| 1: Input: policy π to be evaluated, number of samples N, regularization parameter 入 > 0 |
| 2:Let QH+1(·,·)=O andVH+1(-)= 0 |
| forh=H,H-1,...,1do |
| 4: | Take samples (s,a)~ μh,rh ~ r(s,a) and sh ~P(s,a) for eachi∈ [N] |
| 5: | LetAh=∑i∈[N](sn,an）(s,a）T+λI |
| 6: 7: | LetOn=An1（∑N（s,）·(nh+V+1(3)） |
| Let Qn(-,) = (.,·)TOn and Vh(.)=Q(·,π()) |
+
+Remark 4. The key in our construction is the state sdˆ+h in each level, whose feature vector is defined to be $\textstyle \sum _ { c \in { \hat { d } } } e _ { c } / { \hat { d } } ^ { 1 / 2 }$ . In each level, $s _ { h } ^ { \hat { d } + 1 }$ amplifies the $Q$ -value by a $\hat { d } ^ { 1 / 2 }$ factor, due to the linearity of the $Q$ -function. After all the $H$ levels, the value will be amplified by a $\hat { d } ^ { H / 2 }$ factor. Since $s _ { h } ^ { \hat { d } + 1 }$ is not in the support of the data distribution, the only way to estimate the value of the policy istimate the expected reward value in the last level. Our construction forces the estimation error of the last level to be amplified exponentially and thus implies an exponential lower bound.
+
+# 5 UPPER BOUNDS: LOW DISTRIBUTION SHIFT OR POLICY COMPLETENESS ARE SUFFICIENT
+
+In order to illustrate the error amplification issue and discuss conditions that permit sample-efficient offline RL, in this section, we analyze Least-Squares Policy Evaluation when applied to the offline policy evaluation problem under the realizability assumption. The algorithm is presented in Algorithm 1. For simplicity here we assume the policy $\pi$ to be evaluated is deterministic.
+
+Notation. For each $h \in [ H ]$ , define $\Lambda _ { h } = \mathbb { E } _ { ( s , a ) \sim \mu _ { h } } \left[ \phi ( s , a ) \phi ( s , a ) ^ { \top } \right]$ to be the feature covariance matrix at level $h$ . For each $h \in [ H - 1 ]$ , define $\overline { { \Lambda } } _ { h + 1 } = \mathbb { E } _ { ( s , a ) \sim \mu _ { h } , \overline { { s } } \sim P ( \cdot \vert s , a ) } \left[ \phi ( \overline { { s } } , \pi ( \overline { { s } } ) ) \phi ( \overline { { s } } , \pi ( \overline { { s } } ) ) ^ { \top } \right]$ to be the feature covariance matrix of the one-step lookahead distribution at level $h$ . Moreover, define $\overline { { \Lambda } } _ { 1 } = \phi ( s _ { 1 } , \pi ( s _ { 1 } ) ) \phi ( s _ { 1 } , \pi ( s _ { 1 } ) ) ^ { \top }$ . We define $\Phi _ { h }$ to be a $N \times d$ matrix, whose $i$ -th row is $\phi ( s _ { h } ^ { i } , a _ { h } ^ { i } )$ , and define $\overline { { \Phi } } _ { h + 1 }$ to be another $N \times d$ matrix whose $i$ -th row is $\phi ( \overline { { s } } _ { h } ^ { i } , \pi ( \overline { { s } } _ { h } ^ { i } ) )$ . For each $h \in [ H ]$ and $i \in [ N ]$ , define $\xi _ { h } ^ { i } = r _ { h } ^ { i } + V ( \overline { { s } } _ { h } ^ { i } ) - Q ( s _ { h } ^ { i } , a _ { h } ^ { i } )$ . We use $\xi _ { h }$ to denote a vector whose $i$ -th entry is $\xi _ { h } ^ { i }$ .
+
+Now we present a general lemma that characterizes the estimation error of Algorithm 1 by an equality. The proof can be found in Appendix D. Later, we apply this general lemma to special cases.
+
+Lemma 5.1. Suppose $\lambda > 0$ in Algorithm $^ { l }$ , and for the given policy $\pi$ , there exists $\theta _ { 1 } , \theta _ { 2 } , \ldots , \theta _ { d } \in$ $\mathbb { R } ^ { d }$ such that for each $h \in [ H ]$ , $Q _ { h } ^ { \bar { \pi } } ( s , a ) = \phi ( s , \bar { a } ) ^ { \top } \theta _ { h }$ for all $( s , a ) \in S _ { h } \times \mathcal { A }$ . Then we have
+
+$$
+( Q ^ { \pi } ( s _ { 1 } , \pi ( s _ { 1 } ) ) - \hat { Q } ( s _ { 1 } , \pi ( s _ { 1 } ) ) ) ^ { 2 } = \displaystyle \left\| \sum _ { h = 1 } ^ { H } \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \overline { { \Phi } } _ { 2 } \hat { \Lambda } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \cdot \cdot \cdot ( \hat { \Lambda } _ { h } ^ { - 1 } \Phi _ { h } ^ { \top } \xi _ { h } - \lambda \hat { \Lambda } _ { h } ^ { - 1 } \theta _ { h } ) \right\| _ { \overline { { \Lambda } } _ { 1 } } ^ { 2 } .
