhoseini, Hieu Pham, Quoc V. Le, Benoit Steiner, Rasmus Larsen, Yuefeng Zhou, Naveen Kumar, Mohammad Norouzi, Samy Bengio, and Jeff Dean. Device placement optimization with reinforcement learning. In Proceedings of the $3 3 ^ { r d }$ International Conference on Machine Learning (ICML), 2017.
+
+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, Joel Veness, Marc G. Bellemare, Alex Graves, Martin Riedmiller, Andreas K. Fidjeland, Georg Ostrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen King, Dharshan Kumaran, Daan Wierstra, Shane Legg, Demis Hassabis Ioannis Antonoglou, Daan Wierstra, and Martin A. Riedmiller. Human-level control through deep reinforcement learning. Nature, 518:529–533, 2015.
+
+Volodymyr Mnih, Adrià Puigdomènech Badia, Mehdi Mirza, Alex Graves, Tim Harley, Timothy P. Lillicrap, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In Proceedings of the $3 3 ^ { r d }$ International Conference on Machine Learning (ICML), pp. 1928–1937, 2016.
+
+Vinod Nair and Geoffrey E. Hinton. Rectified linear units improve restricted boltzmann machines. In Proceedings of the $2 7 ^ { t h }$ International Conference on Machine Learning (ICML), pp. 807–814, 2010.
+
+Lerrel Pinto, James Davidson, Rahul Sukthankar, and Abhinav Gupta. Robust adversarial reinforcement learning. In Proceedings of the $3 4 ^ { t h }$ International Conference on Machine Learning (ICML), pp. 2817–2826, 2017.
+
+Haakon Riiser, Paul Vigmostad, Carsten Griwodz, and Pål Halvorsen. Commute Path Bandwidth Traces from 3G Networks: Analysis and Applications. In Proceedings of the $4 ^ { t h }$ ACM Multimedia Systems Conference (MMSys), 2013.
+
+John Schulman, Sergey Levine, Philipp Moritz, Michael I. Jordan, and Pieter Abbeel. Trust region policy optimization. arXiv preprint arXiv:1502.05477, 2015a.
+
+John Schulman, Philipp Moritz, Sergey Levine, Michael Jordan, and Pieter Abbeel. High-dimensional continuous control using generalized advantage estimation. arXiv preprint arXiv:1506.02438, 2015b.
+
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.
+
+David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. Nature, 550(7676):354, 2017.
+
+Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: An Introduction, Second Edition. MIT Press, 2017.
+
+Richard S. Sutton, David A. McAllester, Satinder P. Singh, and Yishay Mansour. Policy gradient methods for reinforcement learning with function approximation. In Advances in Neural Information Processing Systems, pp. 1057–1063. 2000.
+
+Gerald Tesauro. Temporal difference learning and td-gammon. Communications of the ACM, 38(3): 58–68, 1995.
+
+Philip Thomas. Bias in natural actor-critic algorithms. In International Conference on Machine Learning, pp. 441–448, 2014.
+
+Emanuel Todorov, Tom Erez, and Yuval Tassa. MuJoCo: A physics engine for model-based control. In Proceedings of the 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 5026–5033, 2012.
+
+George Tucker, Surya Bhupatiraju, Shixiang Gu, Richard E. Turner, Zoubin Ghahramani, and Sergey Levine. The mirage of action-dependent baselines in reinforcement learning. arXiv preprint arXiv:1802.10031, 2018.
+
+Ricardo Vilalta and Youssef Drissi. A perspective view and survey of meta-learning. Artificial Intelligence Review, 18(2):77–95, 2002.
+
+Lex Weaver and Nigel Tao. The optimal reward baseline for gradient-based reinforcement learning. In Proceedings of the $I 7 ^ { t h }$ Conference on Uncertainty in Artificial Intelligence, pp. 538–545. Morgan Kaufmann Publishers Inc., 2001.
+
+Ronald J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992.
+
+Keith Winstein and Hari Balakrishnan. TCP ex machina: Computer-generated congestion control. In ACM SIGCOMM Computer Communication Review, volume 43, pp. 123–134. ACM, 2013.
+
+Cathy Wu, Aboudy Kreidieh, Kanaad Parvate, Eugene Vinitsky, and Alexandre M Bayen. Flow: Architecture and benchmarking for reinforcement learning in traffic control. arXiv preprint arXiv:1710.05465, 2017.
+
+Cathy Wu, Aravind Rajeswaran, Yan Duan, Vikash Kumar, Alexandre M. Bayen, Sham Kakade, Igor Mordatch, and Pieter Abbeel. Variance reduction for policy gradient with action-dependent factorized baselines. In Proceedings of the $\mathit { 6 } ^ { t h }$ International Conference on Learning Representations (ICLR), 2018.
+
+Yuxin Wu and Yuandong Tian. Training agent for first-person shooter game with actor-critic curriculum learning. In Submitted to International Conference on Learning Representations, 2017.
+
+Xiaoqi Yin, Abhishek Jindal, Vyas Sekar, and Bruno Sinopoli. A Control-Theoretic Approach for Dynamic Adaptive Video Streaming over HTTP. In Proceedings of the 2015 ACM SIGCOMM Conference, 2015.
+
+# A ILLUSTRATION OF VARIANCE REDUCTION IN 1D GRID WORLD
+
+Consider a walker in a 1D grid world, where the state $s _ { t } \in \mathbb { Z }$ at time $t$ denotes the position of the walker, and action $a _ { t } \ \in \ \{ - 1 , + 1 \}$ denotes the intent to either move forward or backward. Additionally let $z _ { t } \in \{ - 1 , + 1 \}$ be a uniform i.i.d. “exogenous input” that perturbs the position of the walker. For an action $a _ { t }$ and input $z _ { t }$ , the state of the walker in the next step is given by $s _ { t + 1 } = s _ { t } + a _ { t } + z _ { t }$ . The objective of the game is to move the walker forward; hence, the reward is $\boldsymbol { r } _ { t } = \boldsymbol { a } _ { t } + \boldsymbol { z } _ { t }$ at each time step. $\gamma \in [ 0 , 1 ]$ is a discount factor.
+
+While the optimal policy for this game is clear $a _ { t } = + 1$ for all $t$ ), consider learning such a policy using policy gradient. For simplicity, let the policy be parametrized as $\pi _ { \theta } ( a _ { t } = + 1 | s _ { t } ) { \overset {  } { = } } e ^ { \theta } / ( { \overset { \cdot } { 1 + } } e ^ { \theta } { \overset { \cdot } { ) } }$ , with $\theta$ initialized to 0 at the start of training. In the following, we evaluate the variance of the policy gradient estimate at the start of training under (i) the standard value function baseline, and (ii) a baseline that is the expected cumulative reward conditioned on all future $z _ { t }$ inputs.
+
+is because Variance under standard baseline. The value function in this case is identically 0 at all states. This $\begin{array} { r } { \mathbb { E } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { t } ] = \mathbb { E } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } ( a _ { t } + z _ { t } ) ] = 0 } \end{array}$ since both actions $a _ { t }$ and inputs $z _ { t }$ are i.i.d. with mean 0. Also note that $\nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } = + 1 ) = 1 / 2$ and $\nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } = - 1 ) = - 1 / 2$ ; hence $\nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ) = a _ { t } / 2$ . Therefore the variance of the policy gradient estimate can be written as
+
+$$
+V _ { 1 } = \mathrm { V a r } \left[ \sum _ { t = 0 } ^ { \infty } \frac { a _ { t } } { 2 } \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } r _ { t ^ { \prime } } \right] = \mathrm { V a r } \left[ \sum _ { t = 0 } ^ { \infty } \frac { a _ { t } } { 2 } \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } ( a _ { t ^ { \prime } } + z _ { t ^ { \prime } } ) \right] .
+$$
+
+Variance under input-dependent baseline. Now, consider an alternative “input-dependent” baseline $V ( s _ { t } | z )$ defined as $\mathbb { E } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { t } | z ]$ . Intuitively this baseline captures the average reward incurred when experiencing a particular fixed $_ z$ sequence. We refer the reader to $\ S 4$ for a formal discussion and analysis of input-dependent baselines. Evaluating the baseline we get $\begin{array} { r } { V ( s _ { t } | z ) = \mathbb { E } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { t } | z ] = } \end{array}$ $\textstyle \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } z _ { t }$ . Therefore the variance of the policy gradient estimate in this case is
+
+$$
+V _ { 2 } = \mathrm { V a r } \left[ \sum _ { t = 0 } ^ { \infty } \frac { a _ { t } } { 2 } \left( \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } r _ { t ^ { \prime } } - \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } z _ { t ^ { \prime } } \right) \right] = \mathrm { V a r } \left[ \sum _ { t = 0 } ^ { \infty } \frac { a _ { t } } { 2 } \left( \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } a _ { t ^ { \prime } } \right) \right] .
+$$
+
+Reduction in variance. To analyze the variance reduction between the two cases (Equations (5) and (6)), we note that
+
+$$
+V _ { 1 } = V _ { 2 } + \mathrm { V a r } \left[ \sum _ { t = 0 } ^ { \infty } \frac { a _ { t } } { 2 } \left( \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } z _ { t ^ { \prime } } \right) \right] + 2 \mathrm { C o v } \left( \sum _ { t = 0 } ^ { \infty } \frac { a _ { t } } { 2 } \left( \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } a _ { t ^ { \prime } } \right) , \sum _ { t = 0 } ^ { \infty } \frac { a _ { t } } { 2 } \left( \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } z _ { t ^ { \prime } } \right) \right)
+$$
+
+$$
+= V _ { 2 } + \mathrm { V a r } \left[ \sum _ { t = 0 } ^ { \infty } \frac { a _ { t } } { 2 } \left( \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } z _ { t ^ { \prime } } \right) \right] .
+$$
+
+This follows because
+
+$$
+\begin{array} { r } { \mathbb { E } \left[ \displaystyle \sum _ { t = 0 } ^ { \infty } \frac { a _ { t } } { 2 } \left( \displaystyle \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } z _ { t ^ { \prime } } \right) \right] = \displaystyle \sum _ { t = 0 } ^ { \infty } \sum _ { t ^ { \prime } = t } ^ { \infty } \frac { \gamma ^ { t ^ { \prime } } } { 2 } \mathbb { E } [ a _ { t } z _ { t ^ { \prime } } ] = 0 , \quad \mathrm { a n d } } \\ { \mathbb { E } \left[ \left( \displaystyle \sum _ { t = 0 } ^ { \infty } \frac { a _ { t } } { 2 } \left( \displaystyle \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } a _ { t ^ { \prime } } \right) \right) \left( \displaystyle \sum _ { t = 0 } ^ { \infty } \frac { a _ { t } } { 2 } \left( \displaystyle \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } z _ { t ^ { \prime } } \right) \right) \right] = } \\ { \displaystyle \sum _ { t _ { 1 } = 0 } ^ { \infty } \displaystyle \sum _ { t _ { 1 } ^ { \prime } = t _ { 1 } } ^ { \infty } \sum _ { t _ { 2 } = 0 } ^ { \infty } \sum _ { t _ { 2 } ^ { \prime } = t _ { 2 } } ^ { \infty } \mathbb { E } \left[ \frac { a _ { t _ { 1 } } a _ { t _ { 1 } ^ { \prime } } a _ { t _ { 2 } } z _ { t _ { 2 } ^ { \prime } } } { 4 } \gamma ^ { t _ { 1 } ^ { \prime } + t _ { 2 } ^ { \prime } } \right] = 0 . } \end{array}
+$$
+
+Therefore the covariance term in Equation (7) is 0. Hence the variance reduction from Equation (8) can be written as
+
+$$
+\begin{array} { r l } & { V _ { 1 } - V _ { 2 } = \mathrm { V a r } \left[ \displaystyle \sum _ { t = 0 } ^ { \infty } \frac { a _ { t } } { 2 } \left( \displaystyle \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } z _ { t ^ { \prime } } \right) \right] = \displaystyle \sum _ { t _ { 1 } = 0 } ^ { \infty } \displaystyle \sum _ { t _ { 1 } ^ { \prime } = t _ { 1 } } ^ { \infty } \displaystyle \sum _ { t _ { 2 } = 0 } ^ { \infty } \displaystyle \sum _ { t _ { 2 } ^ { \prime } = t _ { 2 } } ^ { \infty } \mathbb { E } \left[ \frac { a _ { t _ { 1 } } a _ { t _ { 2 } } z _ { t _ { 1 } ^ { \prime } } z _ { t _ { 2 } ^ { \prime } } } { 4 } \gamma ^ { t _ { 1 } ^ { \prime } + t _ { 2 } ^ { \prime } } \right] } \\ & { \quad \quad \quad \quad \quad \quad = \displaystyle \sum _ { t _ { 1 } = 0 } ^ { \infty } \displaystyle \sum _ { t _ { 1 } ^ { \prime } = t _ { 1 } } ^ { \infty } \mathbb { E } \left[ \displaystyle \frac { a _ { t _ { 1 } } ^ { 2 } z _ { t _ { 1 } ^ { \prime } } ^ { 2 } } { 4 } \gamma ^ { 2 t _ { 1 } ^ { \prime } } \right] = \displaystyle \frac { \mathrm { V a r } ( a _ { 0 } ) \mathrm { V a r } ( z _ { 0 } ) } { 4 ( 1 - \gamma ^ { 2 } ) ^ { 2 } } . } \end{array}
+$$
+
+Thus the input-dependent baseline reduces variance of the policy gradient estimate by an amount proportional to the variance of the external input. In this toy example, we have chosen $z _ { t }$ to be binaryvalued, but more generally the variance of $z _ { t }$ could be arbitrarily large and might be a dominating factor of the overall variance in the policy gradient estimation.
+
+# B MARKOV PROPERTIES OF INPUT-DRIVEN DECISION PROCESSES
+
+Proposition 1. An input-driven decision process satisfying the conditions of case 1 in Figure 3 is a fully observable MDP, with state $\tilde { s } _ { t } : = ( s _ { t } , z _ { t } )$ , and action $\tilde { a } _ { t } : = a _ { t }$ .
+
+Proof.
+
+$$
+\begin{array} { r l } & { \operatorname* { P r } \bigl ( \tilde { s } _ { t + 1 } | \tilde { s } _ { 0 : t } , \tilde { a } _ { 0 : t } \bigr ) = \operatorname* { P r } \bigl ( s _ { t + 1 } , z _ { t + 1 } | s _ { 0 : t } , z _ { 0 : t } , a _ { 0 : t } \bigr ) } \\ & { \qquad = \operatorname* { P r } \bigl ( s _ { t + 1 } , z _ { t + 1 } | s _ { t } , z _ { t } , a _ { t } \bigr ) \quad \quad \mathrm { ( b y ~ d e f i n i t i o n ~ o f ~ c a s e ~ l ~ i n ~ F i g u r e ~ 3 a ) } } \\ & { \qquad = \operatorname* { P r } \bigl ( \tilde { s } _ { t + 1 } | \tilde { s } _ { t } , \tilde { a } _ { t } \bigr ) . } \end{array}
+$$
+
+Proposition 2. An input-driven decision process satisfying the conditions of case 2 in Figure 3, with state $\tilde { s } _ { t } : = ( s _ { t } , z _ { 0 : t } )$ and action $\tilde { a } _ { t } : = a _ { t }$ is a fully observable MDP. If only $\omega _ { t } = s _ { t }$ is observed at time $t$ , it is a partially observable MDP (POMDP).
+
+Proof.
+
+$$
+\begin{array} { r l } & { \mathrm { \mathfrak { p } } _ { \mathrm { r } } \big ( \tilde { s } _ { t + 1 } \big | \tilde { s } _ { 0 : t } , \tilde { a } _ { 0 : t } \big ) = \operatorname* { P r } \big ( s _ { t + 1 } , z _ { 0 : t + 1 } \big | s _ { 0 : t } , z _ { 0 : t } , a _ { 0 : t } \big ) } \\ & { = \operatorname* { P r } \big ( s _ { t + 1 } \big | s _ { 0 : t } , z _ { 0 : t + 1 } , a _ { 0 : t } \big ) \operatorname* { P r } \big ( z _ { 0 : t + 1 } \big | s _ { 0 : t } , z _ { 0 : t } , a _ { 0 : t } \big ) } \\ & { = \operatorname* { P r } \big ( s _ { t + 1 } \big | s _ { t } , z _ { 0 : t + 1 } , a _ { t } \big ) \operatorname* { P r } \big ( z _ { 0 : t + 1 } \big | s _ { t } , z _ { 0 : t } , a _ { t } \big ) \mathrm { ~ ( b y ~ d e f i n i t i o n ~ o f ~ c a s e ~ 2 ~ i n ~ F i g u t ~ } } \\ & { = \operatorname* { P r } \big ( s _ { t + 1 } , z _ { 0 : t + 1 } \big | s _ { t } , z _ { 0 : t } , a _ { t } \big ) } \\ & { = \operatorname* { P r } \big ( \tilde { s } _ { t + 1 } \big | \tilde { s } _ { t } , \tilde { a } _ { t } \big ) . } \end{array}
+$$
+
+Therefore, $( \tilde { s } _ { t } , \tilde { a } _ { t } )$ is a fully observable MDP. If only $\omega _ { t } = s _ { t }$ is observed, the decision process is a POMDP, since the $z _ { 0 : t }$ component of the state is not observed. □
+
+# C PROOF OF LEMMA 1
+
+Proof. From the definition of an input-driven MDP (Definition 1), we have
+
+$$
+\begin{array} { r l } & { \mathrm { P r } ( z _ { 0 : \infty } , \omega _ { t } , a _ { t } ) = \mathrm { P r } ( z _ { 0 : t } , \omega _ { t } , a _ { t } ) \mathrm { P r } ( z _ { t + 1 : \infty } | z _ { 0 : t } , \omega _ { t } , a _ { t } ) } \\ & { \qquad = \mathrm { P r } ( z _ { 0 : t } , \omega _ { t } ) \mathrm { P r } ( a _ { t } | z _ { 0 : t } , \omega _ { t } ) \mathrm { P r } ( z _ { t + 1 : \infty } | z _ { 0 : t } ) } \\ & { \qquad = \mathrm { P r } ( z _ { 0 : t } , \omega _ { t } ) \pi _ { \theta } ( a _ { t } | \omega _ { t } ) \mathrm { P r } ( z _ { t + 1 : \infty } | z _ { 0 : t } ) } \\ & { \qquad = \mathrm { P r } ( z _ { 0 : \infty } , \omega _ { t } ) \pi _ { \theta } ( a _ { t } | \omega _ { t } ) . } \end{array}
+$$
+
+Notice that $\operatorname* { P r } ( a _ { t } | z _ { 0 : t } , \omega _ { t } ) = \pi _ { \theta } ( a _ { t } | \omega _ { t } )$ in both the MDP and POMDP cases in Figure 3. By marginalizing over $z _ { 0 : t - 1 }$ on both sides, we obtain the result:
+
+$$
+\operatorname* { P r } ( z _ { t : \infty } , \omega _ { t } , a _ { t } ) = \operatorname* { P r } ( z _ { t : \infty } , \omega _ { t } ) \pi _ { \theta } ( a _ { t } | \omega _ { t } ) .
+$$
+
+# D PROOF OF LEMMA 2
+
+Proof. Expanding the Policy Gradient Theorem (Sutton & Barto, 2017), we have
+
+$$
+\begin{array} { r l } { \zeta _ { \vartheta } \eta ( \pi _ { \vartheta } ) = \mathbb { E } [ \displaystyle \sum _ { t = 0 } ^ { \infty } \nabla _ { \vartheta } \log \pi _ { \vartheta } ( \alpha _ { t } | \omega _ { t } ) \displaystyle \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { \tau } ] } \\ & { = \displaystyle \sum _ { t = 0 } ^ { \infty } \mathbb { E } [ \nabla _ { \vartheta } \log \pi _ { \vartheta } ( \alpha _ { t } | \omega _ { t } ) \displaystyle \sum _ { t ^ { \prime } \geq 1 } \gamma ^ { t ^ { \prime } } r _ { \tau } ] } \\ & { = \displaystyle \sum _ { t = 0 } ^ { \infty } [ \displaystyle \sum _ { s , \alpha , s } \mathbf { P r } ( \omega _ { t } - \omega , \alpha _ { t } = \alpha , z _ { t \leq \infty } = z ) \nabla _ { \vartheta } \log \pi _ { \vartheta } ( \alpha | \omega ) \mathbb { E } [ \displaystyle \sum _ { t ^ { \prime } \geq t } \gamma ^ { t ^ { \prime } } r _ { \tau } | \omega _ { t } = \omega , \alpha _ { t } = \alpha , z _ { t }   } \\ & { \displaystyle  = \displaystyle \sum _ { t = 0 } ^ { \infty } [ \displaystyle \sum _ { s , \alpha , s } \mathbf { P r } ( \omega _ { t } - \omega , z _ { t \leq \infty } = z ) \pi _ { \vartheta } ( \alpha | \omega ) \nabla _ { \vartheta } \log \pi _ { \vartheta } ( \alpha | \omega ) \mathbb { E } [ \displaystyle \sum _ { t ^ { \prime } \geq t } \gamma ^ { t ^ { \prime } } r _ { t ^ { \prime } } | \omega _ { t } = \omega , \alpha _ { t } = \alpha , z _ { t }   } \end{array}
+$$
+
+where the last step uses Lemma 1. Using the definition of $Q ( \omega , a , z )$ , we obtain:
+
+$$
+\begin{array} { l } { \displaystyle \nabla _ { \theta } \eta ( \pi _ { \theta } ) = \sum _ { t = 0 } ^ { \infty } \left[ \sum _ { \omega , a , z } \mathrm { P r } ( \omega _ { t } = \omega , z _ { t : \infty } = z ) \pi _ { \theta } ( a | \omega ) \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) \gamma ^ { t } Q ( \omega , a , z ) \right] } \\ { \displaystyle = \sum _ { \omega , a , z } \left[ \pi _ { \theta } ( a | \omega ) \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) Q ( \omega , a , z ) \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \mathrm { P r } ( \omega _ { t } = \omega , z _ { t : \infty } = z ) \right] \right] } \\ { \displaystyle = \sum _ { \omega , a , z } \pi _ { \theta } ( a | \omega ) \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) Q ( \omega , a , z ) \rho _ { \pi } ( \omega , z ) } \\ { \displaystyle = \mathbb { E } _ { ( \omega , z ) \sim \rho } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) Q ( \omega , a , z ) \right] . } \end{array}
+$$
+
+# E PROOF OF THEOREM 2
+
+Proof. Let $G ( \omega , a )$ denote $\nabla _ { \boldsymbol { \theta } } \log \pi _ { \boldsymbol { \theta } } ( a \vert \omega ) ^ { T } \nabla _ { \boldsymbol { \theta } } \log \pi _ { \boldsymbol { \theta } } ( a \vert \omega )$ . For any input-dependent baseline $b ( \omega , z )$ , the variance of the policy gradient estimate is given by
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { ( \omega , z ) \sim \rho _ { \pi } } [ \| \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) [ Q ( \omega , a , z ) - b ( \omega , z ) ] - \mathbb { E } _ { \rho _ { \pi } , \pi _ { \theta } } [ \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) [ Q ( \omega , a , z ) - b ( \omega , z ) ] ] \| } \\ & { \quad a \sim \mathbb { E } _ { \rho _ { \pi } , \pi _ { \theta } } [ G ( \omega , a ) [ Q ( \omega , a , z ) - b ( \omega , z ) ] ^ { 2 } ] - \| \mathbb { E } _ { \rho _ { \pi } , \pi _ { \theta } } [ \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) [ Q ( \omega , a , z ) - b ( \omega , z ) ] ] \| _ { 2 } ^ { 2 } } \\ & { \quad = \mathbb { E } _ { \rho _ { \pi } , \pi _ { \theta } } [ G ( \omega , a ) [ Q ( \omega , a , z ) - b ( \omega , z ) ] ^ { 2 } ] - \| \mathbb { E } _ { \rho _ { \pi } , \pi _ { \theta } } [ \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) Q ( \omega , a , z ) ] \| _ { 2 } ^ { 2 } \quad \mathrm { ( d u e ~ t o ~ T h e c o n s ~ o f ~ b e c o n s ~ ) } } \\ & { \quad = \mathbb { E } _ { \rho _ { \pi } , \pi _ { \theta } } [ G ( \omega , a ) Q ( \omega , a , z ) ^ { 2 } ] - \| \mathbb { E } _ { \rho _ { \pi } , \pi _ { \theta } } [ \nabla _ { \theta } \log \pi _ { \theta } ( a | \omega ) Q ( \omega , a , z ) ] \| _ { 2 } ^ { 2 } } \\ & { \quad + \mathbb { E } _ { \rho _ { \pi } } [ \mathbb { E } _ { a \sim \pi _ { \theta } } [ G ( \omega , a ) | z , \omega |  } \end{array}
+$$
+
+Notice that the baseline is only involved in the last term in a quadratic form, where the second order term is positive. To minimize the variance, we set baseline to the minimizer of the quadratic equation, i.e., $2 \mathrm { E } _ { a \sim \pi _ { \theta } } ^ {  } [ G ( \omega , a ) | \omega , z ] b ( \omega , z ) - 2 \mathrm { E } _ { a \sim \pi _ { \theta } } [ G ( \omega , a ) Q ( \omega , a , z ) | \omega , z ] = 0 :$ and hence the result follows. □
+
+# F INPUT-DEPENDENT BASELINE FOR TRPO
+
+We show that the input-dependent baselines are bias-free for Trust Region Policy Optimization (TRPO) (Schulman et al., 2015a).
+
+![](images/9f5ed5f8cb5077c4cce80fd61254a5e5a84b2248b289dabe0278a5da1df5deb0.jpg)  
+Figure 6: An LSTM-based input-dependent baseline (green) does not provide significant performance gain over standard state-dependent baseline (red) for load balancing and Walker2d with wind environments.
+
+Preliminaries. Stochastic gradient descent using Equation (1) does not guarantee consistent policy improvement in complex control problems. TRPO is an alternative approach that offers monotonic policy improvements, and derives a practical algorithm with better sample efficiency and performance. TRPO maximizes a surrogate objective, subject to a KL divergence constraint:
+
+$$
+\begin{array} { r l } { \mathrm { ~ \Theta ~ } } & { \varpi \mathrm { ~ \ " ~ } \stackrel { \sim } { \theta } \varpi \varphi _ { \pi _ { 0 \mathrm { l d } } } \left[ \frac { \pi _ { \theta } ( a | s ) } { \pi _ { 0 \mathrm { l d } } ( a | s ) } Q _ { \pi _ { \mathrm { o l d } } } ( s , a ) \right] } \\ & { \mathrm { ~ s u b j e c t ~ t o ~ \ " ~ } \mathbb { E } _ { s \sim \rho _ { \pi _ { 0 \mathrm { l d } } } } \left[ D _ { \mathrm { K L } } \left( \pi _ { \mathrm { o l d } } ( \cdot | s ) | | \pi _ { \theta } ( \cdot | s ) \right) \right] \le \delta , } \end{array}
+$$
+
+in which $\delta$ serves as a step size for policy update. Using a baseline in the TRPO objective, i.e. replacing $Q _ { \pi _ { \mathrm { o l d } } } ( s , a )$ with $Q _ { \pi _ { \mathrm { o l d } } } ( s , a ) - b ( s )$ , empirically improves policy performance (Schulman et al., 2015b).