+$$
+
+Now we consider two special cases where the estimation error in Equation (1) can be upper bounded.
+
+Low Distribution Shift. The first special we focus on is the case where the distribution shift between the data distributions and the distribution induced by the policy to be evaluated is low. To measure the distribution shift formally, our main assumption is as follows.
+
+Assumption 3. We assume that for each $h \in [ H ]$ , there exists $C _ { h } \geq 1$ such that $\overline { { \Lambda _ { h } } } \preceq C _ { h } \Lambda _ { h }$ .
+
+Remark 5. For each Therefore, Assumptio $h \in [ H ]$ , if be r $\sigma _ { \mathrm { m i n } } ( \Lambda _ { h } ) \succeq \frac { 1 } { C _ { h } } I$ for some ssumption $C _ { h } \geq 1$ e . $\overline { { { \Lambda } } } _ { h } \preceq I \preceq C _ { h } \Lambda _ { h }$ $C _ { h } \Lambda _ { h } \succeq I$ to the original version of Assumption 3 as it gives a tighter characterization of the distribution shift.
+
+Now we state the theoretical guarantee of Algorithm 1. The proof can be found in Appendix D.
+
+Theorem 5.2. Suppose for the given policy $\pi$ , there exists $\theta _ { 1 } , \theta _ { 2 } , \ldots , \theta _ { d } \ \in \ \mathbb { R } ^ { d }$ such that for√ each $h \in [ H ] , Q _ { h } ^ { \pi } ( s , a ) \ : = \ : \phi ( s , a ) ^ { \top } \theta _ { h }$ for all $( s , a ) \ \in \ S _ { h } \times { \mathcal A }$ and $\| \theta _ { h } \| _ { 2 } ~ \le ~ H \sqrt { d } .$ 4 Let $\lambda = C H \sqrt { d \log ( d H / \delta ) N }$ for some $C > 0$ . With probability at least $1 - \delta$ , for some $c > 0$ , $\begin{array} { r } { ( Q _ { 1 } ^ { \pi } ( s _ { 1 } , \pi ( s _ { 1 } ) ) - \hat { Q } _ { 1 } ( s _ { 1 } , \pi ( s _ { 1 } ) ) ) ^ { 2 } \leq c \cdot \prod _ { h = 1 } ^ { H } C _ { h } \cdot d H ^ { 5 } \cdot \sqrt { d \log ( d H / \delta ) / N } } \end{array}$ .
+
+Remark 6. The factor $\textstyle \prod _ { h = 1 } ^ { H } C _ { h }$ in Theorem 5.2 implies that the estimation error will be amplified geometrically. Now we discuss how the error is amplified when running Algorithm 1 on the instance in Section 4 to better illustrate the issue. If we run Algorithm 1 on the hard instance in Section 4, when $h = H$ , the estimation error on $V ( s _ { H } ^ { c } )$ would be roughly $N ^ { - 1 / 2 }$ for each $c \in [ \hat { d } ]$ . When using the linear predictor at level $H$ to predict the value of $s _ { H } ^ { * }$ , the error will be amplified by $\hat { d } ^ { 1 / 2 }$ . When $h = H - 1$ , the dataset contains only $s _ { H - 1 } ^ { c }$ for $c \in [ \hat { d } ]$ , and the estimation error on the value of $s _ { H - 1 } ^ { c }$ will be the same as that of $s _ { H } ^ { * }$ , which is roughly $( \hat { d } / N ) ^ { 1 / 2 }$ . Again, the estimation error on the value of $s _ { H - 1 } ^ { * }$ will be $( \hat { d } ^ { 2 } / N ) ^ { 1 / 2 }$ when using the linear predictor at level $H - 1$ . The error will eventually be amplified by a factor of $\hat { d } ^ { H / 2 }$ , which corresponds to the factor $\textstyle \prod _ { h = 1 } ^ { H } C _ { h }$ in Theorem 5.2.
+
+Policy Completeness. In offline RL, another representation condition is closedness under Bellman update (Szepesvari & Munos, 2005; Duan & Wang, 2020), which is stronger than realizability. In ´ the context of offline policy evaluation, we have the following policy completeness assumption.
+
+Assumption 4. For the given policy $\pi$ , for any $h > 1$ and $\theta _ { h } \in \mathbb { R } ^ { d }$ , there exists $\theta ^ { \prime } \in \mathbb { R } ^ { d }$ such that for any $\begin{array} { r } { ( s , a ) \in \mathcal S _ { h - 1 } \times \mathcal A , \mathbb E [ R ( s , a ) ] + \sum _ { s ^ { \prime } \in { \cal S } _ { h } } P ( s ^ { \prime } \mid s , a ) \phi ( s ^ { \prime } , \pi ( s ^ { \prime } ) ) ^ { \top } \theta _ { h } = \phi ( s , a ) ^ { \top } \theta ^ { \prime } . } \end{array}$ .
+
+Under Assumption 4 and the assumption that $\sigma _ { \operatorname* { m i n } } ( \Lambda _ { h } ) \geq \lambda _ { 0 }$ for all $h \in [ H ]$ for some $\lambda _ { 0 } > 0$ , Duan & Wang (2020) have shown that for Algorithm 1, by taking $N = \mathrm { p o l } \bar { \mathrm { y } } ( \bar { H } , d , 1 / \varepsilon , 1 / \lambda _ { 0 } )$ , we have $( Q _ { 1 } ^ { \pi } ( s _ { 1 } , \pi ( s _ { 1 } ) ) - \hat { Q } _ { 1 } ( s _ { 1 } , \pi ( s _ { 1 } ) ) ) ^ { 2 } \leq \varepsilon$ . We refer interested readers to (Duan & Wang, 2020).
+
+We remark that the above analysis again implies that geometric error amplification is a real issue in offline RL, and sample-efficient offline RL is impossible unless the distribution shift is sufficiently low, i.e., $\textstyle \prod _ { h = 1 } ^ { H } C _ { h }$ is bounded, or strong representation condition (e.g. policy completeness) holds.
+
+# 6 CONCLUSION
+
+While the extant body of provable results in the literature largely focus on sufficient conditions for sample-efficient offline RL, this work focuses on obtaining a better understanding of the necessary conditions, where we seek to understand to what extent mild assumptions can imply sample-efficient offline RL. This work shows that for off-policy evaluation, even if we are given a representation that can perfectly represent the value function of the given policy and the data distribution has good coverage over the features, any provable algorithm still requires an exponential number of samples to non-trivially approximate the value of the given policy. These results highlight that provable sample-efficient offline RL is simply not possible unless either the distribution shift condition is sufficiently mild or we have stronger representation conditions that go well beyond realizability.
+
+# ACKNOWLEDGMENTS
+
+The authors would like to thank Akshay Krishnamurthy, Alekh Agarwal, Wen Sun, and Nan Jiang for numerous helpful discussion. Sham M. Kakade gratefully acknowledges funding from the ONR award N00014-18-1-2247, and NSF Awards CCF-1703574 and CCF-1740551. Ruosong Wang was supported in part by the NSF IIS1763562, US Army W911NF1920104, and ONR Grant N000141812861. Research performed while Ruosong Wang was an intern at Microsoft Research.
+
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+# A PROOF OF LEMMA 4.2
+
+Proof. We first verify $Q ^ { \pi }$ is linear for the first $H - 1$ levels. For each $( h , c ) \in [ H - 1 ] \times [ \hat { d } ]$ , we have
+
+$$
+Q _ { h } ^ { \pi } ( s _ { h } ^ { c } , a _ { 1 } ) = R ( s _ { h } ^ { c } , a _ { 1 } ) + R ( s _ { h + 1 } ^ { \hat { d } + 1 } , a _ { 1 } ) + R ( s _ { h + 2 } ^ { \hat { d } + 1 } , a _ { 1 } ) + \ldots + R ( s _ { H } ^ { \hat { d } + 1 } , a _ { 1 } ) = r _ { 0 } \cdot \hat { d } ^ { ( H - h ) / 2 } .