+
+Similar to Theorem 2, we generalize TRPO to input-driven environments, with $\rho _ { \pi } ( \omega , z ) \ =$ $\begin{array} { r } { \sum _ { t = 0 } ^ { \infty } \left[ \gamma ^ { t } \operatorname* { P r } ( \omega _ { t } = \omega , z _ { t : \infty } = \bar { z } ) \right] } \end{array}$ denoting the discounted visitation frequency of the observation $\omega$ and input sequence $_ z$ , and $\begin{array} { r } { Q ( \omega , a , z ) \ = \ \mathbb { E } \left[ \sum _ { l = 0 } ^ { \infty } \gamma ^ { l } r _ { t + l } \ \middle | \ \omega _ { t } = \omega , a _ { t } = a , z _ { t : \infty } = z \right] } \end{array}$ . The TRPO objective becomes $\begin{array} { r l } & { \mathrm { E } _ { ( \omega , z ) \sim \rho _ { \mathrm { o l d } } , a \sim \pi _ { \mathrm { o l d } } } \left[ Q _ { \pi _ { \mathrm { o l d } } } ( \omega , a , z ) \pi _ { \theta } ( a | \omega ) / { \pi _ { \mathrm { o l d } } ( a | \omega ) } \right] } \end{array}$ ], and the constraint is $\mathbb { E } _ { ( \omega , z ) \sim \rho _ { \pi _ { \mathrm { o l d } } } } \left[ D _ { \mathrm { K L } } \left( \pi _ { \mathrm { o l d } } ( \cdot | s ) | | \pi _ { \theta } ( \cdot | s ) \right) \right] \leq \delta$ .
+
+Theorem 3. An input-dependent baseline does not change the optimal solution of the o $\begin{array} { r l } { p t i m i z a t i o n ~ p r o b l e m ~ i n \quad T R P O , ~ t h a t ~ i s ~ a r g m a x _ { \theta } \mathbb { E } _ { ( \omega , z ) \sim \rho _ { o d } , a \sim \pi _ { o d } } \left[ \frac { \pi _ { o } ^ { - } ( a | \omega ) } { \pi _ { o d } ( a | \omega ) } Q _ { \pi _ { o d } } ( \omega , a , z ) \right] } & { = } \\ { r g m a x _ { \theta } \mathbb { E } _ { ( \omega , z ) \sim \rho _ { o d } , a \sim \pi _ { o d } } \left[ \frac { \pi _ { o } ( a | \omega ) } { \pi _ { o d } ( a | \omega ) } \left( Q _ { \pi _ { o d } } ( \omega , a , z ) - b ( \omega , z ) \right) \right] . } \end{array}$
+
+Proof.
+
+$$
+\begin{array} { l } { \displaystyle \mathbb { E } _ { ( \omega , z ) \sim \rho _ { \mathrm { o d d } } , a \sim \pi _ { \mathrm { o d d } } } \left[ \frac { \pi _ { \theta } \left( a | \omega \right) } { \pi _ { \mathrm { o l d } } \left( a | \omega \right) } b ( \omega , z ) \right] = \sum _ { \omega } \sum _ { z } \rho _ { \mathrm { o l d } } ( \omega , z ) \sum _ { a } \pi _ { \mathrm { o l d } } ( a | \omega ) \left[ \frac { \pi _ { \theta } \left( a | \omega \right) } { \pi _ { \mathrm { o l d } } \left( a | \omega \right) } b ( \omega , z ) \right] } \\ { = \sum _ { \omega } \displaystyle \sum _ { z } \sum _ { z } \rho _ { \mathrm { o l d } } ( \omega , z ) \sum _ { a } \pi _ { \theta } ( a | \omega ) b ( \omega , z ) } \\ { = \sum _ { \omega } \displaystyle \sum _ { z } \rho _ { \mathrm { o l d } } ( \omega , z ) b ( \omega , z ) , } \end{array}
+$$
+
+which is independent of $\theta$ . Therefore, $b ( \omega , z )$ does not change the optimal solution to the optimization problem. □
+
+# G INPUT-DEPENDENT BASELINE WITH LSTM
+
+The input-dependent baseline is a function of both the state and the entire future input sequence. A natural approach to approximate such baselines is to use neural models that operate on sequences (e.g., LSTMs (Gers et al., 1999)). However, learning a sequential mapping in a high-dimensional space can be expensive (Bahdanau et al., 2014). For example, consider the LSTM input-dependent baseline with A2C on the load balancing environment (Figure 1a; $\ S 6 . 2 )$ and TRPO on the Walker2d with wind environment (Figure 1c; $\ S 6 . 1 \AA )$ ). As shown in Figure 6, the LSTM input-dependent baseline (green) does not significantly improve the policy performance over a standard state-only baseline (red) in these environments. By contrast, the MAML based input-dependent baseline (§5) reduces the variance in policy gradient estimation much more effectively and achieves a consistently better policy performance.
+
+# H ADDITIONAL POMDP EXPERIMENT
+
+Recall that the HalfCheetah on floating tiles environment (Figure 1d) is a POMDP, since the agent does not observe the buoyancy of the tiles. Figure 4 (middle) shows that the input-dependent baseline significantly improves the TRPO performance for this POMDP environment. We also created a POMDP version of the Walker2d with wind environment (Figure 1c), where the force of the wind is removed from the observation provided to the agent. The results of repeating the same training are shown in Figure 7. We make two key observations: (1) Compared with the MDP case in Figure 4 (left), the performance drops slightly overall. This is expected because the agent can react more agilely if it directly observes the current wind in the MDP case. (2) Input-dependent baselines reduce variance and improve the policy performance, with the MAML-based approach achieving the best performance, similar to the MDP case.
+
+![](images/36c4ac5aca3c6f85f71e318cf5eccccd5f2d7f91455f4f4cca8546fb33999de4.jpg)  
+Figure 7: Input-dependent baseline improves TRPO performance in the POMDP version of the Walker2d with wind environment.
+
+# I PSEUDOCODE FOR TRAINING MULTI-VALUEBASELINES
+
+In $\ S 5$ , we explained the idea of efficiently computing input-dependent baselines $( \ S 4 . 1 )$ using multiple value networks on a fixed set of input sequences. Algorithm 2 depicts the details of this approach.
+
+Algorithm 2 Training multi-value baselines for policy-based methods.
+
+Require: pregenerated input seuqnces $\{ z _ { 1 } , z _ { 2 } , \cdots , z _ { N } \}$ , step sizes $\alpha , \beta$   
+1: Initialize value network parameters $\theta _ { V _ { 1 } } , \theta _ { V _ { 1 } } , \cdot \cdot \cdot , \theta _ { V _ { N } }$ and policy parameters $\theta$   
+2: while not done do   
+3: Sample a input sequence $z _ { i }$   
+4: Sample $k$ rollouts $\mathcal { T } _ { 1 } , \mathcal { T } _ { 2 } , . . . , \mathcal { T } _ { k }$ using policy $\pi _ { \theta }$ and input sequence $z _ { i }$   
+5: Update policy with Equation (2) using baseline estimated with $\theta _ { V _ { i } }$   
+6: Update $i$ -th value network parameters: $\boldsymbol { \theta } _ { V _ { i } } \gets \boldsymbol { \theta } _ { V _ { i } } - \beta \nabla _ { \boldsymbol { \theta } _ { V _ { i } } } \mathcal { L } _ { 1 : k } \left[ V _ { \boldsymbol { \theta } _ { V _ { i } } } \right]$   
+7: end while
+
+# J SETUP FOR DISCRETE-ACTION ENVIRONMENTS
+
+Load balancing across servers (Figure 1a). In this environment, an RL agent balances jobs over $k$ servers to minimize the average job completion time. Similar to $\ S 3$ , the job sizes follow a Pareto distribution (scale $x _ { m } = 1 0 0$ , shape $\alpha = 1 . 5$ ), and jobs arrive in a Poisson process $\lambda = 5 5$ ). We run over 10 simulated servers with different processing rates, ranging linearly from 0.15 to 1.05. In this setting, the load of the system is at $90 \%$ (i.e., on average, $90 \%$ of the queues are non-empty). In each episode, we generate 500 jobs as the exogenous input process. The problem of minimizing average job completion time on servers with heterogeneous processing rates does not have a closed-form solution (Harchol-Balter & Vesilo, 2010); the most widely-used heuristic is to join the shortest queue (Daley, 1987). However, understanding the workload pattern can give a better policy; for example, we can reserve some servers for small jobs. In this environment, the observed state is a vector of $( j , q _ { 1 } , q _ { 2 } , . . . , q _ { k } )$ , where $j$ is the size of the incoming job, $q _ { i }$ is the amount of work currently in each queue. The action $a \in \{ 1 , 2 , . . . , k \}$ schedules the incoming job to a specific queue. The reward is the number of active jobs times the negated time elapsed since the last action.
+
+Bitrate adaptation for video streaming (Figure 1b). Streaming video over variable-bandwidth connections requires the client to adapt the video bitrates to optimize the user experience. This is challenging since the available network bandwidth (the exogenous input process) is hard to predict accurately. We simulate real-world video streaming using public cellular network data (Riiser et al., 2013) and video with seven bitrate levels and 500 chunks (DASH Industry Form, 2016). The reward is a weighted combination of video resolution, time paused for rebuffering, and the number of bitrate changes (Mao et al., 2017). The observed state contains bandwidth history, current video buffer size, and current bitrate. The action is the next video chunk’s bitrate. State-of-the-art heuristics for this problem conservatively estimate the network bandwidth and use model predictive control to choose the optimal bitrate over the near-term horizon (Yin et al., 2015).
+
+![](images/b94bd9bbaa4a14326edec26cd17adbb48a1e9b8e3d022624463b16f2fea0655d.jpg)  
+Figure 8: In continuous-action MuJoCo environments (§6.1), PPO (Schulman et al., 2017) with input-dependent baselines achieves $4 2 \% - 3 . 5 \times$ better testing reward than PPO with a standard state-dependent baseline. Learning curves are on 100 testing episodes with unseen input sequences; shaded area spans one standard deviation.
+
+# K EXPERIMENT DETAILS
+
+In our discrete-action environments (§6.2), we build 10-value networks and a meta-baseline using MAML (Finn et al., 2017), both on top of the OpenAI A2C implementation (Dhariwal et al., 2017). We use $\gamma = 0 . 9 9 5$ for both environments. The actor and the critic networks have 2 hidden layers, with 64 and 32 hidden neurons on each. The activation function is ReLU (Nair & Hinton, 2010) and the optimizer is Adam (Chilimbi et al., 2014). We train the policy with 16 (synchronous) parallel agents. The learning rate is $1 ^ { - 3 }$ . The entropy factor (Mnih et al., 2016) is decayed linearly from 1 to 0.001 over 10,000 training iterations. For the meta-baseline, the meta learning rate is $1 ^ { - 3 }$ and the model specification has five step updates, each with learning rate $1 ^ { - 4 }$ . The model specification step in MAML is performed with vanilla stochastic gradient descent.
+
+We introduce disturbance into our continuous-action robot control environments (§6.1). For the walker with wind (Figure 1c), we randomly sample a wind force in $[ - 1 , 1 ]$ initially and add a Gaussian noise sampled from $\mathcal { N } ( 0 , 1 )$ at each step. The wind is bounded between $[ - 1 0 , 1 0 ]$ . The episode terminates when the walker falls. For the half-cheetah with floating tiles, we extend the number of piers from 10 in the original environment (Clavera et al., 2018a) to 50, so that the agent remains on the pathway for longer. We initialize the tiles with damping sampled uniformly in $[ 0 , 1 0 ]$ . For the 7-DoF robot arm environments, we initialize the target to randomly appear within $( - 0 . 1 , - 0 . 2 , 0 . 5 )$ , $( 0 . 4 , 0 . 2 , - 0 . 5 )$ in 3D. The position of the target is perturbed with a Gaussian noise sampled from $\mathcal { N } ( 0 , 0 . 1 )$ in each coordinate at each step. We bound the position of the target so that it is confined within the arm’s reach. The episode length of all these environments are capped at 1,000.
+
+We build the multi-value networks and meta-baseline on top of the TRPO implementation by OpenAI (Dhariwal et al., 2017). We turned off the GAE enhancement by using $\lambda = 1$ for fair comparison. We found that it makes only a small performance difference (within $\pm 5 \%$ using $\lambda =$ $\{ 0 . 9 5 , 0 . 9 6 , 0 . 9 7 , 0 . 9 8 , 0 . 9 9 , 1 \}$ ) in our environments. We use $\gamma = 0 . 9 9$ for all three environments. The policy network has two hidden layers, with 128 and 64 hidden neurons on each. The activation function is ReLU (Nair & Hinton, 2010). The KL divergence constraint $\delta$ is 0.01. The learning rate for value functions is $1 ^ { - 3 }$ . The hyperparameter of training the meta-baseline is the same as the discrete-action case.
+
+# L INPUT-DEPENDENT BASELINES WITH PPO
+
+Figure 8 shows the results of applying input-dependent baselines on PPO (Schulman et al., 2017) in MuJoCo (Todorov et al., 2012) environments. We make three key observations. First, compared to Figure 4 the best performances of PPO in these environments (blue curves) are better than that of TRPO. This is as expected, because the variance of the reward feedbacks in these environments is generally large and the reward clipping in PPO helps. Second, input-dependent baselines improve the policy performance for all environments. In particular, the meta-learning approach achieves the best performance, as it is not restricted to a fixed set of input sequences during training (§5). Third, the trend of learning curve is similar to that in TRPO (Figure 4), which shows our inputdependent baseline approach is generally applicable to a range of policy gradient based methods (e.g., A2C (§6.2), TRPO (§6.1), and PPO).
+
+![](images/904a0a9eff158528fbfff88da34bb5cf8f5a9a05a0164317e0e6a174e2aea32d.jpg)  
+Figure 9: The input-dependent baseline technique is complementary and orthogonal to RARL (Pinto et al., 2017). The implementation of input-dependent baseline is MAML (§5). Left: learning curves of testing rewards; shaded area spans one standard deviation; the input-dependent baseline improves the policy optimization for both TRPO and RARL, while RARL improves TRPO in the Walker2d environment with wind disturbance. Right: CDF of testing performance; RARL improves the policy especially in the low reward region; applying the input-dependent baseline boosts the performance for both TRPO and RARL significantly (blue, red).
+
+# M INPUT-DEPENDENT BASELINES WITH RARL
+
+Our work is orthogonal and complementary to adversarial and robust reinforcement learning (e.g., RARL (Pinto et al., 2017)). These methods seek to improve policy robustness by co-training an adversary to generate a worst-case noise process, whereas our work improves policy optimization itself in the presence of inputs like noise. Note that if an adversary generates high-variance noise, similar to the inputs we consider in our experiments (§6), techniques such RARL alone are not adequate to train good controllers.
+
+To empirically demonstrate this effect, we repeat the Walker2d with wind experiment described in $\ S 6 . 1$ . In this environment, we add a noise (of the same scale as the original random walk) on the wind and co-train an adversary to control the strength and direction of this noise. We follow the training procedure described in RARL (Pinto et al., 2017, §3.3).
+
+Figure 9 depicts the results. With either the standard state-dependent baseline or our input-dependent baseline, RARL generally improves the robustness of the policy, as RARL achieves better testing rewards especially in the low reward region (i.e., compared the yellow curve to green curve, or red curve to blue curve in CDF of Figure 9). Moreover, input-dependent baseline significantly improves the policy optimization, which boosts the performance of both TRPO and RARL (i.e., comparing the blue curve to the green curve, and the red curve to the yellow curve). Therefore, in this environment, the input-dependent baseline helps improve the policy optimization methods and is complementary to adversarial RL methods such as RARL.
+
+# N INPUT-DEPENDENT BASELINES WITH META-POLICY ADAPTATION
+
+There has been a line of work focusing on fast policy adaptation (Clavera et al., 2018a;b; Harrison et al., 2017). For example, Clavera et al. (2018b) propose a model-based meta-policy optimization approach (MB-MPO). It quickly learns the system dynamics using supervised learning and uses the learned model to perform virtual rollouts for meta-policy adaptation. Conceptually, our work differs because the goal is fundamentally different: our goal is to learn a single policy that performs well in the presence of a stochastic input process, while MB-MPO aims to quickly adapt a policy to new environments.
+
+It is worth noting that the policy adaptation approaches are well-suited to handling model discrepancy between training and testing. However, in our setting, there exists no model discrepancy. In particular, the distribution of the input process is the same during training and testing. For example, in our load balancing environment (Figure 1a, $\ S 6 . 2 )$ , the exogenous workload process is sampled from the same distribution during training and testing.
+
+Therefore our work is conceptually complementary to policy adaptation approaches. Since some of these methods require a policy optimization step (e.g., (Clavera et al., 2018b, $\ S 4 . 2 )$ ), our inputdependent baseline can help these methods by reducing variance during training. We perform an experiment to investigate this. Specifically, we apply the meta-policy adaptation technique proposed by Clavera et al. (2018b) in our Walker2d environment with wind disturbance (Figure 1c, $\ S 6 . 1 )$ . For this environment, although the wind pattern is drawn from the same stochastic process (random walk), we aim to adapt the policy to each particular instantiation of the wind.
+
+![](images/77ca2cee74da09fecf1a11c6f2dcfff3508753760890e43bff0f39e01eeca1b5.jpg)  
+Figure 10: The input-dependent baseline technique is complementary to MPO (Clavera et al., 2018b). The implementation of input-dependent baseline is MAML (§5). Left: learning curves in the testing Walker2d environment with wind disturbance; MPO is tested with adapted policy in each testing instance of the wind input; shaded area spans one standard deviation; the input-dependent baseline improves the policy optimization for both TRPO and MPO, while MPO improves TRPO. Right: meta policy adaptation at training timestep $5 e 7$ ; adapting the policy in specific input instances help boosting the performance (comparing yellow with green, and red with blue); applying input-dependent baseline generally improves the policy performance.
+
+Operationally, to reduce complexity, we bypass the supervised learning step for the system dynamics and use the simulator to generate rollouts directly, since the interaction with the simulator is not costly for our purpose and the state transition in our environment is not deterministic. Following the meta-policy adaptation approach, the policy optimization algorithm is TRPO (Schulman et al., 2015a). The meta-policy adaptation algorithm is MAML (Finn et al., 2017). In particular, we performed ten gradient steps to specialize the meta-policy for each instantiation of the input process. For input-dependent baseline, we inherit our meta-baseline approach from $\ S 5$ . Similar to policy adaptation, we adapt our meta-baseline alongside with the policy adaptation in the ten gradient steps for each input instance.
+
+The results of our experiment is shown in Figure 10. The learning curve (left figure) shows the policy performance for 100 unseen test input sequences at each training checkpoint. We measure the performance of MPO after ten steps of policy adaptation for each of the 100 input sequences. As expected, policy adaptation specializes to the particular instance of the input process and improves policy performance in the learning curve (e.g., MPO improves over TRPO, as shown by the green and yellow learning curve). However, policy adaptation does not solve the problem of variance caused by the input process, since the policy optimization step within policy adaptation suffers from large variance. Using an input-dependent baseline improves performance both for TRPO and MPO. Indeed, MPO trained with the input-dependent baseline (and adapted for each input sequence) outperforms the single TRPO policy, as shown by the red learning curve.
+
+This effect is more evident in the policy adaptation curve (right figure). The policy adaptation curve shows the testing performance of the adapted policy at each adaptation step (the meta-policy is taken from the 5e7 training timestep). With an input-dependent baseline, the meta policy already performs quite well at the $0 ^ { \mathrm { t h } }$ step of policy adaptation (without any adaptation). This is perhaps unsurprising, since a single policy (e.g., the TRPO policy trained with input-dependent baseline) can achieve good performance in this environment. However, specializing the meta-policy for each particular input instance further improves performance.

parse/train/Sk7KsfW0-/Sk7KsfW0-.md

@@ -0,0 +1,227 @@
+# LIFELONG LEARNING WITH DYNAMICALLY EXPANDABLE NETWORKS
+
+Jaehong $\mathbf { V o o n } ^ { 1 \ast }$ , Eunho Yang1,3, Jeongtae Lee2, Sung Ju Hwang1,3   
+KAIST1, Daejeon, South Korea, UNIST2, Ulsan, South Korea, AITrics3, Seoul, South Korea   
+jaehong.yoon93@gmail.com, jtlee@unist.ac.kr   
+{eunhoy, sjhwang82}@kaist.ac.kr
+
+# ABSTRACT
+
+We propose a novel deep network architecture for lifelong learning which we refer to as Dynamically Expandable Network (DEN), that can dynamically decide its network capacity as it trains on a sequence of tasks, to learn a compact overlapping knowledge sharing structure among tasks. DEN is efficiently trained in an online manner by performing selective retraining, dynamically expands network capacity upon arrival of each task with only the necessary number of units, and effectively prevents semantic drift by splitting/duplicating units and timestamping them. We validate DEN on multiple public datasets under lifelong learning scenarios, on which it not only significantly outperforms existing lifelong learning methods for deep networks, but also achieves the same level of performance as the batch counterparts with substantially fewer number of parameters. Further, the obtained network fine-tuned on all tasks obtained siginficantly better performance over the batch models, which shows that it can be used to estimate the optimal network structure even when all tasks are available in the first place.
+
+# 1 INTRODUCTION
+
+Lifelong learning (Thrun, 1995), the problem of continual learning where tasks arrive in sequence, is an important topic in transfer learning. The primary goal of lifelong learning is to leverage knowledge from earlier tasks for obtaining better performance, or faster convergence/training speed on models for later tasks. While there exist many different approaches to tackle this problem, we consider lifelong learning under deep learning to exploit the power of deep neural networks. Fortunately, for deep learning, storing and transferring knowledge can be done in a straightforward manner through the learned network weights. The learned weights can serve as the knowledge for the existing tasks, and the new task can leverage this by simply sharing these weights.
+
+Therefore, we can consider lifelong learning simply as a special case of online or incremental learning, in case of deep neural networks. There are multiple ways to perform such incremental learning (Rusu et al., 2016; Zhou et al., 2012). The simplest way is to incrementally fine-tune the network to new tasks by continuing to train the network with new training data. However, such simple retraining of the network can degenerate the performance for both the new tasks and the old ones. If the new task is largely different from the older ones, such as in the case where previous tasks are classifying images of animals and the new task is to classify images of cars, then the features learned on the previous tasks may not be useful for the new one. At the same time, the retrained representations for the new task could adversely affect the old tasks, as they may have drifted from their original meanings and are no longer optimal for them. For example, the feature describing stripe pattern from zebra, may changes its meaning for the later classification task for classes such as striped t-shirt or fence, which can fit to the feature and drastically change its meaning.
+
+Then how can we ensure that the knowledge sharing through the network is beneficial for all tasks, in the online/incremental learning of a deep neural network? Recent work suggests to either use a regularizer that prevents the parameters from drastic changes in their values yet still enables to find a good solution for the new task (Kirkpatrick et al., 2017), or block any changes to the old task parameters (Rusu et al., 2016). Our strategy is different from both approaches, since we retrain the network at each task $t$ such that each new task utilizes and changes only the relevant part of the previous trained network, while still allowing to expand the network capacity when necessary. In this way, each task $t$ will use a different subnetwork from the previous tasks, while still sharing a considerable part of the subnetwork with them. Figure 1 illustrates our model in comparison with existing deep lifelong learning methods.
+
+![](images/b5c12bb74695a013b25befa9b435a5c5d2bec01656e52c3cf843804c3e51d620.jpg)  
+Figure 1: Concept: (a) Retraining models such as Elastic Weight Consoliation Kirkpatrick et al. (2017) retrains the entire network learned on previous tasks while regularizing it to prevent large deviation from the original model. Units and weights colored in red denote the ones that are retrained, and black ones are ones that remain fixed. (b) Non-retraining models such as Progressive Network (Rusu et al., 2016) expands the network for the new task $t$ , while withholding modification of network weights for previous tasks. (c) Our DEN selectively retrains the old network, expanding its capacity when necessary, and thus dynamically deciding its optimal capacity as it trains on.
+
+There are a number of challenges that need to be tackled for such incremental deep learning setting with selective parameter sharing and dynamic layer expansion.
+
+1) Achieving scalability and efficiency in training: If the network grows in capacity, training cost per task will increasingly grow as well, since the later tasks will establish connections to a much larger network. Thus, we need a way to keep the computational overhead of retraining to be low.
+
+2) Deciding when to expand the network, and how many neurons to add: The network might not need to expand its size, if the old network sufficiently explains the new task. On the other hand, it might need to add in many neurons if the task is very different from the existing ones. Hence, the model needs to dynamically add in only the necessary number of neurons.
+
+3) Preventing semantic drift, or catastrophic forgetting, where the network drifts away from the initial configuration as it trains on, and thus shows degenerate performance for earlier examples/tasks. As our method retrains the network, even partially, to fit to later learned tasks, and add in new neurons which might also negatively affect the prior tasks by establishing connections to old subnetwork, we need a mechanism to prevent potential semantic drift.
+
+To overcome such challenges, we propose a novel deep network model along with an efficient and effective incremental learning algorithm, which we name as Dynamically Expandable Networks (DEN). In a lifelong learning scenario, DEN maximally utilizes the network learned on all previous tasks to efficiently learn to predict for the new task, while dynamically increasing the network capacity by adding in or splitting/duplicating neurons when necessary. Our method is applicable to any generic deep networks, including convolutional networks.
+
+We validate our incremental deep neural network for lifelong learning on multiple public datasets, on which it achieves similar or better performance than the model that trains a separate network for each task, while using only $1 1 . 9 \% p - 6 0 . 3 \% p$ of its parameters. Further, fine-tuning of the learned network on all tasks obtains even better performance, outperforming the batch model by as much as $0 . 0 5 \% p - 4 . 8 \% p$ . Thus, our model can be also used for structure estimation to obtain optimal performance over network capacity even when batch training is possible, which is a more general setup.
+
+# 2 RELATED WORK
+
+Lifelong learning Lifelong learning (Thrun, 1995) is the learning paradigm for continual learning where the model learns from a sequence of tasks while transferring knowledge obtained from earlier tasks to later ones. Since its inception of idea by Thrun (1995), it has been extensively studied due to its practicality in scenarios where the data arrives in streams, such as in autonomous driving or learning of robotic agents. Lifelong learning is often tackled as an online multi-task learning problem, where the focus is on efficient training as well as on knowledge transfer. Eaton & Ruvolo (2013) suggest an online lifelong learning framework (ELLA) that is based on an existing multi-task learning formulation (Kumar & Daume III, 2012) that efficiently updates latent parameter bases for a sequence of tasks, by removing dependency to previous tasks for the learning of each task predictor, and preventing retraining previous task predictors. Recently, lifelong learning is studied in deep learning frameworks; since lifelong learning of a deep network can be straightforwardly done by simple re-training, the primary focus of research is on overcoming catastrophic forgetting (Kirkpatrick et al., 2017; Rusu et al., 2016; Zenke et al., 2017; Lee et al., 2017).
+
+Preventing catastrophic forgetting Incremental or lifelong learning of deep networks results in the problem known as catastrophic forgetting, which describes the case where the retraining of the network for new tasks results in the network forgetting what are learned for previous tasks. One solution to this problem is to use a regularizer that prevents the new model from deviating too much from the previous one, such as $\ell _ { 2 }$ -regularizer. However, use of the simple $\ell _ { 2 }$ -regularizer prevents the model from learning new knowledge for the new tasks, which results in suboptimal performances on later tasks. To overcome this limitation, Kirkpatrick et al. (2017) proposed a method called Elastic Weight Consolidation (EWC) that regularizes the model parameter at each step with the model parameter at previous iteration via the Fisher information matrix for the current task, which enables to find a good solution for both tasks. Zenke et al. (2017) proposed a similar approach, but their approach computes the per-synapse consolidation online, and considers the entire learning trajectory rather than the final parameter value. Another way to prevent catastrophic forgetting is to completely block any modifications to the previous network, as done in Rusu et al. (2016), where at each learning stage the network is expanded with a subnetwork with fixed capacity that is trained with incoming weights from the original network, but without backpropagating to it.
+
+Dynamic network expansion There are few existing works that explored neural networks that can dynamically increase its capacity during training. Zhou et al. (2012) propose to incrementally train a denoising autoencoder by adding in new neurons for a group of difficult examples with high loss, and later merging them with other neurons to prevent redundancy. Recently, Philipp & Carbonell (2017) propose a nonparametric neural network model which not only learns to minimize the loss but also find the minimum dimensionality of each layer that can reduce the loss, with the assumption that each layer has infinite number of neurons. Cortes et al. (2016) also propose a network that can adaptively learn both the structure and the weights to minimize the given loss, based on boosting theory. However, none of these work considered multi-task setting and involves iterative process of adding in neurons (or sets of neurons), while our method only needs to train the network once for each task, to decide how many neurons to add. Xiao et al. (2014) propose a method to incrementally train a network for multi-class classification, where the network not only grows in capacity, but forms a hierarchical structure as new classes arrive at the model. The model, however, grows and branches only the topmost layers, while our method can increase the number of neurons at any layer.
+
+# 3 INCREMENTAL LEARNING OF A DYNAMICALLY EXPANDABLE NETWORK
+
+We consider the problem of incremental training of a deep neural network under the lifelong learning scenario, where unknown number of tasks with unknown distributions of training data arrive at the model in sequence. Specifically, our goal is to learn models for a sequence of $T$ tasks, $t =$ $1 , \ldots , t , \ldots , T$ for unbounded $T$ where the task at time point $t$ comes with training data $\mathcal { D } _ { t } ~ =$ $\{ x _ { i } , y _ { i } \} _ { i = 1 } ^ { N _ { t } }$ }Nt . Note that each task $t$ can be either a single task, or comprised of set of subtasks. While our method is generic to any kinds of tasks, for simplification, we only consider the binary classification task, that is, $y \in \{ 0 , 1 \}$ for input feature $\pmb { x } \in \mathbb { R } ^ { d }$ . It is the main challenge in the lifelong learning setting that all the previous training datasets up to $t - 1$ are not available at the current time $t$ (only the model parameters for the previous tasks are accessible, if any). The lifelong learning agent at time $t$ aims to learn the model parameter $\mathbf { \mathbf { } } W ^ { t }$ by solving following problem:
+
+$$
+\operatorname* { m i n i m i z e } _ { W ^ { t } } \mathcal { L } ( W ^ { t } ; W ^ { t - 1 } , \mathcal { D } _ { t } ) + \lambda \Omega ( W ^ { t } ) , \quad t = 1 , \dots .
+$$
+
+![](images/8238350ec118c4baceeac79fb522eab4689d7e76656397bdfc1686632ac4db74.jpg)  
+Figure 2: Incremental learning of a dynamically expandable network: Left: Selective retraining. DEN first identifies neurons that are relevant to the new tasks, and selectively retrains the network parameters associated with them. Center: Dynamic network expansion. If the selective retraining fails to obtain desired loss below set threshold, we expand the network capacity in a top-down manner, while eliminating any unnecessary neurons using group-sparsity regularization. Right: Network split/duplication. DEN calculates the drift $\rho _ { i } ^ { t }$ for each unit to identify units that have drifted too much from their original values during training and duplicate them.
+
+where $\mathcal { L }$ is task specific loss function, $\mathbf { \mathbf { \mathbf { \mathbf { W } } ^ { t } } }$ is the parameter for task $t$ , and $\Omega ( W ^ { t } )$ is the regularization (e.g. element-wise $\ell _ { 2 }$ norm) to enforce our model $\mathbf { \mathbf { \mathbf { \mathbf { W } } } ^ { t } }$ appropriately. In case of a neural network which is our primary interest, ${ W ^ { t } = \{ W _ { l } \} _ { l = 1 } ^ { L } }$ is the weight tensor.
+
+To tackle these challenges of lifelong learning, we let the network to maximally utilize the knowledge obtained from the previous tasks, while allowing it to dynamically expand its capacity when the accumulated knowledge alone cannot sufficiently explain the new task. Figure 2 and Algorithm 1 describes our incremental learning process.
+
+# Algorithm 1 Incremental Learning of a Dynamically Expandable Network
+
+Input: Dataset $\mathcal { D } = ( \mathcal { D } _ { 1 } , \ldots , \mathcal { D } _ { T } )$ , Thresholds $\tau$ , $\sigma$   
+Output: $W ^ { T }$   
+for $t = 1 , \dots , T$ do if $t = 1$ then Train the network weights $W ^ { 1 }$ using Eq. 2 else W t = SelectiveRetraining $( W ^ { t - 1 } )$ {Selectively retrain the previous network using Algorithm 2 } if $\mathcal { L } _ { t } > \tau$ then $\mathbf { } W ^ { t } = D y n a m i c E x p a n s i o n ( W ^ { t } )$ {Expand the network capacity using Algorithm 3} $\mathbf { } W ^ { t } = S p l i t ( W ^ { t } )$ {Split and duplicate the units using Algorithm 4 }
+
+In following subsections, we describe each component of our incremental learning algorithm in detail: 1) Selective retraining, 2) Dynamic network expansion, and 3) Network split/duplication.
+
+Selective Retraining. A most naive way to train the model for a sequence of tasks would be retraining the entire model every time a new task arrives. However, such retraining will be very costly for a deep neural network. Thus, we suggest to perform selective retraining of the model, by retraining only the weights that are affected by the new task. Initially $\scriptstyle ( t = 1 )$ , we train the network with $\ell _ { 1 }$ -regularization to promote sparsity in the weights, such that each neuron is connected to only few neurons in the layer below:
+
+$$
+\operatorname* { m i n i m i z e } _ { W ^ { t = 1 } } \mathcal { L } ( W ^ { t = 1 } ; \mathcal { D } _ { t } ) + \mu \sum _ { l = 1 } ^ { L } \| W _ { l } ^ { t = 1 } \| _ { 1 }
+$$
+
+where $1 { \le } l { \le } L$ denotes the $l _ { t h }$ layer of the network, $\mathbf { } W _ { l } ^ { t }$ is the network parameter at layer $l$ , and $\mu$ is the regularization parameter of the element-wise $\ell _ { 1 }$ norm for sparsity on $W$ . For convolutional layers, we apply $( 2 , 1 )$ -norm on the filters, to select only few filters from the previous layer.
+
+Throughout our incremental learning procedure, we maintain $W ^ { t - 1 }$ to be sparse, thus we can drastically reduce the computation overheads if we can focus on the subnetwork connected new task. To this end, when a new task $t$ arrives at the model, we first fit a sparse linear model to predict task $t$
+
+using topmost hidden units of the neural network via solving the following problem:
+
+$$
+\operatorname* { m i n i m i z e } _ { W _ { L , t } ^ { t } } \mathcal { L } ( W _ { L , t } ^ { t } ; W _ { 1 : L - 1 } ^ { t - 1 } , \mathcal { D } _ { t } ) + \mu \| W _ { L , t } ^ { t } \| _ { 1 }
+$$
+
+where W t−11:L− denotes the set of all other parameters except $W _ { L , t } ^ { t }$ . That is, we solve this optimization to obtain the connections between output unit $o _ { t }$ and the hidden units at layer $L – 1$ (fixing all other parameters up to layer $L – 1$ as $W ^ { t - 1 }$ ). Once we build the sparse connection at this layer, we can identify all units and weights in the network that are affected by the training, while leaving the part of the network that are not connected to $o _ { t }$ unchanged. Specifically, we perform breadth-first search on the network starting from those selected nodes, to identify all units (and input feature) that have paths to $o _ { t }$ . Then, we train only the weights of the selected subnetwork $S$ , denoted as $\mathbf { } W _ { S } ^ { t }$ :
+
+$$
+\operatorname* { m i n i m i z e } _ { W _ { S } ^ { t } } \mathcal { L } ( W _ { S } ^ { t } ; W _ { S ^ { c } } ^ { t - 1 } , \mathcal { D } _ { t } ) + \mu \| W _ { S } ^ { t } \| _ { 2 }
+$$
+
+We use the element-wise $\ell _ { 2 }$ regularizer since the sparse connections have been already established1. This partial retraining will result in lower computational overhead and also help with avoiding negative transfer, since neurons that are not selected will not get affected by the retraining process. Algorithm 2 describes the selective retraining process.
+
+# Algorithm 2 Selective Retraining
+
+Input: Datatset $\mathcal { D } _ { t }$ , Previous parameter $W ^ { t - 1 }$   
+Output: network parameter $\hat { W } ^ { t }$   
+Initialize Solve Eq. $l \gets L - 1$ , n $S = \{ o _ { t } \}$ $\boldsymbol { W } _ { L , t } ^ { t }$   
+Add neuron $i$ to $S$ if the weight between $i$ and $o _ { t }$ in $W _ { L , t } ^ { t }$ is not zero.   
+for $l = L - 1 , \ldots , 1$ do Add neuron $_ { i }$ to $S$ if there exists some neuron $j \in S$ such that $\mathbf { } W _ { l , i j } ^ { t - 1 } \neq 0$   
+Solve Eq. 4 to obtain
+
+Dynamic Network Expansion. In case where the new task is highly relevant to the old ones, or aggregated partial knowledge obtained from each task is sufficient to explain the new task, selective retraining alone will be sufficient for the new task. However, when the learned features cannot accurately represent the new task, additional neurons need to be introduced to the network, in order to account for the features that are necessary for the new task. Some existing work (Zhou et al., 2012; Rusu et al., 2016) are based on a similar idea. However, they are either inefficient due to iterative training that requires repeated forward pass (Zhou et al., 2012), or adds in constant number of units at each task $t$ without consideration of the task difficulty (Rusu et al., 2016) and thus are suboptimal in terms of performance and network capacity utility.
+
+To overcome these limitations, we instead propose an efficient way of using group sparse regularization to dynamically decide how many neurons to add at which layer, for each task without repeated retraining of the network for each unit. Suppose that we expand the $l _ { t h }$ layer of a network with a constant number of units, say $k$ , inducing two parameter matrices expansions: $\pmb { W _ { l } ^ { t } } = [ \pmb { W } _ { l } ^ { t - 1 } ; \pmb { W } _ { l } ^ { \mathcal { N } } ]$ and $W _ { l - 1 } ^ { t } = [ \boldsymbol { W } _ { l - 1 } ^ { t - 1 } ; \boldsymbol { W } _ { l - 1 } ^ { \mathcal { N } } ]$ for outgoing and incoming layers respectively, where $W ^ { \mathcal { N } }$ is the expanded weight matrix involved with added neurons. Since we do not always want to add in all $k$ units (depending on the relatedness between the new task and the old tasks), we perform group sparsity regularization on the added parameters as follows:
+
+$$
+\operatorname* { m i n i m i z e } _ { \boldsymbol { W } _ { l } ^ { N } } \mathcal { L } ( \boldsymbol { W } _ { l } ^ { N } ; \boldsymbol { W } _ { l } ^ { t - 1 } , \mathcal { D } _ { t } ) + \mu \| \boldsymbol { W } _ { l } ^ { N } \| _ { 1 } + \gamma \sum _ { g } \| \boldsymbol { W } _ { l , g } ^ { N } \| _ { 2 }
+$$
+
+where $g \in { \mathcal { G } }$ is a group defined on the incoming weights for each neuron. For convolutional layers, we defined each group as the activation map for each convolutional filter. This group sparsity regularization was used in Wen et al. (2016) and Alvarez & Salzmann (2016) to find the right number of neurons for a full network, while we apply it to the partial network. Algorithm 3 describes the details on how expansion works.
+
+After selective retraining is done, the network checks if the loss is below certain threshold. If not, then at each layer we expand its capacity by $k$ neurons and solve for Eq. 5. Due to group sparsity regularization in Eq. 5, hidden units (or convolutional filters) that are deemed unnecessary from the training will be dropped altogether. We expect that from this dynamic network expansion process, the model captures new features that were not previously represented by $\mathbf { \big . } W _ { l } ^ { t - 1 }$ to minimize residual errors, while maximizing the utilization of the network capacity by avoiding to add in too many units.
+
+<table><tr><td>Algorithm3Dynamic Network Expansion</td></tr><tr><td>Input: Datatset Dt,Threshold T</td></tr><tr><td>Perform Algorithm 2 and compute L</td></tr><tr><td>if&gt;Tthen</td></tr><tr><td>Add k units hat all layers</td></tr><tr><td>Solve for Eq. 5 at all layers</td></tr><tr><td>forl= L-1,...,1 do</td></tr><tr><td>Remove useless units in h</td></tr></table>
+
+Network Split/Duplication. A crucial challenge in lifelong learning is the problem of semantic drift, or catastrophic forgetting, which describes the problem where the model gradually fits to the later learned tasks and thus forgets what it learned for earlier tasks, resulting in degenerate performance for them. The most popular yet simple way of preventing semantic drift is to regularize the parameters from deviating too much from its original values using $\ell _ { 2 }$ -regularization, as follows:
+
+$$
+\operatorname* { m i n i m i z e } _ { W ^ { t } } \mathcal { L } ( W ^ { t } ; \mathcal { D } _ { t } ) + \lambda \| W ^ { t } - W ^ { t - 1 } \| _ { 2 } ^ { 2 }
+$$
+
+where $t$ is the current task, and $W ^ { t - 1 }$ is the weight tensor of the network trained for tasks $\{ 1 , \ldots , t -$ $1 \}$ , and $\lambda$ is the regularization parameter. This $\ell _ { 2 }$ regularization will enforce the solution $\mathbf { \mathbf { \mathbf { \mathbf { W } } } ^ { t } }$ to be found close to $\pmb { W } ^ { \top - 1 }$ , by the degree given by $\lambda$ ; if $\lambda$ is small, then the network will be learned to reflect the new task more while forgetting about the old tasks, and if $\lambda$ is high, then $\mathbf { \mathbf { } } W ^ { t }$ will try to preserve the knowledge learned at previous tasks as much as possible. Rather than placing simple $\ell _ { 2 }$ regularization, it is also possible to weight each element with Fisher information (Kirkpatrick et al., 2017). Nonetheless, if number of tasks is large, or if the later tasks are semantically disparate from the previous tasks, it may become difficult to find a good solution for both previous and new tasks.
+
+A better solution in such a case, is to split the neuron such that we have features that are optimal for two different tasks. After performing Eq. 6, we measure the amount of semantic drift for each hidden unit $i$ , $\rho _ { i } ^ { t }$ , as the $\ell _ { 2 }$ -distance between the incoming weights at $t \mathrm { - } 1$ and at $t$ . Then if $\rho _ { i } ^ { t } > \sigma$ we consider that the meaning of the feature have significantly changed during training, and split this neuron $i$ into two copies (properly introducing new edges from and to duplicate). This operation can be performed for all hidden units in parallel. After this duplication of the neurons, the network needs to train the weights again by solving Eq. 6 since split changes the overall structure. However, in practice this secondary training usually converges fast due to the reasonable parameter initialization from the initial training. Algorithm 4 describes the algorithm for split operation.
+
+<table><tr><td>Algorithm 4 Network Split/Duplication</td></tr><tr><td>Input: Weight Wt-1, Threshold σ</td></tr><tr><td></td></tr><tr><td>Perform Eq.6 to obtain Wt forall hidden uniti do</td></tr><tr><td>p=|w-w-12</td></tr><tr><td>if p&gt;σthen</td></tr><tr><td>Copy i into i’ (w&#x27; introduction of edges for i&#x27;)</td></tr><tr><td>Perform Eq. 6 with the initialization of Wt to obtain Wt</td></tr></table>
+
+Timestamped Inference. In both the network expansion and network split procedures, we timestamp each newly added unit $j$ by setting $\{ z \} _ { j } = t$ to record the training stage $t$ when it is added to the network, to further prevent semantic drift caused by the introduction of new hidden units. At inference time, each task will only use the parameters that were introduced up to stage $t$ , to prevent the old tasks from using new hidden units added in the training process. This is a more flexible strategy than fixing the weights learned up to each learning stage as in Rusu et al. (2016), since early tasks can still benefit from the learning at later tasks, via units that are further trained, but not split.
+
+# 4 EXPERIMENT
+
+Baselines and our model. 1) DNN-STL. Base deep neural network, either feedforward or convolutional, trained for each task separately.
+
+2) DNN-MTL. Base DNN trained for all tasks at once.
+
+3) DNN. Base DNN. All incremental models use $\ell _ { 2 }$ -regularizations.
+
+3) DNN-L2. Base DNN, where at each task $t$ , $\mathbf { \mathbf { } } W ^ { t }$ is initialized as $W ^ { t - 1 }$ and continuously trained with $\ell _ { 2 }$ -regularization between $\mathbf { \mathbf { \mathbf { \mathbf { W } } } ^ { t } }$ and $W ^ { t - 1 }$ .
+
+4) DNN-EWC. Deep network trained with elastic weight consolidation (Kirkpatrick et al., 2017) for regularization.
+
+5) DNN-Progressive. Our implementation of the progressive network (Rusu et al., 2016), whose network weights for each task remain fixed for the later arrived tasks.
+
+6) DEN. Our dynamically expandable network.
+
+Base network settings. 1) Feedforward networks: We use a two-layer network with 312-128 neurons with ReLU activations. 2) Convolutional networks. For experiments on the CIFAR-100 dataset, we use a modified version of AlexNet (Krizhevsky et al., 2012) that has five convolutional layers (64-128-256-256-128 depth with $5 \times 5$ filter size), and three fully-connected layers (384-192- 100 neurons at each layer).
+
+All models and algorithms are implemented using the Tensorflow (Abadi et al., 2016) library. We will release our codes upon acceptance of our paper, for reproduction of the results.
+
+Datasets. 1) MNIST-Variation. This dataset consists of 62, 000 images of handwritten digits from 0 to 9. Unlike MNIST, the handwritten digits are rotated to arbitrary angles and has noise in the background, which makes the prediction task more challenging. We use $\bar { 1 , 0 0 0 / 2 0 0 / 5 , 0 0 0 }$ 0 images for train/val/test split for each class. We form each task to be one-versus-rest binary classification.
+
+2) CIFAR-100. This dataset consists of 60, 000 images of 100 generic object classes(Krizhevsky & Hinton, 2009). Each class has 500 images for training and 100 images for test. We used a CNN as the base network for the experiments on this dataset, to show that our method is applicable to a CNN. Further, we considered each task as a set of 10 subtasks, each of which is a binary classification task on each class.
+
+3) AWA (Animals with Attributes). This dataset consists of 30, 475 images of 50 animals (Lampert et al., 2009). For features, we use DECAF features provided with the dataset, whose dimensionality is reduced to 500 by PCA. We use random splits of 30/30/30 images for training/validation/test.
+
+# 4.1 QUANTITATIVE EVALUATION
+
+We validate our models for both prediction accuracy and efficiency, where we measure the efficiency by network size at the end of training and training time. We first report the average per-task performance of baselines and our models in the top row of Figure 3. DNN-STL showed best performances on AWA and CIFAR-100 dataset; they are expected to perform well, since they are trained to be optimal for each task, while all other models are trained online which might cause semantic drift. When the number of tasks is small, MTL works the best from knowledge sharing via multi-task learning, but when the number of tasks is large, STL works better since it has larger learning capacity than MTL. Our model, DEN, performs almost the same as these batch models, and even outperforms them on MNIST-Variation dataset. Retraining models combined with regularization, such as L2 and EWC do not perform well, although the latter outperforms the former. This is expected as the two models cannot dynamically increase their capacity. Progressive network works better than the two, but it underperforms DEN on all datasets. The performance gap is most significant on AWA, as larger number of tasks $T = 5 0$ ) may have made it more difficult to find the appropriate network capacity. If the network is too small, then it will not have sufficient learning capacity to represent new tasks, and if too large, it will become prone to overfitting.
+
+![](images/c22c03394cf776bcd0d0798d019152606bd8a0068a8b039b8b2a42afd060af87.jpg)  
+Figure 3: Top row:Average per-task performance of the models over number of task $t$ , averaged over five random splits. The numbers in the legend denote average per-task performance after the model has finished learning $t = T$ ). Bottom row: Accuracy over network capacity. The network capacity is given relative to the capacity of MTL, which we consider as $1 0 0 \%$ .
+
+![](images/803e536fdc7a47a07ece54f0aae03adb5021656eadcabdd95711605d85cf0dc8.jpg)  
+Figure 4: Effect of selective retraining. (a) shows AUROC over actual training time and (b) shows the number of selected neurons by selective retraining. (c) Expansion performance. We report both the prediction AUROC and network capacity measured by the relative number of parameters to that of DNN-MTL on MNIST-Variance dataset. Reported numbers are mean and standard error for five random splits.
+
+We further report the performance of each model over network capacity measured relative to MTL on each dataset, in Figure 3 (bottom row). For baselines, we report the performance of multiple models with different network capacity. DEN obtains much better performance with substantially fewer number of parameters than Progressive network or obtain significantly better performance using similar number of parameters. DEN also obtains the same level of performance as STL using only $1 8 . 0 \%$ , $6 0 . 3 \%$ , and $1 1 . 9 \%$ of its capacity on MNIST-Variation, CIFAR-100, and AWA respectively. This also shows the main advantage of DEN, which is being able to dynamically find its optimal capacity, since it learns a very compact model on MNIST-Variation, whilst learning a substantially large network on CIFAR-100. Further fine-tuning of DEN on all tasks (DEN-Finetune) obtains the best performing model on all datasets, which shows that DEN is not only useful for lifelong learning, but can be also used for network capacity estimation when all tasks are available in the first place.
+
+Effect of selective retraining. We further examine how efficient and effective the selective training is, by measuring the training speed and the area under ROC curve on MNIST-Variation dataset. To this end, we compare the model without network expansion, which we refer to as DNN-Selective, against retraining on DNN-L2 and DNN-L1 (Eq.(2)), for both the accuracy and efficiency. Figure 4(a) shows both the accuracy over training time measured as actual time spent with GPU computation, for each model. We observe that selective retraining takes significantly less time than the full retraining of the network, and even less than DNN-L1 that comes with sparse network weights. Further, whereas DNN-L1 obtained slightly less accuracy than DNN-L2, DNN-Selective improves the accuracy over the base network by $2 \% p$ . This accuracy gain may be due to the suppression of catastrophic forgetting, as DEN trains only a partial subnetwork for each newly introduced task. Figure 4(b) shows the number of selected neurons at each layer with selective retraining. Note that DNN-selective mostly selects less portion of upper level units which are more task-specific, while selecting larger portion of more generic lower layer units.
+
+![](images/f2865983a15b8e4cd7f9cc4d2e727d7b8aed31f3471a17a503159a933a6c6575.jpg)  
+Figure 5: Semantic drift experiment on the MNIST-Variation dataset. We report the AUROC of different models on $t = 1$ , $t = 4$ , and $t = 7$ at each training stage to see how the model performance changes over time for these tasks. Reported AUROC is the average over five random splits.
+
+Effect of network expansion. We also compare the effectiveness of the network expansion against the model with a variant of our model that does selective retraining and layer expansion, but without network split. We refer to this model as DNN-Dynamic. We compare DNN-Dynamic with DNN-L2 used in the main experiment, and DNN-Constant, which is a version of our model that expands its capacity at each layer with fixed number of units, on MNIST-Variation dataset. Figure 4(c) shows the experimetal results. DNN-Dynamic obtains the best mean AU-ROC, significantly outperforming all models including DNN-Constant, while increasing the size of the network substantially less than DNN-Constant $( \mathrm { k } { = } 2 0 )$ ). This may be because having less number of parameters is not only beneficial in terms of training efficiency, but also advantageous in preventing the model from overfitting. We can set the network capacity of DNN-Constant to be similar $( \mathrm { k } { = } 1 3 )$ ) to obtain better accuracy, but it still underperforms DEN which can dynamically adjust the number of neurons at each layer.
+
+Effect of network split/duplication and timestamped inference. To see how network split/duplication and unit timestamping help prevent semantic drift (or catastrophic forgetting), while allowing to obtain good performances on later tasks, we compare the performance of our model against baselines and also a variant of our DEN without timestamped inference (DEN-No-Stamp) at different learning stages. Each figure in Figure 5 (a), (b), and (c) shows how the performance of the model changes at each training stage $t$ , for tasks $\scriptstyle t = 1$ , $\scriptstyle t = 4$ , and $t { = } 7$ .
+
+We observe that DNN-L2 prevents semantic drift of the models learned at early stages, but results in increasingly worse performance on later tasks $( t { = } 4 , 7 )$ . DNN-EWC, on the other hand, has better performance on later tasks than DNN-L2, as reported in Kirkpatrick et al. (2017). However, it shows significantly lower performance than both DNN-Progressive and our model, which may be due to its inability to increase network capacity, that may result in limited expressive power. DNN-Progressive shows no semantic drift on old tasks, which is expected because it does not retrain parameters for them. DEN w/o Timestamping works better than DNN-Progressive on later tasks, with slight performance degeneration over time. Finally, our full model with timestamped inference, DEN, shows no sign of noticeable performance degeneration at any learning stage, while significantly outperforming DNN-Progressive. This results show that DEN is highly effective in preventing semantic drift as well.