+$$
+
+Moreover, for all $a \in { \mathcal { A } }$ ,
+
+$$
+\begin{array} { r } { \mathcal { Q } _ { h } ^ { \pi } ( s _ { h } ^ { \hat { d } + 1 } , a ) = R ( s _ { h } ^ { \hat { d } + 1 } , a ) + R ( s _ { h + 1 } ^ { \hat { d } + 1 } , a _ { 1 } ) + R ( s _ { h + 2 } ^ { \hat { d } + 1 } , a _ { 1 } ) + . . . + R ( s _ { H } ^ { \hat { d } + 1 } , a _ { 1 } ) = r _ { 0 } \cdot \hat { d } ^ { ( H - h + 1 ) / 2 } . } \end{array}
+$$
+
+Therefore, if we define
+
+$$
+\boldsymbol { \theta } _ { h } ^ { \pi } = \sum _ { c = 1 } ^ { \hat { d } } \boldsymbol { r } _ { 0 } \cdot \hat { d } ^ { ( H - h ) / 2 } \cdot \boldsymbol { e } _ { c } + \sum _ { c = 1 } ^ { \hat { d } } Q _ { h } ^ { \pi } ( s _ { h } ^ { c } , a _ { 2 } ) \cdot \boldsymbol { e } _ { c + \hat { d } } ,
+$$
+
+then $Q _ { h } ^ { \pi } ( s , a ) = \left( \theta _ { h } ^ { \pi } \right) ^ { \top } \phi ( s , a )$ for all $( s , a ) \in S _ { h } \times \mathcal { A }$
+
+Now we verify that the $Q$ -function is linear for the last level. Clearly, for all $c \in [ \hat { d } ]$ and $a \in { \mathcal { A } }$ , $Q _ { H } ^ { \pi } ( s _ { H } ^ { c } , a ) = r _ { 0 }$ and $Q _ { H } ^ { \pi } ( s _ { H } ^ { \hat { d } + 1 } , a ) = r _ { 0 } \cdot \sqrt { \hat { d } }$ . Thus by defining $\begin{array} { r } { \theta _ { H } ^ { \pi } = \sum _ { c = 1 } ^ { d } r _ { 0 } \cdot e _ { c } } \end{array}$ , we have $Q _ { H } ^ { \pi } ( s , a ) = \left( \theta _ { H } ^ { \pi } \right) ^ { \top } \phi ( s , a )$ for all $( s , a ) \in { \cal S } _ { H } \times \mathcal { A }$ .
+
+# B A TECHNICAL LEMMA
+
+We need the following lemma in the proof of our hardness results.
+
+Lemma B.1. Let α be a random variable uniformly distributed on $\{ \alpha _ { + } , \alpha _ { - } \}$ , where $\alpha _ { - } = 1 / 2$ and $\alpha _ { + } = 1 / 2 + \varepsilon$ with $0 < \varepsilon < 1$ . Suppose that $\xi _ { 1 } , \xi _ { 2 } , \ldots , \xi _ { m }$ are i.i.d. $\{ + 1 , - 1 \}$ -valued random variables with $\operatorname* { P r } [ \xi _ { i } = + 1 ] = \alpha$ for all $i \in [ m ]$ . Let $f$ be a function from $\bar { \{ + 1 , - 1 \} } ^ { m }$ to $\{ \alpha _ { + } , \alpha _ { - } \}$ . Suppose $m \le C / \varepsilon ^ { 2 } \log ( 1 / \delta )$ for some fixed constant $C$ . Then
+
+$$
+\operatorname* { P r } [ f ( \xi _ { 1 } , \xi _ { 2 } , \ldots , \xi _ { m } ) \neq \alpha ] > \delta .
+$$
+
+To our best knowledge, Lemma B.1 was first proved in (Chernoff, 1972) and has enormous applications in statistical learning theory (see, e.g., Chapter 5 in (Anthony & Bartlett, 2009)) and bandits (Mannor & Tsitsiklis, 2004).
+
+To prove Lemma B.1, one can first prove that the maximum likelihood estimator (MLE) is optimal and then show that MLE requires $\dot { \Omega ( 1 / \varepsilon ^ { 2 } \log ( 1 / \delta ) ) }$ samples to correctly output $\alpha$ with probability $1 - \delta$ by anti-concentration. Lemma B.1 can also be proved by using information theory. See, e.g., (Kaufmann et al., 2016) for such a proof.
+
+# C ANOTHER HARD INSTANCE
+
+In this section, we present another hard case under a weaker version of Assumption 1. Here the transition operator is stochastic and the reward is deterministic and sparse, meaning that the reward value is non-zero only for the last level. Moreover, the data distributions $\{ \mu _ { h } \} _ { h = 1 } ^ { H }$ are induced by a fixed policy $\pi _ { \mathrm { d a t a } }$ . We also illustrate the construction in Figure 2. Throughout this section, we use $d$ the denote the feature dimension, and we assume $d$ is an even integer for simplicity. We use $\hat { d }$ to denote $d / 2 - 1$ .
+
+In this section, we adopt the following realizability assumption, which is a weaker version of Assumption 1.
+
+Assumption 5 (Realizability). For the policy $\pi : { \mathcal { S } } \to \Delta ( { \mathcal { A } } )$ to be evaluated, there exists $\theta _ { 1 } , \ldots \theta _ { H } \in \mathbb { R } ^ { d }$ such that for all $( s , a ) \in S \times A$ and $h \in [ H ]$ ,
+
+$$
+Q _ { h } ^ { \pi } ( s , a ) = \theta _ { h } ^ { \top } \phi ( s , a ) .