+
+# 5 CONCLUSION
+
+We proposed a novel deep neural network for lifelong learning, Dynamically Expandable Network (DEN). DEN performs partial retraining of the network trained on old tasks by exploiting task relatedness, while increasing its capacity when necessary to account for new knowledge required to account for new tasks, to find the optimal capacity for itself, while also effectively preventing semantic drift. We implement both feedforward and convolutional neural network version of our DEN, and validate them on multiple classification datasets under lifelong learning scenarios, on which they significantly outperform the existing lifelong learning methods, achieving almost the same performance as the network trained in batch while using as little as $1 1 . 9 \% p - 6 0 . 3 \% p$ of its capacity. Further fine-tuning of the models on all tasks results in obtaining models that outperform the batch models, which shows that DEN is useful for network structure estimation as well.
+
+Acknowledgements This research was supported by Next-Generation Information Computing Development Program through the National Research Foundation of Korea of the Ministry of Science, ICT & Future Planning (NRF-2016M3C4A7952600), Samsung Research Funding Center of Samsung Electronics (SRFC-IT150203), and the ICT R&D program of MSIP/IITP (2016-0-00563, Research on Adaptive Machine Learning Technology Development for Intelligent Autonomous Digital Companion, and 2017-0-00537, Development of Autonomous IoT Collaboration Framework for Space Intelligence).
+
+# REFERENCES
+
+Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale Machine Learning on Heterogeneous Distributed Systems. arXiv:1603.04467, 2016.
+
+Jose M Alvarez and Mathieu Salzmann. Learning the number of neurons in deep networks. In Advances in Neural Information Processing Systems, pp. 2262–2270, 2016.
+
+Corinna Cortes, Xavi Gonzalvo, Vitaly Kuznetsov, Mehryar Mohri, and Scott Yang. Adanet: Adaptive structural learning of artificial neural networks. arXiv preprint arXiv:1607.01097, 2016.
+
+Eric Eaton and Paul L. Ruvolo. ELLA: An efficient lifelong learning algorithm. In Sanjoy Dasgupta and David Mcallester (eds.), ICML, volume 28, pp. 507–515. JMLR Workshop and Conference Proceedings, 2013.
+
+James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei A Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Grabska-Barwinska, et al. Overcoming catastrophic forgetting in neural networks. Proceedings of the National Academy of Sciences, pp. 201611835, 2017.
+
+Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. 2009.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton. ImageNet Classification with Deep Convolutional Neural Networks. In NIPS, 2012.
+
+Abhishek Kumar and Hal Daume III. Learning task grouping and overlap in multi-task learning. In ICML, 2012.
+
+Christoph Lampert, Hannes Nickisch, and Stefan Harmeling. Learning to Detect Unseen Object Classes by Between-Class Attribute Transfer. In CVPR, 2009.
+
+Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based Learning Applied to Document Recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+
+Sang-Woo Lee, Jin-Hwa Kim, Jung-Woo Ha, and Byoung-Tak Zhang. Overcoming catastrophic forgetting by incremental moment matching. arXiv preprint arXiv:1703.08475, 2017.
+
+George Philipp and Jaime G. Carbonell. Nonparametric neural networks. In ICLR, 2017.
+
+Andrei Rusu, Neil Rabinowitz, Guillaume Desjardins, Hubert Soyer, James Kirkpatrick, Koray Kavukcuoglu, Razvan Pascanu, and Raia Hadsell. Progressive neural networks. arXiv preprint arXiv:1606.04671, 2016.
+
+S. Thrun. A lifelong learning perspective for mobile robot control. In V. Graefe (ed.), Intelligent Robots and Systems. Elsevier, 1995.
+
+Wei Wen, Chunpeng Wu, Yandan Wang, Yiran Chen, and Hai Li. Learning structured sparsity in deep neural networks. In NIPS, pp. 2074–2082, 2016.
+
+Tianjun Xiao, Jiaxing Zhang, Kuiyuan Yang, Yuxin Peng, and Zheng Zhang. Error-driven incremental learning in deep convolutional neural network for large-scale image classification. In Proceedings of the 22nd ACM international conference on Multimedia, pp. 177–186. ACM, 2014.
+
+Friedemann Zenke, Ben Poole, and Surya Ganguli. Continual learning through synaptic intelligence. In ICML, pp. 3987–3995, 2017.
+
+Guanyu Zhou, Kihyuk Sohn, and Honglak Lee. Online incremental feature learning with denoising autoencoders. In International Conference on Artificial Intelligence and Statistics, pp. 1453–1461, 2012.
+
+# APPENDIX A
+
+# A.1 RESULTS ON PERMUTED MNIST
+
+We provide additional experimental results on Permuted MNIST dataset Lecun et al. (1998). This dataset consists of 70, 000 images of handwritten digits from 0 to 9, where 60, 000 images are used for training, and 10, 000 images for test. The difference of this dataset from the original MNIST is that each of the ten tasks is the multi-class classification of a different random permutation of the input pixels. Figure 6 shows the results of this experiment. Our DEN outperforms all lifelong learning baselines while using only 1.39 times of base network capacity. Further, DEN-Finetune achieves the best AUROC among all models, including DNN-STL and DNN-MTL.
+
+![](images/9c2dfb6bb52462b10583b22796642314cbb6c58caca19eef0215078cd13c419b.jpg)  
+Figure 6: Results on the Permuted MNIST. Average per-task AUROC and network capacity of all models relative to MTL on Permuted MNIST.

parse/train/r1GAsjC5Fm/r1GAsjC5Fm.md

@@ -0,0 +1,335 @@
+# SELF-MONITORING NAVIGATION AGENT VIA AUXILIARY PROGRESS ESTIMATION
+
+Chih-Yao $\mathbf { M _ { a } } { * } _ { } ^ { + }$ , Jiasen $\mathbf { L } \mathbf { u } ^ { * \dagger }$ , Zuxuan $\mathbf { W _ { u } } ^ { * \dagger }$ , Ghassan AlRegib†, Zsolt Kira†, Richard Socher§ & Caiming Xiong§
+
+†Georgia Institute of Technology   
+{cyma,jiasenlu,alregib,zkira}@gatech.edu   
+‡University of Maryland, College Park   
+{zxwu}@cs.umd.edu   
+§Salesforce Research   
+{rsocher,cxiong}@salesforce.com
+
+# ABSTRACT
+
+The Vision-and-Language Navigation (VLN) task entails an agent following navigational instruction in photo-realistic unknown environments. This challenging task demands that the agent be aware of which instruction was completed, which instruction is needed next, which way to go, and its navigation progress towards the goal. In this paper, we introduce a self-monitoring agent with two complementary components: (1) visual-textual co-grounding module to locate the instruction completed in the past, the instruction required for the next action, and the next moving direction from surrounding images and (2) progress monitor to ensure the grounded instruction correctly reflects the navigation progress. We test our selfmonitoring agent on a standard benchmark and analyze our proposed approach through a series of ablation studies that elucidate the contributions of the primary components. Using our proposed method, we set the new state of the art by a significant margin ( $8 \%$ absolute increase in success rate on the unseen test set). Code is available at https://github.com/chihyaoma/selfmonitoring-agent.
+
+# 1 INTRODUCTION
+
+Recently, the Vision-and-Language (VLN) navigation task (Anderson et al., 2018b), which requires the agent to follow natural language instructions to navigate through a photo-realistic unknown environment, has received significant attention (Wang et al., 2018b; Fried et al., 2018). In the VLN task, an agent is placed in an unknown realistic environment and is required to follow natural language instructions to navigate from its starting location to a target location. In contrast to some existing navigation tasks (Kempka et al., 2016; Zhu et al., 2017; Mirowski et al., 2017; 2018), we address the class of tasks where the agent does not have an explicit representation of the target (e.g., location in a map or image representation of the goal) to know if the goal has been reached or not (Matuszek et al., 2013; Hemachandra et al., 2015; Duvallet et al., 2016; Arkin et al., 2017). Instead, the agent needs to be aware of its navigation status through the association between the sequence of observed visual inputs to instructions.
+
+Consider an example as shown in Fig. 1, given the instruction ”Exit the bedroom and go towards the table. Go to the stairs on the left of the couch. Wait on the third step.”, the agent first needs to locate which instruction is needed for the next movement, which in turn requires the agent to be aware of (i.e., to explicitly represent or have an attentional focus on) which instructions were completed or ongoing in the previous steps. For instance, the action ”Go to the stairs” should be carried out once the agent has exited the room and moved towards the table. However, there exists inherent ambiguity for ”go towards the table”. Intuitively, the agent is expected to ”Go to the stairs” after completing ”go towards the table”. But, it is not clear what defines the completeness of ”Go towards the table”. The completeness of an ongoing action often depends on the availability of the next action. Since the transition between past and next part of the instructions is a soft boundary, in order to determine when to transit and to follow the instruction correctly the agent is required to keep track of both grounded instructions. On the other hand, assessing the progress made towards the goal has indeed been shown to be important for goal-directed tasks in humans decision-making (Benn et al., 2014; Chatham et al., 2012; Berkman & Lieberman, 2009). While a number of approaches have been proposed for VLN (Anderson et al., 2018b; Wang et al., 2018b; Fried et al., 2018), previous approaches generally are not aware of which instruction is next nor progress towards the goal; indeed, we qualitatively show that even the attentional mechanism of the baseline does not successfully track this information through time.
+
+![](images/03e64e937c93bc83cd207099a38a596235846717160705fb06d7cb2fb36d202e.jpg)  
+Figure 1: Vision-and-Language Navigation task and our proposed self-monitoring agent. The agent is constantly aware of what was completed, what is next, and where to go, as it navigates through unknown environments by following navigational instructions.
+
+In this paper, we propose an agent endowed with the following abilities: (1) identify which direction to go by finding the part of the instruction that corresponds to the observed images—visual grounding, (2) identify which part of the instruction has been completed or ongoing and which part is potentially needed for the next action selection—textual grounding, and (3) ensure that the grounded instruction can correctly be used to estimate the progress made towards the goal, and apply regularization to ensure this —progress monitoring. Therefore, we introduce the self-monitoring agent consisting of two complementary modules: visual-textual co-grounding and progress monitor.
+
+More specifically, we achieve both visual and textual grounding simultaneously by incorporating the full history of grounded instruction, observed images, and selected actions into the agent. We leverage the structural bias between the words in instructions used for action selection and progress made towards the goal and propose a new objective function for the agent to measure how well it can estimate the completeness of instruction-following. We then demonstrate that by conditioning on the positions and weights of grounded instruction as input, the agent can be self-monitoring of its progress and further ensure that the textual grounding accurately reflects the progress made.
+
+Overall, we propose a novel self-monitoring agent for VLN and make the following contributions: (1) We introduce the visual-textual co-grounding module, which performs grounding interdependently across both visual and textual modalities. We show that it can outperform the baseline method by a large margin. (2) We propose to equip the self-monitoring agent with a progress monitor, and for navigation tasks involving instructions instantiate this by introducing a new objective function for training. We demonstrate that, unlike the baseline method, the position of grounded instruction can follow both past and future instructions, thereby tracking progress to the goal. (3) With the proposed self-monitoring agent, we set the new state-of-the-art performance on both seen and unseen environments on the standard benchmark. With $8 \%$ absolute improvement in success rate on the unseen test set, we are ranked #1 on the challenge leaderboard.
+
+# 2 SELF-MONITORING NAVIGATION AGENT
+
+# 2.1 NOTATION
+
+Given a natural language instruction with $L$ words, its representation is denoted by $\begin{array} { r l } { \boldsymbol { X } } & { { } = } \end{array}$ $\left\{ { \pmb x } _ { 1 } , { \pmb x } _ { 2 } , \ldots , { \pmb x } _ { L } \right\}$ , where $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } l }$ is the feature vector for the $l$ -th word encoded by an LSTM language encoder. Following Fried et al. (2018), we enable the agent with panoramic view. At each time step, the agent perceives a set of images at each viewpoint $\pmb { v } _ { t } = \left\{ \pmb { v } _ { t , 1 } , \pmb { v } _ { t , 2 } , . . . , \pmb { v } _ { t , K } \right\}$ , where $K$ is the maximum number of navigable directions1, and $\mathbf { \Delta } \mathbf { v } _ { t , k }$ represents the image feature of direction $k$ . The co-grounding feature of instruction and image are denoted as $\hat { \mathbf { x } } _ { t }$ and $\hat { \pmb { v } } _ { t }$ respectively. The selected action is denoted as $\mathbf { a } _ { t }$ . The learnable weights are denoted with $W$ , with appropriate sub/super-scripts as necessary. We omit the bias term $^ { b }$ to avoid notational clutter in the exposition.
+
+![](images/0fd43ff1d97d0ffc067684d152dcc9d33b4df237f90987f3c55fb7725dbd7362.jpg)  
+Figure 2: Proposed self-monitoring agent consisting of visual-textual co-grounding, progress monitoring, and action selection modules. Textual grounding: identify which part of the instruction has been completed or ongoing and which part is potentially needed for next action. Visual grounding: summarize the observed surrounding images. Progress monitor: regularize and ensure grounded instruction reflects progress towards the goal. Action selection: identify which direction to go.
+
+# 2.2 VISUAL AND TEXTUAL CO-GROUNDING
+
+First, we propose a visual and textual co-grounding model for the vision and language navigation task, as illustrated in Fig. 2. We model the agent with a sequence-to-sequence architecture with attention by using a recurrent neural network. More specifically, we use Long Short Term Memory (LSTM) to carry the flow of information effectively. At each step $t$ , the decoder observes representations of the current attended panoramic image feature $\hat { \mathbf { } } _ { }$ , previous selected action $\mathbf { a } _ { t - 1 }$ and current grounded instruction feature $\hat { \mathbf { x } } _ { t }$ as input, and outputs an encoder context $\boldsymbol { h } _ { t }$ :
+
+$$
+\begin{array} { r } { \pmb { h } _ { t } = L S T M \big ( \big [ \hat { \pmb { x } } _ { t } , \hat { \pmb { v } } _ { t } , \mathbf { a } _ { t - 1 } \big ] \big ) } \end{array}
+$$
+
+where $[ , ]$ denotes concatenation. The previous encoder context $\boldsymbol { h } _ { t - 1 }$ is used to obtain the textual grounding feature $\hat { \mathbf { x } } _ { t }$ and visual grounding feature $\hat { \pmb { v } } _ { t }$ , whereas we use current encoder context $\boldsymbol { h } _ { t }$ to obtain next action $\mathbf { a } _ { t }$ , all of which will be illustrated in the rest of the section.
+
+Textual grounding. When the agent moves from one viewpoint to another, it is required to identify which direction to go by relying on a grounded instruction, i.e. which parts of the instruction should be used. This can either be the instruction matched with the past (ongoing action) or predicted for the future (next action). To capture the relative position between words within an instruction, we incorporate the positional encoding $P E ( \cdot )$ (Vaswani et al., 2017) into the instruction features. We then perform soft-attention on the instruction features $\boldsymbol { X }$ , as shown on the left side of Fig. 2. The attention distribution over $L$ words of the instructions is computed as:
+
+$$
+\begin{array} { r } { z _ { t , l } ^ { \mathrm { t e x t u a l } } = ( W _ { x } { h _ { t - 1 } } ) ^ { \top } P E ( { x _ { l } } ) , \quad \mathrm { a n d } \quad \alpha _ { t } = \mathrm { s o f t m a x } ( z _ { t } ^ { \mathrm { t e x t u a l } } ) , } \end{array}
+$$
+
+wherword $W _ { x }$ are parameters to be learnt. he instruction and previous h $z _ { t , l } ^ { \mathrm { t e x t u a l } }$ is astate r val, and omputed as the correlation betweenis the attention weight over features $l$ $\boldsymbol { h } _ { t - 1 }$ $\alpha _ { t }$   
+in $\boldsymbol { X }$ at time $t$ . Based on the textual attention distribution, the grounded textual feature $\hat { \mathbf { x } } _ { t }$ can be   
+obtained by the weighted sum over the textual features $\hat { \pmb { x } } _ { t } = { \pmb { \alpha } } _ { t } ^ { T } { \pmb { X } }$ .
+
+Visual grounding. In order to locate the completed or ongoing instruction, the agent needs to keep track of the sequence of images observed along the navigation trajectory. We thus perform visual attention over the surrounding views based on its previous hidden vector $\boldsymbol { h } _ { t - 1 }$ . The visual attention weight $\beta _ { t }$ can be obtained as:
+
+$$
+z _ { t , k } ^ { \mathrm { v i s u a l } } = ( W _ { v } \boldsymbol { h } _ { t - 1 } ) ^ { \top } \boldsymbol { g } ( v _ { t , k } ) , \quad \mathrm { a n d } \quad \beta _ { t } = \mathrm { s o f t m a x } ( z _ { t } ^ { \mathrm { v i s u a l } } ) ,
+$$
+
+where $g$ is a one-layer Multi-Layer Perceptron (MLP), $W _ { v }$ are parameters to be learnt. Similar to Eq. 2, the grounded visual feature $\hat { \mathbf { } v } _ { t }$ can be obtained by the weighted sum over the visual features $\hat { \pmb { v } } _ { t } = \beta _ { t } ^ { T } \bar { \pmb { V } }$ .
+
+Action selection. To make a decision on which direction to go, the agent finds the image features on navigable directions with the highest correlation with the grounded navigation instruction $\hat { \mathbf { x } } _ { t }$ and the current hidden state $h _ { t }$ . We use the inner-product to compute the correlation, and the probability of each navigable direction is then computed as:
+
+$$
+o _ { t , k } = ( W _ { a } [ { h _ { t } } , \hat { { \boldsymbol { x } } } _ { t } ] ) ^ { \top } g ( { v _ { t , k } } ) \quad \mathrm { a n d } \quad p _ { t } = \mathrm { s o f t m a x } ( o _ { t } ) ,
+$$
+
+where $W _ { a }$ are the learnt parameters, $g ( \cdot )$ is the same MLP as in Eq. 3, and ${ \pmb p } _ { t }$ is the probability of each navigable direction at time $t$ . We use categorical sampling during training to select the next action $\mathbf { a } _ { t }$ . Unlike the previous method with the panoramic view (Fried et al., 2018), which attends to instructions only based on the history of observed images, we achieve both textual and visual grounding using the shared hidden state output containing grounded information from both textual and visual modalities. During action selection, we rely on both hidden state output and grounded instruction, instead of only relying on grounded instruction.
+
+# 2.3 PROGRESS MONITOR
+
+It is imperative that the textual-grounding correctly reflects the progress towards the goal, since the agent can then implicitly know where it is now and what the next instruction to be completed will be. In the visual-textual co-grounding module, we ensure that the grounded instruction reasonably informs decision making when selecting a navigable direction. This is necessary but not sufficient for ensuring that the notion of progress to the goal is encoded. Thus, we propose to equip the agent with a progress monitor that serves as regularizer during training and prunes unfinished trajectories during inference.
+
+Since the positions of localized instruction can be a strong indication of the navigation progress due to the structural alignment bias between navigation steps and instruction, the progress monitor can estimate how close the current viewpoint is to the final goal by conditioning on the positions and weights of grounded instruction. This can further enforce the result of textual-grounding to align with the progress made towards the goal and to ensure the correctness of the textual-grounding.
+
+The progress monitor aims to estimate the navigation progress by conditioning on three inputs: the history of grounded images and instructions, the current observation of the surrounding images, and the positions of grounded instructions. We therefore represent these inputs by using (1) the previous hidden state $\boldsymbol { h } _ { t - 1 }$ and the current cell state $\mathbf { } c _ { t }$ of the LSTM, (2) the grounded surrounding images $\hat { \pmb { v } } _ { t }$ , and (3) the distribution of attention weights of textual-grounding $\pmb { \alpha } _ { t }$ , as shown at the bottom of Fig. 2 represented by dotted lines.
+
+Our proposed progress monitor first computes an additional hidden state output $h _ { t } ^ { p m }$ by using grounded image representations $\hat { \pmb { v } } _ { t }$ as input, similar to how a regular LSTM computes hidden states except we use concatenation over element-wise addition for empirical reasons2. The hidden state output is then concatenated with the attention weights $\pmb { \alpha } _ { t }$ on textual-grounding to estimate how close the agent is to the $\mathrm { g o a l } ^ { 3 }$ . The output of the progress monitor $p _ { t } ^ { p m }$ , which represents the completeness of instruction-following, is computed as:
+
+$$
+\begin{array} { r } { \begin{array} { l l } { h _ { t } ^ { p m } = \sigma ( W _ { h } ( [ h _ { t - 1 } , \hat { v } _ { t } ] ) \otimes t a n h ( c _ { t } ) ) , } & { p _ { t } ^ { p m } = t a n h ( W _ { p m } ( [ \alpha _ { t } , h _ { t } ^ { p m } ] ) ) } \end{array} } \end{array}
+$$
+
+where $W _ { h }$ and $W _ { p m }$ are the learnt parameters, $c _ { t }$ is the cell state of the LSTM, $\otimes$ denotes the element-wise product, and $\sigma$ is the sigmoid function.
+
+# 2.4 TRAINING AND INFERENCE
+
+Training. We introduce a new objective function to train the proposed progress monitor. The training target $y _ { t } ^ { p m }$ is defined as the normalized distance in units of length from the current viewpoint to the goal, i.e., the target will be 0 at the beginning and closer to 1 as the agent approaches the goal4. Note that the target can also be lower than 0, if the agent’s current distance from the goal is farther than the starting point. Finally, our self-monitoring agent is optimized with a cross-entropy loss and a mean squared error loss, computed with respect to the outputs from both action selection and progress monitor.
+
+$$
+\mathcal { L } _ { l o s s } = - \lambda \sum _ { t = 1 } ^ { T } y _ { t } ^ { n v } l o g ( p _ { k , t } ) - ( 1 - \lambda ) \sum _ { t = 1 } ^ { T } ( y _ { t } ^ { p m } - p _ { t } ^ { p m } ) ^ { 2 }
+$$
+
+where ${ p } _ { k , t }$ is the action probability of each navigable direction, $\lambda = 0 . 5$ is the weight balancing the two losses, and $y _ { t } ^ { n v }$ is the ground-truth navigable direction at step $t$ .
+
+Inference. During inference, we follow Fried et al. (2018) by using beam search. we propose that, while the agent decides which trajectories in the beams to keep, it is equally important to evaluate the state of the beams on actions as well as on the agent’s confidence in completing the given instruction at each traversed viewpoint. We accomplish this idea by integrating the output of our progress monitor into the accumulated probability of beam search. At each step, when candidate trajectories compete based on accumulated probability, we integrate the estimated completeness of instruction-following $p _ { t } ^ { p m }$ (normalized between 0 to 1) with action probability ${ p } _ { k , t }$ to directly evaluate the partial and unfinished candidate routes: $p _ { t } ^ { b e a m } = p _ { t } ^ { p m } \times p _ { k , t }$ .
+
+Without beam search, we use greedy decoding for action selection with one condition. If the progress monitor output decreases $( p _ { t + 1 } ^ { p m } < \dot { p } _ { t } ^ { p m } )$ , the agent is required to move back to the previous viewpoint and select the action with next highest probability. We repeat this process until the selected action leads to increasing progress monitor output. We denote this procedure as progress inference.
+
+# 3 EXPERIMENTS
+
+R2R Dataset. We use the Room-to-Room (R2R) dataset (Anderson et al., 2018b) for evaluating our proposed approach. The R2R dataset is built upon the Matterport3D dataset (Chang et al., 2017) and has 7,189 paths sampled from its navigation graphs. Each path has three ground-truth navigation instructions written by humans. The whole dataset is divided into 4 sets: training, validation seen, validation unseen, and test sets unseen.
+
+Evaluation metrics. We follow the same evaluation metrics used by previous work on the R2R task: (1) Navigation Error (NE), mean of the shortest path distance in meters between the agent’s final position and the goal location. (2) Success Rate (SR), the percentage of final positions less than $3 \mathrm { m }$ away from the goal location. (3) Oracle Success Rate (OSR), the success rate if the agent can stop at the closest point to the goal along its trajectory. In addition, we also include the recently introduced Success rate weighted by (normalized inverse) Path Length (SPL) (Anderson et al., 2018a), which trades-off Success Rate against trajectory length.
+
+Table 1: Performance comparison with the state of arts: Student-forcing (Anderson et al., 2018b), RPA (Wang et al., 2018b), and Speaker-Follower (Fried et al., 2018). \*: with data augmentation. leaderboard: when using beam search, we modify our search procedure to comply with the leaderboard guidelines, i.e., all traversed viewpoints are recorded.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">Validation-Seen</td><td colspan="4">Validation-Unseen</td><td colspan="4">Test (unseen)</td></tr><tr><td>NE↓</td><td>SR个</td><td>OSR个</td><td>SPL个</td><td>NE↓</td><td>SR个</td><td>OSR个</td><td>SPL个</td><td>NE↓</td><td>SR↑</td><td>OSR个</td><td>SPL个</td></tr><tr><td>Random</td><td>9.45</td><td>0.16</td><td>0.21</td><td>1</td><td>9.23</td><td>0.16</td><td>0.22</td><td></td><td>9.77</td><td>0.13</td><td>0.18</td><td>-</td></tr><tr><td>Student-forcing</td><td>6.01</td><td>0.39</td><td>0.53</td><td>1</td><td>7.81</td><td>0.22</td><td>0.28</td><td>=</td><td>7.85</td><td>0.20</td><td>0.27</td><td>=</td></tr><tr><td>RPA</td><td>5.56</td><td>0.43</td><td>0.53</td><td>-</td><td>7.65</td><td>0.25</td><td>0.32</td><td>=</td><td>7.53</td><td>0.25</td><td>0.33</td><td>=</td></tr><tr><td>Speaker-Follower</td><td>3.88</td><td>0.63</td><td>0.71</td><td>-</td><td>5.24</td><td>0.50</td><td>0.63</td><td>-</td><td>-</td><td>1</td><td>1</td><td>=</td></tr><tr><td>Speaker-Follower*</td><td>3.08</td><td>0.70</td><td>0.78</td><td>-</td><td>4.83</td><td>0.55</td><td>0.65</td><td>-</td><td>4.87</td><td>0.53</td><td>0.64</td><td>1</td></tr><tr><td>(leaderboard)</td><td>-</td><td>-</td><td>1</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>4.87</td><td>0.53</td><td>0.96</td><td>0.01</td></tr><tr><td>Ours (beam search)</td><td>3.23</td><td>0.70</td><td>0.78</td><td>0.66</td><td>5.04</td><td>0.57</td><td>0.70</td><td>0.51</td><td>4.99</td><td>0.57</td><td>0.68</td><td>0.51</td></tr><tr><td>(leaderboard)</td><td>1</td><td>1</td><td>-</td><td>1</td><td>1</td><td>·</td><td>1</td><td>-</td><td>4.99</td><td>0.57</td><td>0.95</td><td>0.02</td></tr><tr><td>Ours* (beam search)</td><td>3.04</td><td>0.71</td><td>0.78</td><td>0.67</td><td>4.62</td><td>0.58</td><td>0.68</td><td>0.52</td><td>4.48</td><td>0.61</td><td>0.70</td><td>0.56</td></tr><tr><td>(leaderboard)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>4.48</td><td>0.61</td><td>0.97</td><td>0.02</td></tr></table>
+
+![](images/cdb66695242d8fd58d2f130e1b127c746dbb35450e0b9d1bc182c3a5e7dda336.jpg)  
+Figure 3: The positions and weights of grounded instructions as agents navigate by following instructions. Our self-monitoring agent with progress monitor demonstrates the grounded instruction used for action selection shifts gradually from the beginning of instructions towards the end. This is not true of the baseline method.
+
+# 3.1 COMPARISON WITH PRIOR ART
+
+We first compare the proposed self-monitoring agent with existing approaches. As shown in Table 1, our method achieves significant performance improvement compared to the state of the arts without data augmentation. We achieve $70 \%$ SR on the seen environment and $57 \%$ on the unseen environment while the existing best performing method achieved $63 \%$ and $50 \%$ SR respectively. When trained with synthetic data5, our approach achieves slightly better performance on the seen environments and significantly better performance on both the validation unseen environments and the test unseen environments when submitted to the test server. We achieve $3 \%$ and $8 \%$ improvement on SR on both validation and test unseen environments. Both results with or without data augmentation indicate that our proposed approach is more generalizable to unseen environments. At the time of writing, our self-monitoring agent is ranked #1 on the challenge leader-board among the state of the arts.
+
+Note that both Speaker-Follower and our approach in Table 1 use beam search. For comparison without using beam search, please refer to the Appendix.
+
+Textually grounded agent. Intuitively, an instruction-following agent is required to strongly demonstrate the ability to correctly focus and follow the corresponding part of the instruction as it navigates through an environment. We thus record the distribution of attention weights on instruction at each step as indications of which parts of the instruction being used for action selection. We average all runs across both validation seen and unseen dataset splits. Ideally, we expect to see the distribution of attention weights lies close to a diagonal, where at the beginning, the agent focuses on the beginning of the instruction and shifts its attention towards the end of instruction as it moves closer to the goal.
+
+To demonstrate, we use the method with panoramic action space proposed in Fried et al. (2018) as a baseline for comparison. As shown in Figure 3, our self-monitoring agent with progress monitor demonstrates that the positions of grounded instruction over time form a line similar to a diagonal. This result may further indicate that the agent successfully utilizes the attention on instruction to complete the task sequentially. We can also see that both agents were able to focus on the first part of the instruction at the beginning of navigation consistently. However, as the agent moves further in unknown environments, our self-monitoring agent can still successfully identify the parts of instruction that are potentially useful for action selection, whereas the baseline approach becomes uncertain about which part of the instruction should be used for selecting an action.
+
+Table 2: Ablation study showing the effect of each proposed component. All methods use the panoramic action space. Note that, for methods using beam search during inference, only the last selected trajectory is used for evaluating OSR and SPL. \*: we implemented the model from SpeakerFollower (Fried et al., 2018) with panoramic action space as baseline.   
+
+<table><tr><td></td><td></td><td colspan="4">Inference Mode</td><td colspan="4">Validation-Seen</td><td colspan="4">Validation-Unseen</td></tr><tr><td>#</td><td>Co- Grounding</td><td>Progress Monitor</td><td>Greedy Decoding</td><td>Progress Inference</td><td>Beam Search Aug.</td><td>Data</td><td>NE↓ SR个</td><td>OSR↑</td><td></td><td>SPL↑</td><td>NE↓ SR↑</td><td>OSR↑</td><td>SPL个</td></tr><tr><td>Baseline*</td><td></td><td></td><td></td><td></td><td></td><td>4.36</td><td>0.54</td><td>0.68</td><td>-</td><td>7.22</td><td>0.27</td><td>0.39</td><td>-</td></tr><tr><td>1</td><td>√</td><td></td><td>√</td><td></td><td></td><td>3.65</td><td>0.65</td><td>0.75</td><td>0.56</td><td>6.07</td><td>0.42</td><td>0.57</td><td>0.28</td></tr><tr><td>2</td><td>√</td><td>√</td><td>√</td><td></td><td></td><td>3.72</td><td>0.63</td><td>0.75</td><td>0.56</td><td>5.98</td><td>0.44</td><td>0.58</td><td>0.30</td></tr><tr><td>3</td><td>√</td><td>√</td><td>√</td><td></td><td></td><td>3.22</td><td>0.67</td><td>0.78</td><td>0.58</td><td>5.52</td><td>0.45</td><td>0.56</td><td>0.32</td></tr><tr><td>4</td><td>√</td><td>√</td><td></td><td>√</td><td></td><td>3.56</td><td>0.65</td><td>0.75</td><td>0.58</td><td>5.89</td><td>0.46</td><td>0.60</td><td>0.32</td></tr><tr><td>5</td><td>√</td><td>√</td><td></td><td>√</td><td></td><td>3.18</td><td>0.68</td><td>0.77</td><td>0.58</td><td>5.41</td><td>0.47</td><td>0.59</td><td>0.34</td></tr><tr><td>6</td><td>√</td><td></td><td></td><td></td><td>√</td><td>3.66</td><td>0.66</td><td>0.76</td><td>0.62</td><td>5.70</td><td>0.49</td><td>0.68</td><td>0.42</td></tr><tr><td>7</td><td>√</td><td>√</td><td></td><td></td><td>√</td><td>3.23</td><td>0.70</td><td>0.78</td><td>0.66</td><td>5.04</td><td>0.57</td><td>0.70</td><td>0.51</td></tr><tr><td>8</td><td>√</td><td>√</td><td></td><td></td><td>√</td><td>3.04</td><td>0.71</td><td>0.78</td><td>0.67</td><td>4.62</td><td>0.58</td><td>0.68</td><td>0.52</td></tr></table>
+
+# 3.2 ABLATION STUDY
+
+We now discuss the importance of each component proposed in this work. We begin with the same baseline as before (agent with panoramic action space in Fried et al. $\left( 2 0 1 8 \right) )$ .
+
+Co-grounding. When comparing the baseline with row #1 in our proposed method, we can see that our co-grounding agent outperformed the baseline with a large margin. This is due to the fact that we use the LSTM to carry both the textually and visually grounded content, and the decision on each navigable direction is predicted with both textually grounded instruction and the hidden state output of the LSTM. On the other hand, the baseline agent relies on the LSTM to carry visually grounded content, and uses the hidden state output for predicting the textually grounded instruction. As a result, we observed that instead of predicting the instruction needed for selecting a navigable direction, the textually grounded instruction may match with the past sequence of observed images implicitly saved within the LSTM.
+
+Progress monitor. Given the effective co-grounding, the proposed progress monitor further ensure that the grounded instruction correctly reflects the progress made toward the goal. This further improves the performance especially on the unseen environments as we can see from row #1 and #2.
+
+When using the progress inference, the progress monitor serve as a progress indicator for the agent to decide when to move back to the last viewpoint. We can see from row $\# 2$ and $\# 4$ that the SR performance can be further improved around $2 \%$ on both seen and unseen environments.
+
+Finally, we integrate the output of the progress monitor with the state-factored beam search (Fried et al., 2018), so that the candidate paths compete not only based on the probability of selecting a certain navigable direction but also on the estimated correspondence between the past trajectory and the instruction. As we can see by comparing row $\# 2$ , #6, and $\# 7$ , the progress monitor significantly improved the success rate on both seen and unseen environments and is the key for surpassing the state of the arts even without data augmentation. We can also see that when using beam search without progress monitor, the SR on unseen improved $7 \%$ (row #1 vs #6), while using beam search integrated with progress estimation improved $13 \%$ (row $\# 2$ vs $\# 7$ ).
+
+Data augmentation. In the above, we have shown each row in our approach contributes to the performance. Each of them increases the success rate and reduces the navigation error incrementally. By further combining them with the data augmentation pre-trained from the speaker (Fried et al., 2018), the SR and OSR are further increased, and the NE is also drastically reduced. Interestingly, the performance improvement introduced by data augmentation is smaller than from Speaker-Follower on the validation sets (see Table 1 for comparison). This demonstrates that our proposed method is more data-efficient.
+
+![](images/4012ea4e5f3c7e264ef579d57ce4586dbc712a5603d2e05089f6137cae4c9884.jpg)  
+Figure 4: Successful self-monitoring agent navigates in two unseen environments. The agent is able to correctly follow the grounded instruction and achieve the goal successfully. The percentage of instruction completeness estimated by the proposed progress monitor gradually increases as the agent navigates and approaches the goal. Finally, the agent grounded the word ”Stop” to stop (see the supplementary material for full figures).
+
+# 3.3 QUALITATIVE RESULTS
+
+To further validate the proposed method, we qualitatively show how the agent navigates through unseen environments by following instructions as shown in Fig. 4. In each figure, the agent follows the grounded instruction (at the top of the figure) and decides to move towards a certain direction (green arrow). For the full figures and more examples of successful and failed agents in both unseen and seen environments, please see the supplementary material.
+
+Consider the trajectory on the left side in Fig. 4, at step 3, the grounded instruction illustrated that the agent just completed ”turn right” and focuses mainly on ”walk straight to bedroom”. As the agent entered the bedroom, it then shifts the textual grounding to the next action ”Turn left and walk to bed lamp”. Finally, at step 6, the agent completed another ”turn left” and successfully stop at the rug (see the supplementary material for the importance of dealing with duplicate actions). Consider the example on the right side, the agent has already entered the hallway and now turns right to walk across to another room. However, it is ambiguous that which room the instructor is referring to. At step 5, our agent checked out the room on the left first and realized that it does not match with ”Stop in doorway in front of rug”. It then moves to the next room and successfully stops at the goal.
+
+In both cases, we can see that the completeness estimated by progress monitor gradually increases as the agent steadily navigates toward the goal. We have also observed that the estimated completeness ends up much lower for failure cases (see the supplementary material for further details).
+
+# 4 RELATED WORK
+
+Vision, Language, and Navigation.. There is a plethora work investigating the combination of vision and language for a multitude of applications (Zhou et al., 2018a;b; Antol et al., 2015; Tapaswi et al., 2016; Das et al., 2017), etc. While success has been achieved in these tasks to handle massive corpora of static visual input and text data, a resurgence of interest focuses on equipping an agent with the ability to interact with its surrounding environment for a particular goal such as object manipulation with instructions (Misra et al., 2016; Arkin et al., 2017), grounded language acquisition (Al-Omari et al., 2017; Kollar et al., 2013; Spranger & Steels, 2015; Dubba et al., 2014), embodied question answering (Das et al., 2018; Gordon et al., 2018), and navigation (Matuszek et al., 2013; Hemachandra et al., 2015; Duvallet et al., 2016; Zhu et al., 2017; de Vries et al., 2018; Yuke Zhu, 2017; Mousavian et al., 2018; Wayne et al., 2018; Wang et al., 2018a; Mirowski et al., 2017; 2018; Zamir et al., 2018). In this work, we concentrate on the recently proposed the Visionand-Language Navigation task (Anderson et al., 2018b)—asking an agent to carry out sophisticated natural-language instructions in a 3D environment. This task has application to fields such as robotics; in contrast to traditional map-based navigation systems, navigation with instructions provides a flexible way to generalize across different environments.
+
+A few approaches have been proposed for the VLN task. For example, Anderson et al. (2018b) address the task in the form of a sequence-to-sequence translation model. Yu et al. (2018) introduce a guided feature transformation for textual grounding. Wang et al. (2018b) present a planned-head module by combing model-free and model-based reinforcement learning approaches. Recently, Fried et al. (2018) propose to train a speaker to synthesize new instructions for data augmentation and further use it for pragmatic inference to rank the candidate routes. These approaches leverage attentional mechanisms to select related words from a given instruction when choosing an action, but those agents are deployed to explore the environment without knowing about what progress has been made and how far away the goal is. In this paper, we propose a self-monitoring agent that performs co-grounding on both visual and textual inputs and constantly monitors its own progress toward the goal as a way of regularizing the textual grounding.
+
+Visual and textual grounding. Visual grounding learns to localize the most relevant object or region in an image given linguistic descriptions, and has been demonstrated as an essential component for a variety of vision tasks like image captioning (Hu et al., 2016; Rohrbach et al., 2016; Lu et al., 2018), visual question answering (Lu et al., 2016b; Agrawal et al., 2018), relationship detection (Lu et al., 2016a; Ma et al., 2018) and referral expression (Nagaraja et al., 2016; Gavrilyuk et al., 2018). In contrast to identifying regions or objects, we perform visual grounding to locate relevant images (views) in a panoramic photo constructed by stitching multiple images with the aim of choosing which direction to go. Extensive efforts have been made to ground language instructions into a sequence of actions (MacMahon et al., 2006; Branavan et al., 2009; Vogel & Jurafsky, 2010; Tellex et al., 2011; Artzi & Zettlemoyer, 2013; Andreas & Klein, 2015; Mei et al., 2016; Cohn et al., 2016; Misra et al., 2017). These early approaches mainly emphasize the incorporation of structural alignment biases between the linguistic structure and sequence of actions (Mei et al., 2016; Andreas & Klein, 2015), and assume the agents are in relatively easy environment where limited visual perception is required to fulfill the instructions.
+
+# 5 CONCLUSION
+
+We introduce a self-monitoring agent which consists of two complementary modules: visual-textual co-grounding module and progress monitor. The visual-textual co-grounding module locates the instruction completed in the past, the instruction needed in the next action, and the moving direction from surrounding images. The progress monitor regularizes and ensures the grounded instruction correctly reflects the progress towards the goal by explicitly estimating the completeness of instruction-following. This estimation is conditioned on the positions and weights of grounded instruction. Our approach sets a new state-of-the-art performance on the standard Room-to-Room dataset on both seen and unseen environments. While we present one instantiation of self-monitoring for a decision-making agent, we believe that this concept can be applied to other domains as well.
+
+# ACKNOWLEDGMENTS
+
+This research was partially supported by DARPAs Lifelong Learning Machines (L2M) program, under Cooperative Agreement HR0011-18-2-001. We thank the authors from Fried et al. (2018), Ronghang Hu and Daniel Fried, for communicating with us and providing details of the implementation and synthetic instructions for fair comparison.
+
+# REFERENCES
+
+Aishwarya Agrawal, Dhruv Batra, Devi Parikh, and Aniruddha Kembhavi. Dont just assume; look and answer: Overcoming priors for visual question answering. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4971–4980, 2018.
+
+Muhannad Al-Omari, Paul Duckworth, David C Hogg, and Anthony G Cohn. Natural language acquisition and grounding for embodied robotic systems. In Association for the Advancement of Artificial Intelligence (AAAI), pp. 4349–4356, 2017.
+
+Peter Anderson, Angel Chang, Devendra Singh Chaplot, Alexey Dosovitskiy, Saurabh Gupta, Vladlen Koltun, Jana Kosecka, Jitendra Malik, Roozbeh Mottaghi, Manolis Savva, et al. On evaluation of embodied navigation agents. arXiv preprint arXiv:1807.06757, 2018a.
+
+Peter Anderson, Qi Wu, Damien Teney, Jake Bruce, Mark Johnson, Niko Sunderhauf, Ian Reid, ¨ Stephen Gould, and Anton van den Hengel. Vision-and-language navigation: Interpreting visually-grounded navigation instructions in real environments. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), volume 2, 2018b.
+
+Jacob Andreas and Dan Klein. Alignment-based compositional semantics for instruction following. In Proceedings of the Conference on Empirical Methods in Natural Language Processing (EMNLP), 2015.
+
+Stanislaw Antol, Aishwarya Agrawal, Jiasen Lu, Margaret Mitchell, Dhruv Batra, C Lawrence Zitnick, and Devi Parikh. Vqa: Visual question answering. In In Proceedings of the IEEE International Conference on Computer Vision (ICCV), pp. 2425–2433, 2015.
+
+Jacob Arkin, Matthew R Walter, Adrian Boteanu, Michael E Napoli, Harel Biggie, Hadas KressGazit, and Thomas M Howard. Contextual awareness: Understanding monologic natural language instructions for autonomous robots. In Robot and Human Interactive Communication (RO-MAN), 2017 26th IEEE International Symposium on, pp. 502–509. IEEE, 2017.
+
+Yoav Artzi and Luke Zettlemoyer. Weakly supervised learning of semantic parsers for mapping instructions to actions. Transactions of the Association of Computational Linguistics (ACL), 1: 49–62, 2013.
+
+Yael Benn, Thomas L Webb, Betty PI Chang, Yu-Hsuan Sun, Iain D Wilkinson, and Tom FD Farrow. The neural basis of monitoring goal progress. Frontiers in human neuroscience, 8:688, 2014.
+
+Elliot T Berkman and Matthew D Lieberman. The neuroscience of goal pursuit. The psychology of goals, pp. 98–126, 2009.
+
+Satchuthananthavale RK Branavan, Harr Chen, Luke S Zettlemoyer, and Regina Barzilay. Reinforcement learning for mapping instructions to actions. In Proceedings of the Joint Conference of the 47th Annual Meeting of the ACL and the 4th International Joint Conference on Natural Language Processing of the AFNLP: Volume 1-Volume 1, pp. 82–90. Association for Computational Linguistics, 2009.
+
+Angel Chang, Angela Dai, Thomas Funkhouser, Maciej Halber, Matthias Niessner, Manolis Savva, Shuran Song, Andy Zeng, and Yinda Zhang. Matterport3D: Learning from RGB-D data in indoor environments. International Conference on 3D Vision (3DV), 2017.
+
+Christopher H Chatham, Eric D Claus, Albert Kim, Tim Curran, Marie T Banich, and Yuko Munakata. Cognitive control reflects context monitoring, not motoric stopping, in response inhibition. PloS one, 7(2):e31546, 2012.
+
+Trevor Cohn, Cong Duy Vu Hoang, Ekaterina Vymolova, Kaisheng Yao, Chris Dyer, and Gholamreza Haffari. Incorporating structural alignment biases into an attentional neural translation model. In Proceedings of North American Chapter of the Association for Computational Linguistics (NAACL), 2016.
+
+Abhishek Das, Satwik Kottur, Khushi Gupta, Avi Singh, Deshraj Yadav, Jose M.F. Moura, Devi ´ Parikh, and Dhruv Batra. Visual Dialog. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017.
+
+Abhishek Das, Samyak Datta, Georgia Gkioxari, Stefan Lee, Devi Parikh, and Dhruv Batra. Embodied question answering. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.
+
+Harm de Vries, Kurt Shuster, Dhruv Batra, Devi Parikh, Jason Weston, and Douwe Kiela. Talk the walk: Navigating new york city through grounded dialogue. arXiv preprint arXiv:1807.03367, 2018.
+
+Krishna SR Dubba, Miguel R De Oliveira, Gi Hyun Lim, Hamidreza Kasaei, L Seabra Lopes, Ana Tome, and Anthony G Cohn. Grounding language in perception for scene conceptualization in ´ autonomous robots. In Qualitative Representations for Robots: Papers from the AAAI Spring Symposium. Technical report, pp. 26–33. AI Access Foundation, 2014.
+
+Felix Duvallet, Matthew R Walter, Thomas Howard, Sachithra Hemachandra, Jean Oh, Seth Teller, Nicholas Roy, and Anthony Stentz. Inferring maps and behaviors from natural language instructions. In Experimental Robotics, pp. 373–388. Springer, 2016.
+
+Daniel Fried, Ronghang Hu, Volkan Cirik, Anna Rohrbach, Jacob Andreas, Louis-Philippe Morency, Taylor Berg-Kirkpatrick, Kate Saenko, Dan Klein, and Trevor Darrell. Speaker-follower models for vision-and-language navigation. In Advances in Neural Information Processing Systems (NIPS), 2018.
+
+Kirill Gavrilyuk, Amir Ghodrati, Zhenyang Li, and Cees GM Snoek. Actor and action video segmentation from a sentence. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5958–5966, 2018.
+
+Daniel Gordon, Aniruddha Kembhavi, Mohammad Rastegari, Joseph Redmon, Dieter Fox, and Ali Farhadi. Iqa: Visual question answering in interactive environments. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4089–4098, 2018.
+
+Sachithra Hemachandra, Felix Duvallet, Thomas M. Howard, Nicholas Roy, Anthony Stentz, and Matthew R. Walter. Learning models for following natural language directions in unknown environments. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, May 2015.
+
+Ronghang Hu, Huazhe Xu, Marcus Rohrbach, Jiashi Feng, Kate Saenko, and Trevor Darrell. Natural language object retrieval. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4555–4564, 2016.
+
+Michał Kempka, Marek Wydmuch, Grzegorz Runc, Jakub Toczek, and Wojciech Jaskowski. Viz- ´ doom: A doom-based ai research platform for visual reinforcement learning. In In Proceedings of the IEEE Conference on Computational Intelligence and Games (CIG), pp. 1–8. IEEE, 2016.
+
+Thomas Kollar, Jayant Krishnamurthy, and Grant P Strimel. Toward interactive grounded language acqusition. In Robotics: Science and systems, volume 1, pp. 721–732, 2013.
+
+Cewu Lu, Ranjay Krishna, Michael Bernstein, and Li Fei-Fei. Visual relationship detection with language priors. In European Conference on Computer Vision (ECCV), pp. 852–869. Springer, 2016a.
+
+Jiasen Lu, Jianwei Yang, Dhruv Batra, and Devi Parikh. Hierarchical question-image co-attention for visual question answering. In Advances In Neural Information Processing Systems, pp. 289– 297, 2016b.
+
+Jiasen Lu, Jianwei Yang, Dhruv Batra, and Devi Parikh. Neural baby talk. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 7219–7228, 2018.
+
+Chih-Yao Ma, Asim Kadav, Iain Melvin, Zsolt Kira, Ghassan AlRegib, and Hans Peter Graf. Attend and interact: Higher-order object interactions for video understanding. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.
+
+Matt MacMahon, Brian Stankiewicz, and Benjamin Kuipers. Walk the talk: Connecting language, knowledge, and action in route instructions. In Association for the Advancement of Artificial Intelligence (AAAI), 2006.
+
+Cynthia Matuszek, Evan Herbst, Luke Zettlemoyer, and Dieter Fox. Learning to parse natural language commands to a robot control system. In Experimental Robotics, pp. 403–415. Springer, 2013.
+
+Hongyuan Mei, Mohit Bansal, and Matthew R Walter. Listen, attend, and walk: Neural mapping of navigational instructions to action sequences. In Association for the Advancement of Artificial Intelligence (AAAI), volume 1, pp. 2, 2016.
+
+Piotr Mirowski, Razvan Pascanu, Fabio Viola, Hubert Soyer, Andrew J Ballard, Andrea Banino, Misha Denil, Ross Goroshin, Laurent Sifre, Koray Kavukcuoglu, et al. Learning to navigate in complex environments. In Proceedings of the International Conference on Learning Representations (ICLR), 2017.
+
+Piotr Mirowski, Matthew Koichi Grimes, Mateusz Malinowski, Karl Moritz Hermann, Keith Anderson, Denis Teplyashin, Karen Simonyan, Koray Kavukcuoglu, Andrew Zisserman, and Raia Hadsell. Learning to navigate in cities without a map. arXiv preprint arXiv:1804.00168, 2018.
+
+Dipendra Misra, John Langford, and Yoav Artzi. Mapping instructions and visual observations to actions with reinforcement learning. In Proceedings of the Conference on Empirical Methods in Natural Language Processing (EMNLP), 2017.
+
+Dipendra K Misra, Jaeyong Sung, Kevin Lee, and Ashutosh Saxena. Tell me dave: Context-sensitive grounding of natural language to manipulation instructions. The International Journal of Robotics Research, 35(1-3):281–300, 2016.
+
+Arsalan Mousavian, Alexander Toshev, Marek Fiser, Jana Kosecka, and James Davidson. Visual representations for semantic target driven navigation. arXiv preprint arXiv:1805.06066, 2018.
+
+Varun K Nagaraja, Vlad I Morariu, and Larry S Davis. Modeling context between objects for referring expression understanding. In European Conference on Computer Vision (ECCV), pp. 792–807. Springer, 2016.
+
+Anna Rohrbach, Marcus Rohrbach, Ronghang Hu, Trevor Darrell, and Bernt Schiele. Grounding of textual phrases in images by reconstruction. In European Conference on Computer Vision (ECCV), pp. 817–834. Springer, 2016.
+
+Michael Spranger and Luc Steels. Co-acquisition of syntax and semantics-an investigation in spatial language. In Association for the Advancement of Artificial Intelligence (AAAI), 2015.
+
+Makarand Tapaswi, Yukun Zhu, Rainer Stiefelhagen, Antonio Torralba, Raquel Urtasun, and Sanja Fidler. Movieqa: Understanding stories in movies through question-answering. In In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (CVPR), pp. 4631–4640, 2016.
+
+Stefanie Tellex, Thomas Kollar, Steven Dickerson, Matthew R Walter, Ashis Gopal Banerjee, Seth J Teller, and Nicholas Roy. Understanding natural language commands for robotic navigation and mobile manipulation. In Association for the Advancement of Artificial Intelligence (AAAI), volume 1, pp. 2, 2011.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems (NIPS), pp. 5998–6008, 2017.
+
+Adam Vogel and Dan Jurafsky. Learning to follow navigational directions. In Proceedings of the 48th Annual Meeting of the Association for Computational Linguistics (ACL), pp. 806–814. Association for Computational Linguistics, 2010.
+
+Tsun-Hsuan Wang, Hung-Jui Huang, Juan-Ting Lin, Chan-Wei Hu, Kuo-Hao Zeng, and Min Sun. Omnidirectional cnn for visual place recognition and navigation. In In Proceedings of the IEEE Conference on Robotics and Automation (ICRA), 2018a.
+
+Xin Wang, Wenhan Xiong, Hongmin Wang, and William Yang Wang. Look before you leap: Bridging model-free and model-based reinforcement learning for planned-ahead vision-and-language navigation. In European Conference on Computer Vision (ECCV), 2018b.
+
+Greg Wayne, Chia-Chun Hung, David Amos, Mehdi Mirza, Arun Ahuja, Agnieszka GrabskaBarwinska, Jack Rae, Piotr Mirowski, Joel Z Leibo, Adam Santoro, et al. Unsupervised predictive memory in a goal-directed agent. arXiv preprint arXiv:1803.10760, 2018.
+
+Haonan Yu, Xiaochen Lian, Haichao Zhang, and Wei Xu. Guided feature transformation (gft): A neural language grounding module for embodied agents. arXiv preprint arXiv:1805.08329, 2018.
+
+Eric Kolve Dieter Fox Li Fei-Fei Abhinav Gupta Roozbeh Mottaghi Ali Farhadi Yuke Zhu, Daniel Gordon. Visual Semantic Planning using Deep Successor Representations. In In Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2017.
+
+AR Zamir, F Xia, J He, A Sax, J Malik, and S Savarese. Gibson env: Real-world perception for embodied agents. In In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), volume 4, 2018.
+
+Luowei Zhou, Chenliang Xu, and Jason J Corso. Towards automatic learning of procedures from web instructional videos. In Association for the Advancement of Artificial Intelligence (AAAI), 2018a.
+
+Luowei Zhou, Yingbo Zhou, Jason J Corso, Richard Socher, and Caiming Xiong. End-to-end dense video captioning with masked transformer. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 8739–8748, 2018b.