+$$
+
+
+Figure 2: An illustration of the hard instance. Recall that $\hat { d } = d / 2 - 1$ . States on the top are those in the first level $( h = 1 )$ ), while states at the bottom are those in the last level ( $h = H$ ). Dotted line (with arrow) corresponds to transitions associated with actions $a _ { 1 } , a _ { 2 } , \ldots , a _ { \hat { d } }$ , while solid line (with arrow) corresponds to transitions associated with actions $a _ { \hat { d } + 1 } , a _ { \hat { d } + 2 } , \dotsc , a _ { d }$ . We omit the transition associated with $a _ { 1 } , a _ { 2 } , \ldots , a _ { \hat { d } }$ in the figure if all actions give the same transition. For each level $h \in [ H ]$ , $Q$ -values associated with $s _ { h } ^ { 1 } , s _ { h } ^ { 2 } , \ldots , s _ { h } ^ { \hat { d } } , s _ { h } ^ { + } , s _ { h } ^ { - }$ , sdˆh, s+h , s−h are marked on the left, while transition distributions and $Q$ -values associated with + − $s _ { h } ^ { \hat { d } + 1 }$ are marked on the right. Rewards are all deterministic, and the only two states ) with non-zero reward values are marked in black and grey. Consider the fixed policy that returns $\mathit { a } _ { d }$ for all input states. When $r _ { 0 } = 0$ , the value of the policy is 0. When $r _ { 0 } = \hat { d } ^ { - ( \bar { H } - 2 ) / 2 }$ , the value of the policy $s = r _ { 0 } \hat { d } ^ { ( H - 2 ) / 2 } = 1$ .
+
+State Space, Action Space and Transition Operator. In this hard case, the action space ${ \mathcal { A } } =$ $\{ a _ { 1 } , a _ { 2 } , \ldots , a _ { d } \}$ contains $d$ elements. $S _ { 1 }$ contains a single state $s _ { 1 }$ . For each $h \geq 2$ , $S _ { h }$ contains ${ \hat { d } } + 3$ states $s _ { h } ^ { 1 } , s _ { h } ^ { 2 } , \ldots , s _ { h } ^ { \hat { d } } , s _ { h } ^ { \hat { d } + 1 } , s _ { h } ^ { + }$ and $s _ { h } ^ { - }$ .
+
+Let $0 \leq r _ { 0 } \leq \hat { d } ^ { - ( H - 2 ) / 2 }$ be a parameter to be determined. We first define the transition operator for the first level. We have
+
+$$
+P ( s _ { 1 } , a ) = \left\{ \begin{array} { l l } { s _ { 2 } ^ { c } } & { a = a _ { c } , c \in [ \hat { d } ] } \\ { s _ { 2 } ^ { + } } & { a = a _ { \hat { d } + 1 } } \\ { s _ { 2 } ^ { - } } & { a = a _ { \hat { d } + 2 } } \\ { s _ { 2 } ^ { \hat { d } + 1 } } & { a \in \{ a _ { \hat { d } + 3 } , a _ { \hat { d } + 4 } , \dots , a _ { d } \} } \end{array} \right. .
+$$
+
+Now we define the transition operator when $h \in \{ 2 , 3 , \ldots , H - 2 \}$ . For each $h \in \{ 2 , 3 , \ldots , H - 2 \}$ , $a \in { \mathcal { A } }$ and $c \in [ \hat { d } ]$ , we have $P ( s _ { h } ^ { c } , a ) = s _ { h + 1 } ^ { \hat { d } + 1 }$ , $P ( s _ { h } ^ { + } , a ) = s _ { h + 1 } ^ { + }$ and $P ( s _ { h } ^ { - } , a ) = s _ { h + 1 } ^ { - }$ . For each $h \in \{ 2 , 3 , \ldots , H - 2 \}$ and $c \in [ \hat { d } ]$ , we have $P ( s _ { h } ^ { \hat { d } + 1 } , a _ { c } ) = s _ { h + 1 } ^ { c }$ . For all $a \in \left\{ a _ { \hat { d } + 1 } , a _ { \hat { d } + 2 } , \dotsc , a _ { d } \right\}$ ,
+
+we have
+
+$$
+P \left( s _ { h } ^ { \hat { d } + 1 } , a \right) = \left\{ \begin{array} { l l } { s _ { h + 1 } ^ { + } } & { \mathrm { w i t h ~ p r o b a b i l i t y ~ } ( 1 + r _ { 0 } \cdot \hat { d } ^ { ( H - h ) / 2 } ) / 2 } \\ { s _ { h + 1 } ^ { - } } & { \mathrm { w i t h ~ p r o b a b i l i t y ~ } ( 1 - r _ { 0 } \cdot \hat { d } ^ { ( H - h ) / 2 } ) / 2 } \end{array} \right. .
+$$
+
+Now we define the transition operator for the second last level. For all $c \in [ \hat { d } ]$ and $a \in { \mathcal { A } }$ , we have
+
+$$
+P ( s _ { H - 1 } ^ { c } , a ) = \left\{ \begin{array} { l l } { s _ { H } ^ { + } } & { \mathrm { w i t h ~ p r o b a b i l i t y ~ } ( 1 + r _ { 0 } ) / 2 } \\ { s _ { H } ^ { - } } & { \mathrm { w i t h ~ p r o b a b i l i t y ~ } ( 1 - r _ { 0 } ) / 2 } \end{array} \right. .