+
+Yuke Zhu, Roozbeh Mottaghi, Eric Kolve, Joseph J Lim, Abhinav Gupta, Li Fei-Fei, and Ali Farhadi. Target-driven visual navigation in indoor scenes using deep reinforcement learning. In In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pp. 3357–3364. IEEE, 2017.
+
+Table 3: Performance comparison with the state of arts without beam search: Student-forcing (Anderson et al., 2018b), RPA (Wang et al., 2018b), and Speaker-Follower (Fried et al., 2018). \*: with data augmentation.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">Validation-Seen</td><td colspan="4">Validation-Unseen</td><td colspan="4">Test (unseen)</td></tr><tr><td>NE←</td><td>SR↑</td><td>OSR个</td><td>SPL个</td><td>NE↓</td><td>SR↑</td><td>OSR个</td><td>SPL个</td><td>NE↓</td><td>SR个</td><td>OSR个</td><td>SPL个</td></tr><tr><td>Random</td><td>9.45</td><td>0.16</td><td>0.21</td><td>1</td><td>9.23</td><td>0.16</td><td>0.22</td><td>-</td><td>9.77</td><td>0.13</td><td>0.18</td><td>1</td></tr><tr><td>Student-forcing</td><td>6.01</td><td>0.39</td><td>0.53</td><td>=</td><td>7.81</td><td>0.22</td><td>0.28</td><td>-</td><td>7.85</td><td>0.20</td><td>0.27</td><td>-</td></tr><tr><td>RPA</td><td>5.56</td><td>0.43</td><td>0.53</td><td>-</td><td>7.65</td><td>0.25</td><td>0.32</td><td>1</td><td>7.53</td><td>0.25</td><td>0.33</td><td>1</td></tr><tr><td>Speaker-Follower*</td><td>3.36</td><td>0.66</td><td>0.74</td><td>-</td><td>6.62</td><td>0.36</td><td>0.45</td><td>1</td><td>6.62</td><td>0.35</td><td>0.44</td><td>0.28</td></tr><tr><td>Ours* (Greedy Decoding)</td><td>3.22</td><td>0.67</td><td>0.78</td><td>0.58</td><td>5.52</td><td>0.45</td><td>0.56</td><td>0.32</td><td>5.99</td><td>0.43</td><td>0.55</td><td>0.32</td></tr><tr><td>Ours* (Progress Inference)</td><td>3.18</td><td>0.68</td><td>0.77</td><td>0.58</td><td>5.41</td><td>0.47</td><td>0.59</td><td>0.34</td><td>5.67</td><td>0.48</td><td>0.59</td><td>0.35</td></tr></table>
+
+# SUPPLEMENTARY MATERIALS
+
+# COMPARISON WITH PRIOR ART WITHOUT BEAM SEARCH
+
+We provide the comparison with state of the arts without using beam search. The results are shown in Table 3. We can see that our proposed method outperformed existing approaches with a large margin on both validation unseen and test sets. Our method with greedy decoding for action selection improved the SR by $9 \%$ and $8 \%$ on validation unseen and test set. When using progress inference for action selection, the performance on the test set significantly improved by $5 \%$ compared to using greedy decoding, yielding $13 \%$ improvement over the best existing approach.
+
+# IMPLEMENTATION DETAILS
+
+Image feature. Similar to previous work, we use the pre-trained ResNet-152 on ImageNet to extract image features. Each image feature is thus a 2048-d vector. The embedded feature vector for each navigable direction is obtained by concatenating an appearance feature with a 4-d orientation feature $\left[ s i n \phi ; c o s \phi ; s i n \theta ; c o s \theta \right]$ , where $\phi$ and $\theta$ are the heading and elevation angles. Following the work in Fried et al. (2018), the 4-dim orientation features are tiled 32 times, resulting a embedding feature vector with 2176 dimension.
+
+Network architecture. The embedding dimension for encoding the navigation instruction is 256. We use a dropout layer with ratio 0.5 after the embedding layer. We then encode the instruction using a regular LSTM, and the hidden state is 512 dimensional. The MLP $g$ used for projecting the raw image feature is $B N  F C  B N  D r o p o u t  R e L U$ . The FC layer projects the 2176-d input vector to a 1024-d vector, and the dropout ratio is set to be 0.5. The hidden state of the LSTM used for carrying the textual and visual information through time in Eq. 1 is 512. We set the maximum length of instruction to be 80, thus the dimension of the attention weights of textual grounding $\pmb { \alpha } _ { t }$ is also 80. The dimension of the learnable matrices from Eq. 2 to 5 are: $W _ { x } \in \mathbb { R } ^ { 5 1 \breve { 2 } \times 5 1 2 }$ , $\breve { W _ { v } } \in \mathbb { R } ^ { 5 1 2 \times 1 0 2 4 }$ , $W _ { a } \in \mathbb { R } ^ { 1 0 2 4 \times 1 0 2 4 }$ , $W _ { h } \in \mathbb { R } ^ { 1 5 3 6 \times 5 1 2 }$ , and $W _ { p m } \in \mathbb { R } ^ { 5 9 2 \times 1 }$ .
+
+Training. We use ADAM as the optimizer. The learning rate is $1 e - 4$ with batch size of 64 consistently through out all experiments. When using beam search, we set the beam size to be 15. We perform categorical sampling during training for action selection.
+
+# SUBMISSION TO VISION AND LANGUAGE NAVIGATION CHALLENGE
+
+For evaluating our proposed approach on the unseen test set, we participate in the Vision and Language Navigation challenge and submitted our result with the full proposed approach to the test server. We achieved $61 \%$ success rate and ranked $\# 1$ on the test server at the time of writing.
+
+We follow the submission guidelines, where picking the highest confidence trajectory from multiple trials for each instruction is not permissible. This means that using the beam search for competing and selecting a final trajectory is not allow directly. Similar to the submission from SpeakerFollower (Fried et al., 2018), we record all the viewpoints traversed during the beam search process. The final agent traverses through all recorded trajectories by first reaching the end of one trajectory and backtracking to the shared viewpoint with the next trajectory. This means that the agent could backtrack to the start point during this process. The trajectories are however logged according to the closest previous trajectory, so that when a single agent traverses through all recorded trajectories, the overhead for switching from one trajectory to another can be reduced significantly. The final selected trajectory from beam search is then lastly logged to the trajectory. This therefore yields exactly the same success rate and navigation error, as the metrics are computed according to the last viewpoint from a trajectory.
+
+# QUALITATIVE RESULTS
+
+We provide and discuss additional qualitative results on the self-monitoring agent navigating on seen and unseen environments. We first discuss four successful examples in Fig. 5 and 6, and followed by two failure examples in Fig. 7.
+
+# SUCCESSFUL EXAMPLES
+
+In Fig. 5 (a), at the beginning, the agent mostly focuses on ”walk up” for making the first movement. While the agent keeps its attention on ”walk up” as completed instruction or ongoing action, it shifts the attention on instruction to ”turn right” as it walks up the stairs. Once it reached the top of the stairs, it decides to turn right according to the grounded instruction. Once turned right, we can again see that the agent pays attention on both the past action ”turn right” and next action ”walk straight to bedroom”. The agent continues to do so until it decides to stop by grounding on the word ”stop”.
+
+In Fig. 5 (b), the agent starts by focusing on both ”enter bedroom from balcony” and ”turn left” to navigate. It correctly shifts the attention on textual grounding on the following instruction. Interestingly, the given instruction ”walk straight across rug to room” at step 3 is ambiguous since there are two rooms across the rug. Our agent decided to sneak out of the first room on the left and noticed that it does not match with the description from instruction. It then moved to another room across the rug and decided to stop because there is a rug inside the room as described.
+
+In Fig. 6 (a), the given instruction is ambiguous as it only asks the agent to take actions around the stairs. Since there are multiple duplicated actions described in the instruction, e.g. ”walk up” and ”turn left”, only an agent that is able to precisely follow the instruction step-by-step can successfully complete the task. Otherwise, the agent is likely to stop early before it reaches the goal. The agent also needs to demonstrate its ability to assess the completeness of instruction-following task in order to correctly stop at the right amount of repeated actions as described in the instruction.
+
+In Fig. 6 (b), at the beginning (step 0), the agent only focuses on ’left’ for making the first movement (the agent is originally facing the painting). We can see that at each step, the agent correctly focuses on parts of the instruction for making every movements, and it finally believes that the instruction is completed (attention on the last sentence period) and stopped.
+
+# FAILURE EXAMPLES
+
+In Fig. 7 (a) step 1, although the attention on instruction correctly focused on ”take a left” and ”go down”, the agent failed to follow the instruction and was not able to complete the task. We can however see that the progress monitor correctly reflected that the agent did not follow the given instruction successfully. The agent ended up stopping with progress monitor reporting that only $16 \%$ of the instruction was completed.
+
+In Fig. 7 (b) step 2, the attention on instruction only focuses on ”go down” and thus failed to associate the ”go down steps” with the stairs previously mentioned in ”turn right to stairs”. The agent was however able to follow the rest of the instruction correctly by turning right and stopping near a mirror. Note that, different from Fig. 7 (a), the final estimated completeness of instruction-following from progress monitor is much higher $( 1 6 \% )$ , which indicates that the agent failed to be aware that it was not correctly following the instruction.
+
+![](images/87198f7498d1d696f00ef3cfd51e2dfaa7b1d21850ef91ba6dab95ab45c7b00b.jpg)  
+Figure 5: Successful self-monitoring agent navigates in two different unseen environments. Given the navigational instruction located at the top of the figure, the agent starts from starting position and follows the instruction towards the goal. The percentage of instruction completeness estimated by the proposed progress monitor gradually increases as the agent navigates and approaches the goal.
+
+![](images/65dc0f00e3c9691d8de7217920ca9179144403a32bbba16b7dc2a06c051ad49b.jpg)  
+Figure 6: Successful self-monitoring agent navigates in (a) unseen and (b) seen environments. (a) The given instruction is ambiguous as it only asks the agent to take actions around the stairs. Since there are multiple duplicated actions described in the instruction, e.g. ”walk up” and ”turn left”, only an agent that is able to precisely follow the instruction step-by-step can successfully complete the task. Otherwise, the agent is likely to stop early before it reaches the goal. (b) The agent correctly pays attention to parts of the instruction for making decisions on selecting navigable directions. Both the agents decide to stop when shifting the textual grounding on the last sentence period.
+
+![](images/91680abed5178b13fa492098d8e349fa1676ffe10db949d546e8aeaa0fbcbc8a.jpg)  
+Figure 7: Failed self-monitoring agent navigates in unseen environments. (a) The agent missed the ”take a left” at step 1, and consequently unable to follow the following instruction correctly. However, note that the progress monitor correctly reflected that the instruction was not completed. When the agent decides to end the navigation, it reports that only $16 \%$ of the instruction was completed. (b) At step 2, the attention on instruction only focuses on ”go down” and thus failed to associate the ”go down steps” with the stairs previously mentioned in ”turn right to stairs”. The agent was however able to follow the rest of the instruction correctly by turning right and stopping near a mirror. Note that, different from (a), the final estimated completeness of instruction-following is much higher, which suggests that the agent failed to correctly be aware of its progress towards the goal.

parse/train/r1GAsjC5Fm/r1GAsjC5Fm_content_list.json

@@ -0,0 +1,1807 @@
+[
+    {
+        "type": "text",
+        "text": "SELF-MONITORING NAVIGATION AGENT VIA AUXILIARY PROGRESS ESTIMATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            821,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chih-Yao $\\mathbf { M _ { a } } { * } _ { } ^ { + }$ , Jiasen $\\mathbf { L } \\mathbf { u } ^ { * \\dagger }$ , Zuxuan $\\mathbf { W _ { u } } ^ { * \\dagger }$ , Ghassan AlRegib†, Zsolt Kira†, Richard Socher§ & Caiming Xiong§ ",
+        "bbox": [
+            184,
+            169,
+            714,
+            199
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "†Georgia Institute of Technology   \n{cyma,jiasenlu,alregib,zkira}@gatech.edu   \n‡University of Maryland, College Park   \n{zxwu}@cs.umd.edu   \n§Salesforce Research   \n{rsocher,cxiong}@salesforce.com ",
+        "bbox": [
+            184,
+            200,
+            573,
+            285
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            321,
+            544,
+            337
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The Vision-and-Language Navigation (VLN) task entails an agent following navigational instruction in photo-realistic unknown environments. This challenging task demands that the agent be aware of which instruction was completed, which instruction is needed next, which way to go, and its navigation progress towards the goal. In this paper, we introduce a self-monitoring agent with two complementary components: (1) visual-textual co-grounding module to locate the instruction completed in the past, the instruction required for the next action, and the next moving direction from surrounding images and (2) progress monitor to ensure the grounded instruction correctly reflects the navigation progress. We test our selfmonitoring agent on a standard benchmark and analyze our proposed approach through a series of ablation studies that elucidate the contributions of the primary components. Using our proposed method, we set the new state of the art by a significant margin ( $8 \\%$ absolute increase in success rate on the unseen test set). Code is available at https://github.com/chihyaoma/selfmonitoring-agent. ",
+        "bbox": [
+            233,
+            353,
+            764,
+            547
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            571,
+            336,
+            588
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, the Vision-and-Language (VLN) navigation task (Anderson et al., 2018b), which requires the agent to follow natural language instructions to navigate through a photo-realistic unknown environment, has received significant attention (Wang et al., 2018b; Fried et al., 2018). In the VLN task, an agent is placed in an unknown realistic environment and is required to follow natural language instructions to navigate from its starting location to a target location. In contrast to some existing navigation tasks (Kempka et al., 2016; Zhu et al., 2017; Mirowski et al., 2017; 2018), we address the class of tasks where the agent does not have an explicit representation of the target (e.g., location in a map or image representation of the goal) to know if the goal has been reached or not (Matuszek et al., 2013; Hemachandra et al., 2015; Duvallet et al., 2016; Arkin et al., 2017). Instead, the agent needs to be aware of its navigation status through the association between the sequence of observed visual inputs to instructions. ",
+        "bbox": [
+            174,
+            603,
+            825,
+            756
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Consider an example as shown in Fig. 1, given the instruction ”Exit the bedroom and go towards the table. Go to the stairs on the left of the couch. Wait on the third step.”, the agent first needs to locate which instruction is needed for the next movement, which in turn requires the agent to be aware of (i.e., to explicitly represent or have an attentional focus on) which instructions were completed or ongoing in the previous steps. For instance, the action ”Go to the stairs” should be carried out once the agent has exited the room and moved towards the table. However, there exists inherent ambiguity for ”go towards the table”. Intuitively, the agent is expected to ”Go to the stairs” after completing ”go towards the table”. But, it is not clear what defines the completeness of ”Go towards the table”. The completeness of an ongoing action often depends on the availability of the next action. Since the transition between past and next part of the instructions is a soft boundary, in order to determine when to transit and to follow the instruction correctly the agent is required to keep track of both grounded instructions. On the other hand, assessing the progress made towards the goal has indeed been shown to be important for goal-directed tasks in humans decision-making (Benn et al., 2014; Chatham et al., 2012; Berkman & Lieberman, 2009). While a number of approaches have been proposed for VLN (Anderson et al., 2018b; Wang et al., 2018b; Fried et al., 2018), previous approaches generally are not aware of which instruction is next nor progress towards the goal; indeed, we qualitatively show that even the attentional mechanism of the baseline does not successfully track this information through time. ",
+        "bbox": [
+            174,
+            763,
+            825,
+            902
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/03e64e937c93bc83cd207099a38a596235846717160705fb06d7cb2fb36d202e.jpg",
+        "image_caption": [
+            "Figure 1: Vision-and-Language Navigation task and our proposed self-monitoring agent. The agent is constantly aware of what was completed, what is next, and where to go, as it navigates through unknown environments by following navigational instructions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            101,
+            821,
+            222
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            313,
+            825,
+            425
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose an agent endowed with the following abilities: (1) identify which direction to go by finding the part of the instruction that corresponds to the observed images—visual grounding, (2) identify which part of the instruction has been completed or ongoing and which part is potentially needed for the next action selection—textual grounding, and (3) ensure that the grounded instruction can correctly be used to estimate the progress made towards the goal, and apply regularization to ensure this —progress monitoring. Therefore, we introduce the self-monitoring agent consisting of two complementary modules: visual-textual co-grounding and progress monitor. ",
+        "bbox": [
+            174,
+            431,
+            825,
+            530
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "More specifically, we achieve both visual and textual grounding simultaneously by incorporating the full history of grounded instruction, observed images, and selected actions into the agent. We leverage the structural bias between the words in instructions used for action selection and progress made towards the goal and propose a new objective function for the agent to measure how well it can estimate the completeness of instruction-following. We then demonstrate that by conditioning on the positions and weights of grounded instruction as input, the agent can be self-monitoring of its progress and further ensure that the textual grounding accurately reflects the progress made. ",
+        "bbox": [
+            174,
+            536,
+            825,
+            633
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Overall, we propose a novel self-monitoring agent for VLN and make the following contributions: (1) We introduce the visual-textual co-grounding module, which performs grounding interdependently across both visual and textual modalities. We show that it can outperform the baseline method by a large margin. (2) We propose to equip the self-monitoring agent with a progress monitor, and for navigation tasks involving instructions instantiate this by introducing a new objective function for training. We demonstrate that, unlike the baseline method, the position of grounded instruction can follow both past and future instructions, thereby tracking progress to the goal. (3) With the proposed self-monitoring agent, we set the new state-of-the-art performance on both seen and unseen environments on the standard benchmark. With $8 \\%$ absolute improvement in success rate on the unseen test set, we are ranked #1 on the challenge leaderboard. ",
+        "bbox": [
+            173,
+            641,
+            825,
+            780
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 SELF-MONITORING NAVIGATION AGENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            805,
+            542,
+            820
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 NOTATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            839,
+            287,
+            854
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Given a natural language instruction with $L$ words, its representation is denoted by $\\begin{array} { r l } { \\boldsymbol { X } } & { { } = } \\end{array}$ $\\left\\{ { \\pmb x } _ { 1 } , { \\pmb x } _ { 2 } , \\ldots , { \\pmb x } _ { L } \\right\\}$ , where $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } l }$ is the feature vector for the $l$ -th word encoded by an LSTM language encoder. Following Fried et al. (2018), we enable the agent with panoramic view. At each time step, the agent perceives a set of images at each viewpoint $\\pmb { v } _ { t } = \\left\\{ \\pmb { v } _ { t , 1 } , \\pmb { v } _ { t , 2 } , . . . , \\pmb { v } _ { t , K } \\right\\}$ , where $K$ is the maximum number of navigable directions1, and $\\mathbf { \\Delta } \\mathbf { v } _ { t , k }$ represents the image feature of direction $k$ . The co-grounding feature of instruction and image are denoted as $\\hat { \\mathbf { x } } _ { t }$ and $\\hat { \\pmb { v } } _ { t }$ respectively. The selected action is denoted as $\\mathbf { a } _ { t }$ . The learnable weights are denoted with $W$ , with appropriate sub/super-scripts as necessary. We omit the bias term $^ { b }$ to avoid notational clutter in the exposition. ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/0fd43ff1d97d0ffc067684d152dcc9d33b4df237f90987f3c55fb7725dbd7362.jpg",
+        "image_caption": [
+            "Figure 2: Proposed self-monitoring agent consisting of visual-textual co-grounding, progress monitoring, and action selection modules. Textual grounding: identify which part of the instruction has been completed or ongoing and which part is potentially needed for next action. Visual grounding: summarize the observed surrounding images. Progress monitor: regularize and ensure grounded instruction reflects progress towards the goal. Action selection: identify which direction to go. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            103,
+            816,
+            316
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            435,
+            825,
+            492
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 VISUAL AND TEXTUAL CO-GROUNDING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            508,
+            488,
+            522
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "First, we propose a visual and textual co-grounding model for the vision and language navigation task, as illustrated in Fig. 2. We model the agent with a sequence-to-sequence architecture with attention by using a recurrent neural network. More specifically, we use Long Short Term Memory (LSTM) to carry the flow of information effectively. At each step $t$ , the decoder observes representations of the current attended panoramic image feature $\\hat { \\mathbf { } } _ { }$ , previous selected action $\\mathbf { a } _ { t - 1 }$ and current grounded instruction feature $\\hat { \\mathbf { x } } _ { t }$ as input, and outputs an encoder context $\\boldsymbol { h } _ { t }$ : ",
+        "bbox": [
+            173,
+            534,
+            825,
+            618
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/de5519e47da65dc2fd829c4147efba6d9679e596aa0a10556087ca3664b7f5fe.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { h } _ { t } = L S T M \\big ( \\big [ \\hat { \\pmb { x } } _ { t } , \\hat { \\pmb { v } } _ { t } , \\mathbf { a } _ { t - 1 } \\big ] \\big ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            625,
+            596,
+            642
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $[ , ]$ denotes concatenation. The previous encoder context $\\boldsymbol { h } _ { t - 1 }$ is used to obtain the textual grounding feature $\\hat { \\mathbf { x } } _ { t }$ and visual grounding feature $\\hat { \\pmb { v } } _ { t }$ , whereas we use current encoder context $\\boldsymbol { h } _ { t }$ to obtain next action $\\mathbf { a } _ { t }$ , all of which will be illustrated in the rest of the section. ",
+        "bbox": [
+            174,
+            648,
+            825,
+            690
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Textual grounding. When the agent moves from one viewpoint to another, it is required to identify which direction to go by relying on a grounded instruction, i.e. which parts of the instruction should be used. This can either be the instruction matched with the past (ongoing action) or predicted for the future (next action). To capture the relative position between words within an instruction, we incorporate the positional encoding $P E ( \\cdot )$ (Vaswani et al., 2017) into the instruction features. We then perform soft-attention on the instruction features $\\boldsymbol { X }$ , as shown on the left side of Fig. 2. The attention distribution over $L$ words of the instructions is computed as: ",
+        "bbox": [
+            173,
+            696,
+            825,
+            795
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dcf4e118968470e9223800b4a01828d1b1ffa389e682b20581180e168caf0bdb.jpg",
+        "text": "$$\n\\begin{array} { r } { z _ { t , l } ^ { \\mathrm { t e x t u a l } } = ( W _ { x } { h _ { t - 1 } } ) ^ { \\top } P E ( { x _ { l } } ) , \\quad \\mathrm { a n d } \\quad \\alpha _ { t } = \\mathrm { s o f t m a x } ( z _ { t } ^ { \\mathrm { t e x t u a l } } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            289,
+            801,
+            705,
+            821
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "wherword $W _ { x }$ are parameters to be learnt. he instruction and previous h $z _ { t , l } ^ { \\mathrm { t e x t u a l } }$ is astate r val, and omputed as the correlation betweenis the attention weight over features $l$ $\\boldsymbol { h } _ { t - 1 }$ $\\alpha _ { t }$   \nin $\\boldsymbol { X }$ at time $t$ . Based on the textual attention distribution, the grounded textual feature $\\hat { \\mathbf { x } } _ { t }$ can be   \nobtained by the weighted sum over the textual features $\\hat { \\pmb { x } } _ { t } = { \\pmb { \\alpha } } _ { t } ^ { T } { \\pmb { X } }$ . ",
+        "bbox": [
+            174,
+            828,
+            825,
+            887
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Visual grounding. In order to locate the completed or ongoing instruction, the agent needs to keep track of the sequence of images observed along the navigation trajectory. We thus perform visual attention over the surrounding views based on its previous hidden vector $\\boldsymbol { h } _ { t - 1 }$ . The visual attention weight $\\beta _ { t }$ can be obtained as: ",
+        "bbox": [
+            174,
+            103,
+            825,
+            160
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a40b9f272490f789ee924cb5cce03de81b942e2dbb3c73353d44191dd2d8f4e6.jpg",
+        "text": "$$\nz _ { t , k } ^ { \\mathrm { v i s u a l } } = ( W _ { v } \\boldsymbol { h } _ { t - 1 } ) ^ { \\top } \\boldsymbol { g } ( v _ { t , k } ) , \\quad \\mathrm { a n d } \\quad \\beta _ { t } = \\mathrm { s o f t m a x } ( z _ { t } ^ { \\mathrm { v i s u a l } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            165,
+            699,
+            186
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $g$ is a one-layer Multi-Layer Perceptron (MLP), $W _ { v }$ are parameters to be learnt. Similar to Eq. 2, the grounded visual feature $\\hat { \\mathbf { } v } _ { t }$ can be obtained by the weighted sum over the visual features $\\hat { \\pmb { v } } _ { t } = \\beta _ { t } ^ { T } \\bar { \\pmb { V } }$ . ",
+        "bbox": [
+            173,
+            191,
+            825,
+            234
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Action selection. To make a decision on which direction to go, the agent finds the image features on navigable directions with the highest correlation with the grounded navigation instruction $\\hat { \\mathbf { x } } _ { t }$ and the current hidden state $h _ { t }$ . We use the inner-product to compute the correlation, and the probability of each navigable direction is then computed as: ",
+        "bbox": [
+            173,
+            241,
+            825,
+            297
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aa3287e001e1572e34a44a56a922f66150e0101a0d876f4560b733682a2f223e.jpg",
+        "text": "$$\no _ { t , k } = ( W _ { a } [ { h _ { t } } , \\hat { { \\boldsymbol { x } } } _ { t } ] ) ^ { \\top } g ( { v _ { t , k } } ) \\quad \\mathrm { a n d } \\quad p _ { t } = \\mathrm { s o f t m a x } ( o _ { t } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            304,
+            686,
+            324
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $W _ { a }$ are the learnt parameters, $g ( \\cdot )$ is the same MLP as in Eq. 3, and ${ \\pmb p } _ { t }$ is the probability of each navigable direction at time $t$ . We use categorical sampling during training to select the next action $\\mathbf { a } _ { t }$ . Unlike the previous method with the panoramic view (Fried et al., 2018), which attends to instructions only based on the history of observed images, we achieve both textual and visual grounding using the shared hidden state output containing grounded information from both textual and visual modalities. During action selection, we rely on both hidden state output and grounded instruction, instead of only relying on grounded instruction. ",
+        "bbox": [
+            173,
+            329,
+            825,
+            428
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 PROGRESS MONITOR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            444,
+            361,
+            459
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It is imperative that the textual-grounding correctly reflects the progress towards the goal, since the agent can then implicitly know where it is now and what the next instruction to be completed will be. In the visual-textual co-grounding module, we ensure that the grounded instruction reasonably informs decision making when selecting a navigable direction. This is necessary but not sufficient for ensuring that the notion of progress to the goal is encoded. Thus, we propose to equip the agent with a progress monitor that serves as regularizer during training and prunes unfinished trajectories during inference. ",
+        "bbox": [
+            173,
+            470,
+            825,
+            569
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since the positions of localized instruction can be a strong indication of the navigation progress due to the structural alignment bias between navigation steps and instruction, the progress monitor can estimate how close the current viewpoint is to the final goal by conditioning on the positions and weights of grounded instruction. This can further enforce the result of textual-grounding to align with the progress made towards the goal and to ensure the correctness of the textual-grounding. ",
+        "bbox": [
+            174,
+            574,
+            825,
+            645
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The progress monitor aims to estimate the navigation progress by conditioning on three inputs: the history of grounded images and instructions, the current observation of the surrounding images, and the positions of grounded instructions. We therefore represent these inputs by using (1) the previous hidden state $\\boldsymbol { h } _ { t - 1 }$ and the current cell state $\\mathbf { } c _ { t }$ of the LSTM, (2) the grounded surrounding images $\\hat { \\pmb { v } } _ { t }$ , and (3) the distribution of attention weights of textual-grounding $\\pmb { \\alpha } _ { t }$ , as shown at the bottom of Fig. 2 represented by dotted lines. ",
+        "bbox": [
+            173,
+            651,
+            825,
+            736
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our proposed progress monitor first computes an additional hidden state output $h _ { t } ^ { p m }$ by using grounded image representations $\\hat { \\pmb { v } } _ { t }$ as input, similar to how a regular LSTM computes hidden states except we use concatenation over element-wise addition for empirical reasons2. The hidden state output is then concatenated with the attention weights $\\pmb { \\alpha } _ { t }$ on textual-grounding to estimate how close the agent is to the $\\mathrm { g o a l } ^ { 3 }$ . The output of the progress monitor $p _ { t } ^ { p m }$ , which represents the completeness of instruction-following, is computed as: ",
+        "bbox": [
+            173,
+            742,
+            825,
+            827
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ff2bf070eb46c00dac3f110efc209bf942b2ca1bbe27ef896b93446268891279.jpg",
+        "text": "$$\n\\begin{array} { r } { \\begin{array} { l l } { h _ { t } ^ { p m } = \\sigma ( W _ { h } ( [ h _ { t - 1 } , \\hat { v } _ { t } ] ) \\otimes t a n h ( c _ { t } ) ) , } & { p _ { t } ^ { p m } = t a n h ( W _ { p m } ( [ \\alpha _ { t } , h _ { t } ^ { p m } ] ) ) } \\end{array} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            833,
+            743,
+            851
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $W _ { h }$ and $W _ { p m }$ are the learnt parameters, $c _ { t }$ is the cell state of the LSTM, $\\otimes$ denotes the element-wise product, and $\\sigma$ is the sigmoid function. ",
+        "bbox": [
+            176,
+            857,
+            825,
+            886
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 TRAINING AND INFERENCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            405,
+            117
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Training. We introduce a new objective function to train the proposed progress monitor. The training target $y _ { t } ^ { p m }$ is defined as the normalized distance in units of length from the current viewpoint to the goal, i.e., the target will be 0 at the beginning and closer to 1 as the agent approaches the goal4. Note that the target can also be lower than 0, if the agent’s current distance from the goal is farther than the starting point. Finally, our self-monitoring agent is optimized with a cross-entropy loss and a mean squared error loss, computed with respect to the outputs from both action selection and progress monitor. ",
+        "bbox": [
+            173,
+            128,
+            825,
+            227
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/543e20358ba8ee73997ed1f469b6c599c291ecd4b27275e8eedae852c36a919a.jpg",
+        "text": "$$\n\\mathcal { L } _ { l o s s } = - \\lambda \\sum _ { t = 1 } ^ { T } y _ { t } ^ { n v } l o g ( p _ { k , t } ) - ( 1 - \\lambda ) \\sum _ { t = 1 } ^ { T } ( y _ { t } ^ { p m } - p _ { t } ^ { p m } ) ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            229,
+            692,
+            273
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where ${ p } _ { k , t }$ is the action probability of each navigable direction, $\\lambda = 0 . 5$ is the weight balancing the two losses, and $y _ { t } ^ { n v }$ is the ground-truth navigable direction at step $t$ . ",
+        "bbox": [
+            176,
+            295,
+            823,
+            324
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Inference. During inference, we follow Fried et al. (2018) by using beam search. we propose that, while the agent decides which trajectories in the beams to keep, it is equally important to evaluate the state of the beams on actions as well as on the agent’s confidence in completing the given instruction at each traversed viewpoint. We accomplish this idea by integrating the output of our progress monitor into the accumulated probability of beam search. At each step, when candidate trajectories compete based on accumulated probability, we integrate the estimated completeness of instruction-following $p _ { t } ^ { p m }$ (normalized between 0 to 1) with action probability ${ p } _ { k , t }$ to directly evaluate the partial and unfinished candidate routes: $p _ { t } ^ { b e a m } = p _ { t } ^ { p m } \\times p _ { k , t }$ . ",
+        "bbox": [
+            173,
+            329,
+            825,
+            444
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Without beam search, we use greedy decoding for action selection with one condition. If the progress monitor output decreases $( p _ { t + 1 } ^ { p m } < \\dot { p } _ { t } ^ { p m } )$ , the agent is required to move back to the previous viewpoint and select the action with next highest probability. We repeat this process until the selected action leads to increasing progress monitor output. We denote this procedure as progress inference. ",
+        "bbox": [
+            174,
+            449,
+            825,
+            506
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            525,
+            326,
+            541
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "R2R Dataset. We use the Room-to-Room (R2R) dataset (Anderson et al., 2018b) for evaluating our proposed approach. The R2R dataset is built upon the Matterport3D dataset (Chang et al., 2017) and has 7,189 paths sampled from its navigation graphs. Each path has three ground-truth navigation instructions written by humans. The whole dataset is divided into 4 sets: training, validation seen, validation unseen, and test sets unseen. ",
+        "bbox": [
+            173,
+            556,
+            825,
+            626
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Evaluation metrics. We follow the same evaluation metrics used by previous work on the R2R task: (1) Navigation Error (NE), mean of the shortest path distance in meters between the agent’s final position and the goal location. (2) Success Rate (SR), the percentage of final positions less than $3 \\mathrm { m }$ away from the goal location. (3) Oracle Success Rate (OSR), the success rate if the agent can stop at the closest point to the goal along its trajectory. In addition, we also include the recently introduced Success rate weighted by (normalized inverse) Path Length (SPL) (Anderson et al., 2018a), which trades-off Success Rate against trajectory length. ",
+        "bbox": [
+            174,
+            632,
+            823,
+            661
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/44105d8d8dedf373cfa77cb66ce3b35f0b960c856f7dc9a9e3a75674715f6f3e.jpg",
+        "table_caption": [
+            "Table 1: Performance comparison with the state of arts: Student-forcing (Anderson et al., 2018b), RPA (Wang et al., 2018b), and Speaker-Follower (Fried et al., 2018). \\*: with data augmentation. leaderboard: when using beam search, we modify our search procedure to comply with the leaderboard guidelines, i.e., all traversed viewpoints are recorded. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">Validation-Seen</td><td colspan=\"4\">Validation-Unseen</td><td colspan=\"4\">Test (unseen)</td></tr><tr><td>NE↓</td><td>SR个</td><td>OSR个</td><td>SPL个</td><td>NE↓</td><td>SR个</td><td>OSR个</td><td>SPL个</td><td>NE↓</td><td>SR↑</td><td>OSR个</td><td>SPL个</td></tr><tr><td>Random</td><td>9.45</td><td>0.16</td><td>0.21</td><td>1</td><td>9.23</td><td>0.16</td><td>0.22</td><td></td><td>9.77</td><td>0.13</td><td>0.18</td><td>-</td></tr><tr><td>Student-forcing</td><td>6.01</td><td>0.39</td><td>0.53</td><td>1</td><td>7.81</td><td>0.22</td><td>0.28</td><td>=</td><td>7.85</td><td>0.20</td><td>0.27</td><td>=</td></tr><tr><td>RPA</td><td>5.56</td><td>0.43</td><td>0.53</td><td>-</td><td>7.65</td><td>0.25</td><td>0.32</td><td>=</td><td>7.53</td><td>0.25</td><td>0.33</td><td>=</td></tr><tr><td>Speaker-Follower</td><td>3.88</td><td>0.63</td><td>0.71</td><td>-</td><td>5.24</td><td>0.50</td><td>0.63</td><td>-</td><td>-</td><td>1</td><td>1</td><td>=</td></tr><tr><td>Speaker-Follower*</td><td>3.08</td><td>0.70</td><td>0.78</td><td>-</td><td>4.83</td><td>0.55</td><td>0.65</td><td>-</td><td>4.87</td><td>0.53</td><td>0.64</td><td>1</td></tr><tr><td>(leaderboard)</td><td>-</td><td>-</td><td>1</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>4.87</td><td>0.53</td><td>0.96</td><td>0.01</td></tr><tr><td>Ours (beam search)</td><td>3.23</td><td>0.70</td><td>0.78</td><td>0.66</td><td>5.04</td><td>0.57</td><td>0.70</td><td>0.51</td><td>4.99</td><td>0.57</td><td>0.68</td><td>0.51</td></tr><tr><td>(leaderboard)</td><td>1</td><td>1</td><td>-</td><td>1</td><td>1</td><td>·</td><td>1</td><td>-</td><td>4.99</td><td>0.57</td><td>0.95</td><td>0.02</td></tr><tr><td>Ours* (beam search)</td><td>3.04</td><td>0.71</td><td>0.78</td><td>0.67</td><td>4.62</td><td>0.58</td><td>0.68</td><td>0.52</td><td>4.48</td><td>0.61</td><td>0.70</td><td>0.56</td></tr><tr><td>(leaderboard)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>4.48</td><td>0.61</td><td>0.97</td><td>0.02</td></tr></table>",
+        "bbox": [
+            176,
+            771,
+            821,
+            921
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/cdb66695242d8fd58d2f130e1b127c746dbb35450e0b9d1bc182c3a5e7dda336.jpg",
+        "image_caption": [
+            "Figure 3: The positions and weights of grounded instructions as agents navigate by following instructions. Our self-monitoring agent with progress monitor demonstrates the grounded instruction used for action selection shifts gradually from the beginning of instructions towards the end. This is not true of the baseline method. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            99,
+            821,
+            176
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            291,
+            825,
+            362
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.1 COMPARISON WITH PRIOR ART ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            396,
+            431,
+            410
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We first compare the proposed self-monitoring agent with existing approaches. As shown in Table 1, our method achieves significant performance improvement compared to the state of the arts without data augmentation. We achieve $70 \\%$ SR on the seen environment and $57 \\%$ on the unseen environment while the existing best performing method achieved $63 \\%$ and $50 \\%$ SR respectively. When trained with synthetic data5, our approach achieves slightly better performance on the seen environments and significantly better performance on both the validation unseen environments and the test unseen environments when submitted to the test server. We achieve $3 \\%$ and $8 \\%$ improvement on SR on both validation and test unseen environments. Both results with or without data augmentation indicate that our proposed approach is more generalizable to unseen environments. At the time of writing, our self-monitoring agent is ranked #1 on the challenge leader-board among the state of the arts. ",
+        "bbox": [
+            174,
+            429,
+            825,
+            582
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that both Speaker-Follower and our approach in Table 1 use beam search. For comparison without using beam search, please refer to the Appendix. ",
+        "bbox": [
+            174,
+            589,
+            823,
+            617
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Textually grounded agent. Intuitively, an instruction-following agent is required to strongly demonstrate the ability to correctly focus and follow the corresponding part of the instruction as it navigates through an environment. We thus record the distribution of attention weights on instruction at each step as indications of which parts of the instruction being used for action selection. We average all runs across both validation seen and unseen dataset splits. Ideally, we expect to see the distribution of attention weights lies close to a diagonal, where at the beginning, the agent focuses on the beginning of the instruction and shifts its attention towards the end of instruction as it moves closer to the goal. ",
+        "bbox": [
+            174,
+            625,
+            825,
+            736
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate, we use the method with panoramic action space proposed in Fried et al. (2018) as a baseline for comparison. As shown in Figure 3, our self-monitoring agent with progress monitor demonstrates that the positions of grounded instruction over time form a line similar to a diagonal. This result may further indicate that the agent successfully utilizes the attention on instruction to complete the task sequentially. We can also see that both agents were able to focus on the first part of the instruction at the beginning of navigation consistently. However, as the agent moves further in unknown environments, our self-monitoring agent can still successfully identify the parts of instruction that are potentially useful for action selection, whereas the baseline approach becomes uncertain about which part of the instruction should be used for selecting an action. ",
+        "bbox": [
+            174,
+            742,
+            825,
+            868
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/af217e535f6e95b4538bb11edf2466eae2809f0dba173e82a48636419461418b.jpg",
+        "table_caption": [
+            "Table 2: Ablation study showing the effect of each proposed component. All methods use the panoramic action space. Note that, for methods using beam search during inference, only the last selected trajectory is used for evaluating OSR and SPL. \\*: we implemented the model from SpeakerFollower (Fried et al., 2018) with panoramic action space as baseline. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td colspan=\"4\">Inference Mode</td><td colspan=\"4\">Validation-Seen</td><td colspan=\"4\">Validation-Unseen</td></tr><tr><td>#</td><td>Co- Grounding</td><td>Progress Monitor</td><td>Greedy Decoding</td><td>Progress Inference</td><td>Beam Search Aug.</td><td>Data</td><td>NE↓ SR个</td><td>OSR↑</td><td></td><td>SPL↑</td><td>NE↓ SR↑</td><td>OSR↑</td><td>SPL个</td></tr><tr><td>Baseline*</td><td></td><td></td><td></td><td></td><td></td><td>4.36</td><td>0.54</td><td>0.68</td><td>-</td><td>7.22</td><td>0.27</td><td>0.39</td><td>-</td></tr><tr><td>1</td><td>√</td><td></td><td>√</td><td></td><td></td><td>3.65</td><td>0.65</td><td>0.75</td><td>0.56</td><td>6.07</td><td>0.42</td><td>0.57</td><td>0.28</td></tr><tr><td>2</td><td>√</td><td>√</td><td>√</td><td></td><td></td><td>3.72</td><td>0.63</td><td>0.75</td><td>0.56</td><td>5.98</td><td>0.44</td><td>0.58</td><td>0.30</td></tr><tr><td>3</td><td>√</td><td>√</td><td>√</td><td></td><td></td><td>3.22</td><td>0.67</td><td>0.78</td><td>0.58</td><td>5.52</td><td>0.45</td><td>0.56</td><td>0.32</td></tr><tr><td>4</td><td>√</td><td>√</td><td></td><td>√</td><td></td><td>3.56</td><td>0.65</td><td>0.75</td><td>0.58</td><td>5.89</td><td>0.46</td><td>0.60</td><td>0.32</td></tr><tr><td>5</td><td>√</td><td>√</td><td></td><td>√</td><td></td><td>3.18</td><td>0.68</td><td>0.77</td><td>0.58</td><td>5.41</td><td>0.47</td><td>0.59</td><td>0.34</td></tr><tr><td>6</td><td>√</td><td></td><td></td><td></td><td>√</td><td>3.66</td><td>0.66</td><td>0.76</td><td>0.62</td><td>5.70</td><td>0.49</td><td>0.68</td><td>0.42</td></tr><tr><td>7</td><td>√</td><td>√</td><td></td><td></td><td>√</td><td>3.23</td><td>0.70</td><td>0.78</td><td>0.66</td><td>5.04</td><td>0.57</td><td>0.70</td><td>0.51</td></tr><tr><td>8</td><td>√</td><td>√</td><td></td><td></td><td>√</td><td>3.04</td><td>0.71</td><td>0.78</td><td>0.67</td><td>4.62</td><td>0.58</td><td>0.68</td><td>0.52</td></tr></table>",
+        "bbox": [
+            179,
+            167,
+            818,
+            319
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.2 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            354,
+            341,
+            368
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We now discuss the importance of each component proposed in this work. We begin with the same baseline as before (agent with panoramic action space in Fried et al. $\\left( 2 0 1 8 \\right) )$ . ",
+        "bbox": [
+            176,
+            386,
+            823,
+            414
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Co-grounding. When comparing the baseline with row #1 in our proposed method, we can see that our co-grounding agent outperformed the baseline with a large margin. This is due to the fact that we use the LSTM to carry both the textually and visually grounded content, and the decision on each navigable direction is predicted with both textually grounded instruction and the hidden state output of the LSTM. On the other hand, the baseline agent relies on the LSTM to carry visually grounded content, and uses the hidden state output for predicting the textually grounded instruction. As a result, we observed that instead of predicting the instruction needed for selecting a navigable direction, the textually grounded instruction may match with the past sequence of observed images implicitly saved within the LSTM. ",
+        "bbox": [
+            173,
+            420,
+            825,
+            545
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Progress monitor. Given the effective co-grounding, the proposed progress monitor further ensure that the grounded instruction correctly reflects the progress made toward the goal. This further improves the performance especially on the unseen environments as we can see from row #1 and #2. ",
+        "bbox": [
+            176,
+            553,
+            821,
+            594
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "When using the progress inference, the progress monitor serve as a progress indicator for the agent to decide when to move back to the last viewpoint. We can see from row $\\# 2$ and $\\# 4$ that the SR performance can be further improved around $2 \\%$ on both seen and unseen environments. ",
+        "bbox": [
+            176,
+            602,
+            821,
+            643
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, we integrate the output of the progress monitor with the state-factored beam search (Fried et al., 2018), so that the candidate paths compete not only based on the probability of selecting a certain navigable direction but also on the estimated correspondence between the past trajectory and the instruction. As we can see by comparing row $\\# 2$ , #6, and $\\# 7$ , the progress monitor significantly improved the success rate on both seen and unseen environments and is the key for surpassing the state of the arts even without data augmentation. We can also see that when using beam search without progress monitor, the SR on unseen improved $7 \\%$ (row #1 vs #6), while using beam search integrated with progress estimation improved $13 \\%$ (row $\\# 2$ vs $\\# 7$ ). ",
+        "bbox": [
+            173,
+            651,
+            825,
+            762
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Data augmentation. In the above, we have shown each row in our approach contributes to the performance. Each of them increases the success rate and reduces the navigation error incrementally. By further combining them with the data augmentation pre-trained from the speaker (Fried et al., 2018), the SR and OSR are further increased, and the NE is also drastically reduced. Interestingly, the performance improvement introduced by data augmentation is smaller than from Speaker-Follower on the validation sets (see Table 1 for comparison). This demonstrates that our proposed method is more data-efficient. ",
+        "bbox": [
+            173,
+            768,
+            825,
+            866
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/4012ea4e5f3c7e264ef579d57ce4586dbc712a5603d2e05089f6137cae4c9884.jpg",
+        "image_caption": [
+            "Figure 4: Successful self-monitoring agent navigates in two unseen environments. The agent is able to correctly follow the grounded instruction and achieve the goal successfully. The percentage of instruction completeness estimated by the proposed progress monitor gradually increases as the agent navigates and approaches the goal. Finally, the agent grounded the word ”Stop” to stop (see the supplementary material for full figures). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            101,
+            825,
+            381
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.3 QUALITATIVE RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            498,
+            375,
+            512
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To further validate the proposed method, we qualitatively show how the agent navigates through unseen environments by following instructions as shown in Fig. 4. In each figure, the agent follows the grounded instruction (at the top of the figure) and decides to move towards a certain direction (green arrow). For the full figures and more examples of successful and failed agents in both unseen and seen environments, please see the supplementary material. ",
+        "bbox": [
+            174,
+            523,
+            825,
+            593
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Consider the trajectory on the left side in Fig. 4, at step 3, the grounded instruction illustrated that the agent just completed ”turn right” and focuses mainly on ”walk straight to bedroom”. As the agent entered the bedroom, it then shifts the textual grounding to the next action ”Turn left and walk to bed lamp”. Finally, at step 6, the agent completed another ”turn left” and successfully stop at the rug (see the supplementary material for the importance of dealing with duplicate actions). Consider the example on the right side, the agent has already entered the hallway and now turns right to walk across to another room. However, it is ambiguous that which room the instructor is referring to. At step 5, our agent checked out the room on the left first and realized that it does not match with ”Stop in doorway in front of rug”. It then moves to the next room and successfully stops at the goal. ",
+        "bbox": [
+            174,
+            599,
+            825,
+            726
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In both cases, we can see that the completeness estimated by progress monitor gradually increases as the agent steadily navigates toward the goal. We have also observed that the estimated completeness ends up much lower for failure cases (see the supplementary material for further details). ",
+        "bbox": [
+            174,
+            733,
+            825,
+            775
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            795,
+            341,
+            810
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Vision, Language, and Navigation.. There is a plethora work investigating the combination of vision and language for a multitude of applications (Zhou et al., 2018a;b; Antol et al., 2015; Tapaswi et al., 2016; Das et al., 2017), etc. While success has been achieved in these tasks to handle massive corpora of static visual input and text data, a resurgence of interest focuses on equipping an agent with the ability to interact with its surrounding environment for a particular goal such as object manipulation with instructions (Misra et al., 2016; Arkin et al., 2017), grounded language acquisition (Al-Omari et al., 2017; Kollar et al., 2013; Spranger & Steels, 2015; Dubba et al., 2014), embodied question answering (Das et al., 2018; Gordon et al., 2018), and navigation (Matuszek et al., 2013; Hemachandra et al., 2015; Duvallet et al., 2016; Zhu et al., 2017; de Vries et al., 2018; Yuke Zhu, 2017; Mousavian et al., 2018; Wayne et al., 2018; Wang et al., 2018a; Mirowski et al., 2017; 2018; Zamir et al., 2018). In this work, we concentrate on the recently proposed the Visionand-Language Navigation task (Anderson et al., 2018b)—asking an agent to carry out sophisticated natural-language instructions in a 3D environment. This task has application to fields such as robotics; in contrast to traditional map-based navigation systems, navigation with instructions provides a flexible way to generalize across different environments. ",
+        "bbox": [
+            173,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            214
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "A few approaches have been proposed for the VLN task. For example, Anderson et al. (2018b) address the task in the form of a sequence-to-sequence translation model. Yu et al. (2018) introduce a guided feature transformation for textual grounding. Wang et al. (2018b) present a planned-head module by combing model-free and model-based reinforcement learning approaches. Recently, Fried et al. (2018) propose to train a speaker to synthesize new instructions for data augmentation and further use it for pragmatic inference to rank the candidate routes. These approaches leverage attentional mechanisms to select related words from a given instruction when choosing an action, but those agents are deployed to explore the environment without knowing about what progress has been made and how far away the goal is. In this paper, we propose a self-monitoring agent that performs co-grounding on both visual and textual inputs and constantly monitors its own progress toward the goal as a way of regularizing the textual grounding. ",
+        "bbox": [
+            174,
+            222,
+            825,
+            375
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Visual and textual grounding. Visual grounding learns to localize the most relevant object or region in an image given linguistic descriptions, and has been demonstrated as an essential component for a variety of vision tasks like image captioning (Hu et al., 2016; Rohrbach et al., 2016; Lu et al., 2018), visual question answering (Lu et al., 2016b; Agrawal et al., 2018), relationship detection (Lu et al., 2016a; Ma et al., 2018) and referral expression (Nagaraja et al., 2016; Gavrilyuk et al., 2018). In contrast to identifying regions or objects, we perform visual grounding to locate relevant images (views) in a panoramic photo constructed by stitching multiple images with the aim of choosing which direction to go. Extensive efforts have been made to ground language instructions into a sequence of actions (MacMahon et al., 2006; Branavan et al., 2009; Vogel & Jurafsky, 2010; Tellex et al., 2011; Artzi & Zettlemoyer, 2013; Andreas & Klein, 2015; Mei et al., 2016; Cohn et al., 2016; Misra et al., 2017). These early approaches mainly emphasize the incorporation of structural alignment biases between the linguistic structure and sequence of actions (Mei et al., 2016; Andreas & Klein, 2015), and assume the agents are in relatively easy environment where limited visual perception is required to fulfill the instructions. ",
+        "bbox": [
+            174,
+            382,
+            825,
+            575
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            616,
+            318,
+            632
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We introduce a self-monitoring agent which consists of two complementary modules: visual-textual co-grounding module and progress monitor. The visual-textual co-grounding module locates the instruction completed in the past, the instruction needed in the next action, and the moving direction from surrounding images. The progress monitor regularizes and ensures the grounded instruction correctly reflects the progress towards the goal by explicitly estimating the completeness of instruction-following. This estimation is conditioned on the positions and weights of grounded instruction. Our approach sets a new state-of-the-art performance on the standard Room-to-Room dataset on both seen and unseen environments. While we present one instantiation of self-monitoring for a decision-making agent, we believe that this concept can be applied to other domains as well. ",
+        "bbox": [
+            174,
+            659,
+            825,
+            785
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            825,
+            356,
+            840
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This research was partially supported by DARPAs Lifelong Learning Machines (L2M) program, under Cooperative Agreement HR0011-18-2-001. We thank the authors from Fried et al. (2018), Ronghang Hu and Daniel Fried, for communicating with us and providing details of the implementation and synthetic instructions for fair comparison. ",
+        "bbox": [
+            176,
+            867,
+            823,
+            922
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            287,
+            117
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Aishwarya Agrawal, Dhruv Batra, Devi Parikh, and Aniruddha Kembhavi. Dont just assume; look and answer: Overcoming priors for visual question answering. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4971–4980, 2018. ",
+        "bbox": [
+            174,
+            126,
+            823,
+            169
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Muhannad Al-Omari, Paul Duckworth, David C Hogg, and Anthony G Cohn. Natural language acquisition and grounding for embodied robotic systems. In Association for the Advancement of Artificial Intelligence (AAAI), pp. 4349–4356, 2017. ",
+        "bbox": [
+            174,
+            176,
+            820,
+            219
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Peter Anderson, Angel Chang, Devendra Singh Chaplot, Alexey Dosovitskiy, Saurabh Gupta, Vladlen Koltun, Jana Kosecka, Jitendra Malik, Roozbeh Mottaghi, Manolis Savva, et al. On evaluation of embodied navigation agents. arXiv preprint arXiv:1807.06757, 2018a. ",
+        "bbox": [
+            174,
+            227,
+            823,
+            270
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Peter Anderson, Qi Wu, Damien Teney, Jake Bruce, Mark Johnson, Niko Sunderhauf, Ian Reid, ¨ Stephen Gould, and Anton van den Hengel. Vision-and-language navigation: Interpreting visually-grounded navigation instructions in real environments. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), volume 2, 2018b. ",
+        "bbox": [
+            173,
+            277,
+            825,
+            335
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jacob Andreas and Dan Klein. Alignment-based compositional semantics for instruction following. In Proceedings of the Conference on Empirical Methods in Natural Language Processing (EMNLP), 2015. ",
+        "bbox": [
+            173,
+            343,
+            823,
+            386
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Stanislaw Antol, Aishwarya Agrawal, Jiasen Lu, Margaret Mitchell, Dhruv Batra, C Lawrence Zitnick, and Devi Parikh. Vqa: Visual question answering. In In Proceedings of the IEEE International Conference on Computer Vision (ICCV), pp. 2425–2433, 2015. ",
+        "bbox": [
+            174,
+            393,
+            825,
+            438
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jacob Arkin, Matthew R Walter, Adrian Boteanu, Michael E Napoli, Harel Biggie, Hadas KressGazit, and Thomas M Howard. Contextual awareness: Understanding monologic natural language instructions for autonomous robots. In Robot and Human Interactive Communication (RO-MAN), 2017 26th IEEE International Symposium on, pp. 502–509. IEEE, 2017. ",