+$$
+
+For all $a \in { \mathcal { A } }$ , we have $P ( s _ { H - 1 } ^ { + } , a ) = s _ { H } ^ { + }$ and $P ( s _ { H - 1 } ^ { - } , a ) \ = \ s _ { H } ^ { - }$ . For each $c \in [ \hat { d } ]$ , we have $P ( s _ { H - 1 } ^ { \hat { d } + 1 } , a _ { c } ) = s _ { H } ^ { c }$ . For all $a \in \left\{ a _ { \hat { d } + 1 } , a _ { \hat { d } + 2 } , \dotsc , a _ { d } \right\}$ , we have
+
+$$
+P ( s _ { H - 1 } ^ { \hat { d } + 1 } , a ) = \left\{ \begin{array} { l l } { s _ { H } ^ { + } } & { \mathrm { w i t h } \mathrm { p r o b a b i l i t y } \left( 1 + r _ { 0 } \cdot \sqrt { \hat { d } } \right) / 2 } \\ { s _ { H } ^ { - } } & { \mathrm { w i t h } \mathrm { p r o b a b i l i t y } \left( 1 - r _ { 0 } \cdot \sqrt { \hat { d } } \right) / 2 } \end{array} \right. .
+$$
+
+Reward Values. In this hard case, all reward values are deterministic, and reward values can be non-zero only for the last level. Formally, we have
+
+$$
+R ( s , a ) = \left\{ { \begin{array} { l l } { 1 } & { s = s _ { H } ^ { + } } \\ { - 1 } & { s = s _ { H } ^ { - } } \\ { 0 } & { { \mathrm { o t h e r w i s e } } } \end{array} } \right. .
+$$
+
+Feature Mapping. As in the in hard instance in Section 4, let $e _ { 1 } , e _ { 2 } , \ldots , e _ { d }$ be a set of of orthonormal vectors in $\mathbb { R } ^ { d }$ . For the initial state, for each $c \in [ d ]$ , we have $\phi ( s _ { 1 } , a _ { c } ) = e _ { c }$ .
+
+Now we define the feature mapping when $h \in \{ 2 , 3 , \ldots , H \}$ . For each $h \in \{ 2 , 3 , \ldots , H \}$ , $a \in { \mathcal { A } }$ and $c \in [ \hat { d } ] , \phi ( s _ { h } ^ { c } , a ) = e _ { c } , \phi ( s _ { h } ^ { + } , a ) = e _ { \hat { d } + 1 }$ and $\phi ( s _ { h } ^ { - } , a ) = e _ { \hat { d } + 2 }$ . Moreover, for all actions $a \in { \mathcal { A } }$ ,
+
+$$
+\phi ( s _ { h } ^ { \hat { d } + 1 } , a ) = \left\{ { \begin{array} { l l } { e _ { \hat { d } + 2 + c } } & { a = a _ { c } , c \in [ { \hat { d } } ] } \\ { { \frac { 1 } { d ^ { 1 / 2 } } } \left( e _ { 1 } + e _ { 2 } + \ldots + e _ { \hat { d } } \right) } & { a \in \left\{ a _ { \hat { d } + 1 } , a _ { \hat { d } + 2 } , \ldots , a _ { d } \right\} } \end{array} } \right. .
+$$
+
+Clearly, for all (s, a) ∈ S × A, $\| \phi ( s , a ) \| _ { 2 } \leq 1$ .
+
+Verifying Assumption 5. Now we consider the deterministic policy $\pi : { \mathcal { S } } A$ , which is defined to be $\pi ( s ) = a _ { d }$ for all $s \in S$ . We show that Assumption 5 holds.
+
+When $h = 1$ , define
+
+$$
+\theta _ { 1 } = \sum _ { c = 1 } ^ { \hat { d } } r _ { 0 } \cdot \hat { d } ^ { ( H - 3 ) / 2 } \cdot e _ { c } + e _ { \hat { d } + 1 } - e _ { \hat { d } + 2 } + \sum _ { c = 1 } ^ { \hat { d } } r _ { 0 } \cdot \hat { d } ^ { ( H - 2 ) / 2 } \cdot e _ { \hat { d } + 2 + c } .
+$$
+
+For each $h \in \{ 2 , 3 , \ldots , H - 2 \}$ , define
+
+$$
+\theta _ { h } = \sum _ { c = 1 } ^ { \hat { d } } r _ { 0 } \cdot \hat { d } ^ { ( H - h - 1 ) / 2 } \cdot e _ { c } + e _ { \hat { d } + 1 } - e _ { \hat { d } + 2 } + \sum _ { c = 1 } ^ { \hat { d } } r _ { 0 } \cdot \hat { d } ^ { ( H - h - 2 ) / 2 } \cdot e _ { \hat { d } + 2 + c } .