+        "bbox": [
+            173,
+            445,
+            825,
+            502
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yoav Artzi and Luke Zettlemoyer. Weakly supervised learning of semantic parsers for mapping instructions to actions. Transactions of the Association of Computational Linguistics (ACL), 1: 49–62, 2013. ",
+        "bbox": [
+            174,
+            510,
+            825,
+            553
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yael Benn, Thomas L Webb, Betty PI Chang, Yu-Hsuan Sun, Iain D Wilkinson, and Tom FD Farrow. The neural basis of monitoring goal progress. Frontiers in human neuroscience, 8:688, 2014. ",
+        "bbox": [
+            174,
+            560,
+            821,
+            590
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Elliot T Berkman and Matthew D Lieberman. The neuroscience of goal pursuit. The psychology of goals, pp. 98–126, 2009. ",
+        "bbox": [
+            173,
+            598,
+            823,
+            627
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Satchuthananthavale RK Branavan, Harr Chen, Luke S Zettlemoyer, and Regina Barzilay. Reinforcement learning for mapping instructions to actions. In Proceedings of the Joint Conference of the 47th Annual Meeting of the ACL and the 4th International Joint Conference on Natural Language Processing of the AFNLP: Volume 1-Volume 1, pp. 82–90. Association for Computational Linguistics, 2009. ",
+        "bbox": [
+            173,
+            635,
+            826,
+            707
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Angel Chang, Angela Dai, Thomas Funkhouser, Maciej Halber, Matthias Niessner, Manolis Savva, Shuran Song, Andy Zeng, and Yinda Zhang. Matterport3D: Learning from RGB-D data in indoor environments. International Conference on 3D Vision (3DV), 2017. ",
+        "bbox": [
+            173,
+            713,
+            825,
+            757
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Christopher H Chatham, Eric D Claus, Albert Kim, Tim Curran, Marie T Banich, and Yuko Munakata. Cognitive control reflects context monitoring, not motoric stopping, in response inhibition. PloS one, 7(2):e31546, 2012. ",
+        "bbox": [
+            173,
+            765,
+            823,
+            808
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Trevor Cohn, Cong Duy Vu Hoang, Ekaterina Vymolova, Kaisheng Yao, Chris Dyer, and Gholamreza Haffari. Incorporating structural alignment biases into an attentional neural translation model. In Proceedings of North American Chapter of the Association for Computational Linguistics (NAACL), 2016. ",
+        "bbox": [
+            174,
+            816,
+            825,
+            872
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Abhishek Das, Satwik Kottur, Khushi Gupta, Avi Singh, Deshraj Yadav, Jose M.F. Moura, Devi ´ Parikh, and Dhruv Batra. Visual Dialog. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Abhishek Das, Samyak Datta, Georgia Gkioxari, Stefan Lee, Devi Parikh, and Dhruv Batra. Embodied question answering. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Harm de Vries, Kurt Shuster, Dhruv Batra, Devi Parikh, Jason Weston, and Douwe Kiela. Talk the walk: Navigating new york city through grounded dialogue. arXiv preprint arXiv:1807.03367, 2018. ",
+        "bbox": [
+            173,
+            154,
+            823,
+            196
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Krishna SR Dubba, Miguel R De Oliveira, Gi Hyun Lim, Hamidreza Kasaei, L Seabra Lopes, Ana Tome, and Anthony G Cohn. Grounding language in perception for scene conceptualization in ´ autonomous robots. In Qualitative Representations for Robots: Papers from the AAAI Spring Symposium. Technical report, pp. 26–33. AI Access Foundation, 2014. ",
+        "bbox": [
+            174,
+            204,
+            825,
+            262
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Felix Duvallet, Matthew R Walter, Thomas Howard, Sachithra Hemachandra, Jean Oh, Seth Teller, Nicholas Roy, and Anthony Stentz. Inferring maps and behaviors from natural language instructions. In Experimental Robotics, pp. 373–388. Springer, 2016. ",
+        "bbox": [
+            173,
+            270,
+            825,
+            313
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Daniel Fried, Ronghang Hu, Volkan Cirik, Anna Rohrbach, Jacob Andreas, Louis-Philippe Morency, Taylor Berg-Kirkpatrick, Kate Saenko, Dan Klein, and Trevor Darrell. Speaker-follower models for vision-and-language navigation. In Advances in Neural Information Processing Systems (NIPS), 2018. ",
+        "bbox": [
+            173,
+            320,
+            825,
+            377
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Kirill Gavrilyuk, Amir Ghodrati, Zhenyang Li, and Cees GM Snoek. Actor and action video segmentation from a sentence. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5958–5966, 2018. ",
+        "bbox": [
+            174,
+            385,
+            825,
+            429
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Daniel Gordon, Aniruddha Kembhavi, Mohammad Rastegari, Joseph Redmon, Dieter Fox, and Ali Farhadi. Iqa: Visual question answering in interactive environments. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4089–4098, 2018. ",
+        "bbox": [
+            174,
+            436,
+            825,
+            479
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sachithra Hemachandra, Felix Duvallet, Thomas M. Howard, Nicholas Roy, Anthony Stentz, and Matthew R. Walter. Learning models for following natural language directions in unknown environments. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, May 2015. ",
+        "bbox": [
+            174,
+            487,
+            825,
+            544
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ronghang Hu, Huazhe Xu, Marcus Rohrbach, Jiashi Feng, Kate Saenko, and Trevor Darrell. Natural language object retrieval. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4555–4564, 2016. ",
+        "bbox": [
+            173,
+            551,
+            823,
+            594
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Michał Kempka, Marek Wydmuch, Grzegorz Runc, Jakub Toczek, and Wojciech Jaskowski. Viz- ´ doom: A doom-based ai research platform for visual reinforcement learning. In In Proceedings of the IEEE Conference on Computational Intelligence and Games (CIG), pp. 1–8. IEEE, 2016. ",
+        "bbox": [
+            176,
+            603,
+            823,
+            647
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Thomas Kollar, Jayant Krishnamurthy, and Grant P Strimel. Toward interactive grounded language acqusition. In Robotics: Science and systems, volume 1, pp. 721–732, 2013. ",
+        "bbox": [
+            171,
+            654,
+            823,
+            683
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Cewu Lu, Ranjay Krishna, Michael Bernstein, and Li Fei-Fei. Visual relationship detection with language priors. In European Conference on Computer Vision (ECCV), pp. 852–869. Springer, 2016a. ",
+        "bbox": [
+            174,
+            690,
+            823,
+            734
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jiasen Lu, Jianwei Yang, Dhruv Batra, and Devi Parikh. Hierarchical question-image co-attention for visual question answering. In Advances In Neural Information Processing Systems, pp. 289– 297, 2016b. ",
+        "bbox": [
+            173,
+            742,
+            825,
+            785
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jiasen Lu, Jianwei Yang, Dhruv Batra, and Devi Parikh. Neural baby talk. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 7219–7228, 2018. ",
+        "bbox": [
+            168,
+            792,
+            823,
+            821
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chih-Yao Ma, Asim Kadav, Iain Melvin, Zsolt Kira, Ghassan AlRegib, and Hans Peter Graf. Attend and interact: Higher-order object interactions for video understanding. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018. ",
+        "bbox": [
+            173,
+            829,
+            821,
+            872
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Matt MacMahon, Brian Stankiewicz, and Benjamin Kuipers. Walk the talk: Connecting language, knowledge, and action in route instructions. In Association for the Advancement of Artificial Intelligence (AAAI), 2006. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Cynthia Matuszek, Evan Herbst, Luke Zettlemoyer, and Dieter Fox. Learning to parse natural language commands to a robot control system. In Experimental Robotics, pp. 403–415. Springer, 2013. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Hongyuan Mei, Mohit Bansal, and Matthew R Walter. Listen, attend, and walk: Neural mapping of navigational instructions to action sequences. In Association for the Advancement of Artificial Intelligence (AAAI), volume 1, pp. 2, 2016. ",
+        "bbox": [
+            176,
+            154,
+            821,
+            196
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Piotr Mirowski, Razvan Pascanu, Fabio Viola, Hubert Soyer, Andrew J Ballard, Andrea Banino, Misha Denil, Ross Goroshin, Laurent Sifre, Koray Kavukcuoglu, et al. Learning to navigate in complex environments. In Proceedings of the International Conference on Learning Representations (ICLR), 2017. ",
+        "bbox": [
+            174,
+            205,
+            825,
+            261
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Piotr Mirowski, Matthew Koichi Grimes, Mateusz Malinowski, Karl Moritz Hermann, Keith Anderson, Denis Teplyashin, Karen Simonyan, Koray Kavukcuoglu, Andrew Zisserman, and Raia Hadsell. Learning to navigate in cities without a map. arXiv preprint arXiv:1804.00168, 2018. ",
+        "bbox": [
+            174,
+            270,
+            825,
+            313
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Dipendra Misra, John Langford, and Yoav Artzi. Mapping instructions and visual observations to actions with reinforcement learning. In Proceedings of the Conference on Empirical Methods in Natural Language Processing (EMNLP), 2017. ",
+        "bbox": [
+            174,
+            320,
+            825,
+            364
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Dipendra K Misra, Jaeyong Sung, Kevin Lee, and Ashutosh Saxena. Tell me dave: Context-sensitive grounding of natural language to manipulation instructions. The International Journal of Robotics Research, 35(1-3):281–300, 2016. ",
+        "bbox": [
+            174,
+            372,
+            825,
+            415
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Arsalan Mousavian, Alexander Toshev, Marek Fiser, Jana Kosecka, and James Davidson. Visual representations for semantic target driven navigation. arXiv preprint arXiv:1805.06066, 2018. ",
+        "bbox": [
+            171,
+            422,
+            823,
+            452
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Varun K Nagaraja, Vlad I Morariu, and Larry S Davis. Modeling context between objects for referring expression understanding. In European Conference on Computer Vision (ECCV), pp. 792–807. Springer, 2016. ",
+        "bbox": [
+            174,
+            459,
+            825,
+            502
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Anna Rohrbach, Marcus Rohrbach, Ronghang Hu, Trevor Darrell, and Bernt Schiele. Grounding of textual phrases in images by reconstruction. In European Conference on Computer Vision (ECCV), pp. 817–834. Springer, 2016. ",
+        "bbox": [
+            174,
+            511,
+            825,
+            554
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Michael Spranger and Luc Steels. Co-acquisition of syntax and semantics-an investigation in spatial language. In Association for the Advancement of Artificial Intelligence (AAAI), 2015. ",
+        "bbox": [
+            173,
+            561,
+            823,
+            590
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Makarand Tapaswi, Yukun Zhu, Rainer Stiefelhagen, Antonio Torralba, Raquel Urtasun, and Sanja Fidler. Movieqa: Understanding stories in movies through question-answering. In In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (CVPR), pp. 4631–4640, 2016. ",
+        "bbox": [
+            173,
+            598,
+            825,
+            655
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Stefanie Tellex, Thomas Kollar, Steven Dickerson, Matthew R Walter, Ashis Gopal Banerjee, Seth J Teller, and Nicholas Roy. Understanding natural language commands for robotic navigation and mobile manipulation. In Association for the Advancement of Artificial Intelligence (AAAI), volume 1, pp. 2, 2011. ",
+        "bbox": [
+            174,
+            662,
+            825,
+            719
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems (NIPS), pp. 5998–6008, 2017. ",
+        "bbox": [
+            173,
+            728,
+            823,
+            772
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Adam Vogel and Dan Jurafsky. Learning to follow navigational directions. In Proceedings of the 48th Annual Meeting of the Association for Computational Linguistics (ACL), pp. 806–814. Association for Computational Linguistics, 2010. ",
+        "bbox": [
+            174,
+            779,
+            825,
+            821
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tsun-Hsuan Wang, Hung-Jui Huang, Juan-Ting Lin, Chan-Wei Hu, Kuo-Hao Zeng, and Min Sun. Omnidirectional cnn for visual place recognition and navigation. In In Proceedings of the IEEE Conference on Robotics and Automation (ICRA), 2018a. ",
+        "bbox": [
+            178,
+            830,
+            823,
+            872
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Xin Wang, Wenhan Xiong, Hongmin Wang, and William Yang Wang. Look before you leap: Bridging model-free and model-based reinforcement learning for planned-ahead vision-and-language navigation. In European Conference on Computer Vision (ECCV), 2018b. ",
+        "bbox": [
+            176,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Greg Wayne, Chia-Chun Hung, David Amos, Mehdi Mirza, Arun Ahuja, Agnieszka GrabskaBarwinska, Jack Rae, Piotr Mirowski, Joel Z Leibo, Adam Santoro, et al. Unsupervised predictive memory in a goal-directed agent. arXiv preprint arXiv:1803.10760, 2018. ",
+        "bbox": [
+            176,
+            103,
+            821,
+            146
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Haonan Yu, Xiaochen Lian, Haichao Zhang, and Wei Xu. Guided feature transformation (gft): A neural language grounding module for embodied agents. arXiv preprint arXiv:1805.08329, 2018. ",
+        "bbox": [
+            171,
+            155,
+            823,
+            184
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Eric Kolve Dieter Fox Li Fei-Fei Abhinav Gupta Roozbeh Mottaghi Ali Farhadi Yuke Zhu, Daniel Gordon. Visual Semantic Planning using Deep Successor Representations. In In Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2017. ",
+        "bbox": [
+            174,
+            193,
+            823,
+            236
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "AR Zamir, F Xia, J He, A Sax, J Malik, and S Savarese. Gibson env: Real-world perception for embodied agents. In In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), volume 4, 2018. ",
+        "bbox": [
+            173,
+            243,
+            823,
+            286
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Luowei Zhou, Chenliang Xu, and Jason J Corso. Towards automatic learning of procedures from web instructional videos. In Association for the Advancement of Artificial Intelligence (AAAI), 2018a. ",
+        "bbox": [
+            173,
+            295,
+            826,
+            338
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Luowei Zhou, Yingbo Zhou, Jason J Corso, Richard Socher, and Caiming Xiong. End-to-end dense video captioning with masked transformer. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 8739–8748, 2018b. ",
+        "bbox": [
+            173,
+            347,
+            825,
+            390
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Yuke Zhu, Roozbeh Mottaghi, Eric Kolve, Joseph J Lim, Abhinav Gupta, Li Fei-Fei, and Ali Farhadi. Target-driven visual navigation in indoor scenes using deep reinforcement learning. In In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pp. 3357–3364. IEEE, 2017. ",
+        "bbox": [
+            174,
+            398,
+            823,
+            455
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/0b1b1a5cc99d5de8c2cb8fce4ce476816a6fd7f7c3c13712455f820ccb6f2bd3.jpg",
+        "table_caption": [
+            "Table 3: Performance comparison with the state of arts without beam search: Student-forcing (Anderson et al., 2018b), RPA (Wang et al., 2018b), and Speaker-Follower (Fried et al., 2018). \\*: with data augmentation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">Validation-Seen</td><td colspan=\"4\">Validation-Unseen</td><td colspan=\"4\">Test (unseen)</td></tr><tr><td>NE←</td><td>SR↑</td><td>OSR个</td><td>SPL个</td><td>NE↓</td><td>SR↑</td><td>OSR个</td><td>SPL个</td><td>NE↓</td><td>SR个</td><td>OSR个</td><td>SPL个</td></tr><tr><td>Random</td><td>9.45</td><td>0.16</td><td>0.21</td><td>1</td><td>9.23</td><td>0.16</td><td>0.22</td><td>-</td><td>9.77</td><td>0.13</td><td>0.18</td><td>1</td></tr><tr><td>Student-forcing</td><td>6.01</td><td>0.39</td><td>0.53</td><td>=</td><td>7.81</td><td>0.22</td><td>0.28</td><td>-</td><td>7.85</td><td>0.20</td><td>0.27</td><td>-</td></tr><tr><td>RPA</td><td>5.56</td><td>0.43</td><td>0.53</td><td>-</td><td>7.65</td><td>0.25</td><td>0.32</td><td>1</td><td>7.53</td><td>0.25</td><td>0.33</td><td>1</td></tr><tr><td>Speaker-Follower*</td><td>3.36</td><td>0.66</td><td>0.74</td><td>-</td><td>6.62</td><td>0.36</td><td>0.45</td><td>1</td><td>6.62</td><td>0.35</td><td>0.44</td><td>0.28</td></tr><tr><td>Ours* (Greedy Decoding)</td><td>3.22</td><td>0.67</td><td>0.78</td><td>0.58</td><td>5.52</td><td>0.45</td><td>0.56</td><td>0.32</td><td>5.99</td><td>0.43</td><td>0.55</td><td>0.32</td></tr><tr><td>Ours* (Progress Inference)</td><td>3.18</td><td>0.68</td><td>0.77</td><td>0.58</td><td>5.41</td><td>0.47</td><td>0.59</td><td>0.34</td><td>5.67</td><td>0.48</td><td>0.59</td><td>0.35</td></tr></table>",
+        "bbox": [
+            174,
+            154,
+            823,
+            256
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "SUPPLEMENTARY MATERIALS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            281,
+            426,
+            296
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "COMPARISON WITH PRIOR ART WITHOUT BEAM SEARCH ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            313,
+            573,
+            327
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We provide the comparison with state of the arts without using beam search. The results are shown in Table 3. We can see that our proposed method outperformed existing approaches with a large margin on both validation unseen and test sets. Our method with greedy decoding for action selection improved the SR by $9 \\%$ and $8 \\%$ on validation unseen and test set. When using progress inference for action selection, the performance on the test set significantly improved by $5 \\%$ compared to using greedy decoding, yielding $13 \\%$ improvement over the best existing approach. ",
+        "bbox": [
+            174,
+            339,
+            825,
+            422
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            443,
+            364,
+            455
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Image feature. Similar to previous work, we use the pre-trained ResNet-152 on ImageNet to extract image features. Each image feature is thus a 2048-d vector. The embedded feature vector for each navigable direction is obtained by concatenating an appearance feature with a 4-d orientation feature $\\left[ s i n \\phi ; c o s \\phi ; s i n \\theta ; c o s \\theta \\right]$ , where $\\phi$ and $\\theta$ are the heading and elevation angles. Following the work in Fried et al. (2018), the 4-dim orientation features are tiled 32 times, resulting a embedding feature vector with 2176 dimension. ",
+        "bbox": [
+            174,
+            468,
+            825,
+            550
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Network architecture. The embedding dimension for encoding the navigation instruction is 256. We use a dropout layer with ratio 0.5 after the embedding layer. We then encode the instruction using a regular LSTM, and the hidden state is 512 dimensional. The MLP $g$ used for projecting the raw image feature is $B N  F C  B N  D r o p o u t  R e L U$ . The FC layer projects the 2176-d input vector to a 1024-d vector, and the dropout ratio is set to be 0.5. The hidden state of the LSTM used for carrying the textual and visual information through time in Eq. 1 is 512. We set the maximum length of instruction to be 80, thus the dimension of the attention weights of textual grounding $\\pmb { \\alpha } _ { t }$ is also 80. The dimension of the learnable matrices from Eq. 2 to 5 are: $W _ { x } \\in \\mathbb { R } ^ { 5 1 \\breve { 2 } \\times 5 1 2 }$ , $\\breve { W _ { v } } \\in \\mathbb { R } ^ { 5 1 2 \\times 1 0 2 4 }$ , $W _ { a } \\in \\mathbb { R } ^ { 1 0 2 4 \\times 1 0 2 4 }$ , $W _ { h } \\in \\mathbb { R } ^ { 1 5 3 6 \\times 5 1 2 }$ , and $W _ { p m } \\in \\mathbb { R } ^ { 5 9 2 \\times 1 }$ . ",
+        "bbox": [
+            174,
+            558,
+            825,
+            684
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Training. We use ADAM as the optimizer. The learning rate is $1 e - 4$ with batch size of 64 consistently through out all experiments. When using beam search, we set the beam size to be 15. We perform categorical sampling during training for action selection. ",
+        "bbox": [
+            176,
+            690,
+            821,
+            733
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "SUBMISSION TO VISION AND LANGUAGE NAVIGATION CHALLENGE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            751,
+            645,
+            765
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For evaluating our proposed approach on the unseen test set, we participate in the Vision and Language Navigation challenge and submitted our result with the full proposed approach to the test server. We achieved $61 \\%$ success rate and ranked $\\# 1$ on the test server at the time of writing. ",
+        "bbox": [
+            176,
+            777,
+            820,
+            819
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We follow the submission guidelines, where picking the highest confidence trajectory from multiple trials for each instruction is not permissible. This means that using the beam search for competing and selecting a final trajectory is not allow directly. Similar to the submission from SpeakerFollower (Fried et al., 2018), we record all the viewpoints traversed during the beam search process. The final agent traverses through all recorded trajectories by first reaching the end of one trajectory and backtracking to the shared viewpoint with the next trajectory. This means that the agent could backtrack to the start point during this process. The trajectories are however logged according to the closest previous trajectory, so that when a single agent traverses through all recorded trajectories, the overhead for switching from one trajectory to another can be reduced significantly. The final selected trajectory from beam search is then lastly logged to the trajectory. This therefore yields exactly the same success rate and navigation error, as the metrics are computed according to the last viewpoint from a trajectory. ",
+        "bbox": [
+            174,
+            825,
+            823,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            174
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "QUALITATIVE RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            190,
+            338,
+            205
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We provide and discuss additional qualitative results on the self-monitoring agent navigating on seen and unseen environments. We first discuss four successful examples in Fig. 5 and 6, and followed by two failure examples in Fig. 7. ",
+        "bbox": [
+            176,
+            215,
+            825,
+            258
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "SUCCESSFUL EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            273,
+            348,
+            287
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 5 (a), at the beginning, the agent mostly focuses on ”walk up” for making the first movement. While the agent keeps its attention on ”walk up” as completed instruction or ongoing action, it shifts the attention on instruction to ”turn right” as it walks up the stairs. Once it reached the top of the stairs, it decides to turn right according to the grounded instruction. Once turned right, we can again see that the agent pays attention on both the past action ”turn right” and next action ”walk straight to bedroom”. The agent continues to do so until it decides to stop by grounding on the word ”stop”. ",
+        "bbox": [
+            174,
+            297,
+            825,
+            382
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 5 (b), the agent starts by focusing on both ”enter bedroom from balcony” and ”turn left” to navigate. It correctly shifts the attention on textual grounding on the following instruction. Interestingly, the given instruction ”walk straight across rug to room” at step 3 is ambiguous since there are two rooms across the rug. Our agent decided to sneak out of the first room on the left and noticed that it does not match with the description from instruction. It then moved to another room across the rug and decided to stop because there is a rug inside the room as described. ",
+        "bbox": [
+            174,
+            388,
+            825,
+            472
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 6 (a), the given instruction is ambiguous as it only asks the agent to take actions around the stairs. Since there are multiple duplicated actions described in the instruction, e.g. ”walk up” and ”turn left”, only an agent that is able to precisely follow the instruction step-by-step can successfully complete the task. Otherwise, the agent is likely to stop early before it reaches the goal. The agent also needs to demonstrate its ability to assess the completeness of instruction-following task in order to correctly stop at the right amount of repeated actions as described in the instruction. ",
+        "bbox": [
+            174,
+            479,
+            823,
+            563
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 6 (b), at the beginning (step 0), the agent only focuses on ’left’ for making the first movement (the agent is originally facing the painting). We can see that at each step, the agent correctly focuses on parts of the instruction for making every movements, and it finally believes that the instruction is completed (attention on the last sentence period) and stopped. ",
+        "bbox": [
+            174,
+            570,
+            825,
+            626
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "FAILURE EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            641,
+            318,
+            655
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 7 (a) step 1, although the attention on instruction correctly focused on ”take a left” and ”go down”, the agent failed to follow the instruction and was not able to complete the task. We can however see that the progress monitor correctly reflected that the agent did not follow the given instruction successfully. The agent ended up stopping with progress monitor reporting that only $16 \\%$ of the instruction was completed. ",
+        "bbox": [
+            174,
+            665,
+            823,
+            736
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 7 (b) step 2, the attention on instruction only focuses on ”go down” and thus failed to associate the ”go down steps” with the stairs previously mentioned in ”turn right to stairs”. The agent was however able to follow the rest of the instruction correctly by turning right and stopping near a mirror. Note that, different from Fig. 7 (a), the final estimated completeness of instruction-following from progress monitor is much higher $( 1 6 \\% )$ , which indicates that the agent failed to be aware that it was not correctly following the instruction. ",
+        "bbox": [
+            174,
+            742,
+            825,
+            825
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/87198f7498d1d696f00ef3cfd51e2dfaa7b1d21850ef91ba6dab95ab45c7b00b.jpg",
+        "image_caption": [
+            "Figure 5: Successful self-monitoring agent navigates in two different unseen environments. Given the navigational instruction located at the top of the figure, the agent starts from starting position and follows the instruction towards the goal. The percentage of instruction completeness estimated by the proposed progress monitor gradually increases as the agent navigates and approaches the goal. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            167,
+            823,
+            784
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/65dc0f00e3c9691d8de7217920ca9179144403a32bbba16b7dc2a06c051ad49b.jpg",
+        "image_caption": [
+            "Figure 6: Successful self-monitoring agent navigates in (a) unseen and (b) seen environments. (a) The given instruction is ambiguous as it only asks the agent to take actions around the stairs. Since there are multiple duplicated actions described in the instruction, e.g. ”walk up” and ”turn left”, only an agent that is able to precisely follow the instruction step-by-step can successfully complete the task. Otherwise, the agent is likely to stop early before it reaches the goal. (b) The agent correctly pays attention to parts of the instruction for making decisions on selecting navigable directions. Both the agents decide to stop when shifting the textual grounding on the last sentence period. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            145,
+            825,
+            765
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/91680abed5178b13fa492098d8e349fa1676ffe10db949d546e8aeaa0fbcbc8a.jpg",
+        "image_caption": [
+            "Figure 7: Failed self-monitoring agent navigates in unseen environments. (a) The agent missed the ”take a left” at step 1, and consequently unable to follow the following instruction correctly. However, note that the progress monitor correctly reflected that the instruction was not completed. When the agent decides to end the navigation, it reports that only $16 \\%$ of the instruction was completed. (b) At step 2, the attention on instruction only focuses on ”go down” and thus failed to associate the ”go down steps” with the stairs previously mentioned in ”turn right to stairs”. The agent was however able to follow the rest of the instruction correctly by turning right and stopping near a mirror. Note that, different from (a), the final estimated completeness of instruction-following is much higher, which suggests that the agent failed to correctly be aware of its progress towards the goal. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            174,
+            823,
+            705
+        ],
+        "page_idx": 17
+    }
+]