+$$
+
+For the second last level $h = H - 1$ , define
+
+$$
+\theta _ { H - 1 } = \sum _ { c = 1 } ^ { \hat { d } } r _ { 0 } \cdot e _ { c } + e _ { \hat { d } + 1 } - e _ { \hat { d } + 2 } .
+$$
+
+Finally, for the last level $h = H$ , define
+
+$$
+\theta _ { H } = e _ { \hat { d } + 1 } - e _ { \hat { d } + 2 } .
+$$
+
+It can be verified that for each $h \in [ H ]$ , $Q _ { h } ^ { \pi } ( s , a ) = \theta _ { h } ^ { \top } \phi ( s , a )$ for all $( s , a ) \in S _ { h } \times \mathcal { A }$ .
+
+The Data Distributions. For the first level, the data distribution $\mu _ { 1 }$ is defined to be the uniform distribution over $\{ ( s _ { 1 } , a _ { c } ) ~ | ~ c ~ \in ~ [ d ] \}$ . For each $h \geq 2$ , the data distribution $\mu _ { h }$ is a uniform distribution over
+
+$$
+\{ ( s _ { h } ^ { 1 } , a _ { 1 } ) , ( s _ { h } ^ { 2 } , a _ { 1 } ) , \ldots , ( s _ { h } ^ { \hat { d } } , a _ { 1 } ) , ( s _ { h } ^ { + } , a _ { 1 } ) , ( s _ { h } ^ { - } , a _ { 1 } ) , ( s _ { h } ^ { \hat { d } + 1 } , a _ { 1 } ) , ( s _ { h } ^ { \hat { d } + 1 } , a _ { 2 } ) , \ldots , ( s _ { h } ^ { \hat { d } + 1 } , a _ { \hat { d } } ) \} .
+$$
+
+Notice that again $( s _ { h } ^ { \hat { d } + 1 } , a )$ is not in the support of $\mu _ { h }$ for all actions $a \in \left\{ a _ { \hat { d } + 1 } , a _ { \hat { d } + 2 } , \dotsc , a _ { d } \right\}$ . It can be seen that for all $h \in [ H ]$ ,
+
+$$
+\mathbb { E } _ { ( s , a ) \sim \mu _ { h } } [ \phi ( s , a ) \phi ( s , a ) ^ { \top } ] = \frac { 1 } { d } \sum _ { c = 1 } ^ { d } e _ { c } e _ { c } ^ { \top } = \frac { 1 } { d } I .
+$$
+
+Moreover, by defining
+
+$$
+\pi _ { \operatorname* { d a t a } } ( s ) = \left\{ \begin{array} { l l } { \operatorname { U n i f o r m } ( A ) } & { s = s _ { 1 } } \\ { a _ { 1 } } & { s \in \{ s _ { h } ^ { c } \mid h \in \{ 2 , 3 , \ldots , H \} , c \in [ \hat { d } ] \} } \\ { a _ { 1 } } & { s \in \{ s _ { h } ^ { + } \mid h \in \{ 2 , 3 , \ldots , H \} \} } \\ { a _ { 1 } } & { s \in \{ s _ { h } ^ { - } \mid h \in \{ 2 , 3 , \ldots , H \} \} } \\ { \operatorname { U n i f o r m } ( \{ a _ { 1 } , a _ { 2 } , \ldots , a _ { \hat { d } } \} ) } & { s \in \{ s _ { h } ^ { \hat { d } + 1 } \mid h \in \{ 2 , 3 , \ldots , H \} \} } \end{array} \right. ,
+$$
+
+we have $\mu _ { h } = \mu _ { h } ^ { \pi _ { \mathrm { d a t a } } }$ for all $h \in [ H ]$
+
+The Lower Bound. Now we show that it is information-theoretically hard for any algorithm to distinguish the case $r _ { 0 } = 0$ and $r _ { 0 } = \hat { d } ^ { - ( H - 2 ) / 2 }$ in the offline setting by taking samples from the data distributions $\{ \mu _ { h } \} _ { h = 1 } ^ { H }$ . Here we consider the above policy $\pi$ defined above which returns action $a _ { d }$ for all input states. Notice that when $r _ { 0 } = 0$ , the value of the policy would be zero. On the other hand, when $\stackrel { \cdot } { r } _ { 0 } = \hat { d } ^ { - ( H - 2 ) / 2 }$ , the value of the policy would be $r _ { 0 } \cdot \hat { d } ^ { ( \overleftarrow { H } - 2 ) / 2 } = 1 .$ . Therefore, if the algorithm approximates the value of the policy up to an approximation error of $1 / 2$ , then it must distinguish the case that $r _ { 0 } = 0$ and $r _ { 0 } = \hat { d } ^ { - ( H - 2 ) / 2 }$ .
+
+We first notice that for the case $r _ { 0 } = 0$ and $r _ { 0 } = \hat { d } ^ { - ( H - 2 ) / 2 }$ , the data distributions $\{ \mu _ { h } \} _ { h = 1 } ^ { H }$ , the feature mapping $\phi : \mathcal { S } \times \mathcal { A } \mathbb { R } ^ { d }$ , the policy $\pi$ to be evaluated and the reward distributions $R$ are the same. Thus, in order to distinguish the case $r _ { 0 } = 0$ and $r _ { 0 } = \hat { d } ^ { - ( H - 2 ) / 2 }$ , the only way is to query the transition operator $P$ by using sampling taken from the data distributions.
+
+Now, for all state-action pairs $( s , a )$ in the support of the data distributions of the first $H - 2$ lev
+elsing amely only $\mu _ { 1 } , \mu _ { 2 } , \ldots , \mu _ { H - 2 } )$ , the transition opeion distributions of ang-, and $r _ { 0 }$ $( s _ { h } ^ { \hat { d } + 1 } , a _ { \hat { d } + 1 } ) , ( s _ { h } ^ { \hat { d } + 1 } , a _ { \hat { d } + 2 } ) , \dots , ( s _ { h } ^ { \hat { d } + 1 } , a _ { d } )$ support of in the supp f of $h ~ \in ~ [ H \mathrm { ~ - ~ } 2 ]$ . Moreover, for anyso be identical no mat
+$( s , a ) \in \{ s _ { H - 1 } ^ { + } , s _ { H - 1 } ^ { - } , s _ { H - 1 } ^ { \hat { d } + 1 } \} \times \mathcal { A }$ $\mu _ { H - 1 }$ $P ( s , a )$
+ter $r _ { 0 } = 0$ or $r _ { 0 } = \hat { d } ^ { - ( H - 2 ) / 2 }$ . For those state-action pairs $( s , a )$ in the support of $\mu _ { H - 1 }$ with
+$s \not \in \{ s _ { H - 1 } ^ { + } , s _ { H - 1 } ^ { - } , s _ { H - 1 } ^ { \hat { d } + 1 } \}$ , we have
+
+$$
+P ( s , a ) = \left\{ \begin{array} { l l } { s _ { H } ^ { + } } & { \mathrm { w i t h } \ p r o b a b i l i t y \ ( 1 + r _ { 0 } ) / 2 } \\ { s _ { H } ^ { - } } & { \mathrm { w i t h } \ p r o b a b i l i t y \ ( 1 - r _ { 0 } ) / 2 } \end{array} \right. .
+$$
+
+Again, this is because $( s _ { H - 1 } ^ { \hat { d } + 1 } , a )$ is not in the support of $\mu _ { H - 1 }$ for all $a \in \left\{ a _ { \hat { d } + 1 } , a _ { \hat { d } + 2 } , \dotsc , a _ { d } \right\}$
+
+Therefore, in order to distinguish the case $r _ { 0 } = 0$ and $r _ { 0 } = \hat { d } ^ { - ( H - 2 ) / 2 }$ , the agent needs distinguish two transition distributions
+
+$$
+p _ { 1 } = { \left\{ \begin{array} { l l } { s _ { H } ^ { + } } & { { \mathrm { w i t h ~ p r o b a b i l i t y ~ } } 1 / 2 } \\ { s _ { H } ^ { - } } & { { \mathrm { w i t h ~ p r o b a b i l i t y ~ } } 1 / 2 } \end{array} \right. }
+$$
+
+and
+
+$$
+p _ { 2 } = \left\{ s _ { H } ^ { + } \quad \mathrm { w i t h } \mathrm { p r o b a b i l i t y } ( 1 + \hat { d } ^ { - ( H - 2 ) / 2 } ) / 2 \right.
+$$
+
+Again, by Lemma B.1, in order to distinguish $p _ { 1 }$ and $p _ { 2 }$ with probability at least 0.9, one needs $\Omega ( \hat { d } ^ { H - 2 } )$ samples. Formally, we have the following theorem.
+
+Theorem C.1. Suppose Assumption 2 holds, and rewards are deterministic and could be nonezero only for state-action pairs in the last level. Fix an algorithm that takes as input both a policy and a feature mapping. There exists an MDP satisfying Assumption 5, such that for a fixed policy $\pi : { \mathcal { S } } A$ , the algorithm requires $\Omega ( ( d / 2 - 1 ) ^ { H / 2 } )$ samples to output the value of π up to constant additive approximation error with probability at least 0.9.
+
+# D ANALYSIS OF ALGORITHM 1
+
+D.1 PROOF OF LEMMA 5.1
+
+Clearly,
+
+For the first te
+
+$$
+\begin{array} { r l } & { \frac { 1 } { \sqrt { 3 } } \sum _ { k \geq 0 , k \leq 1 } \bar { \zeta } _ { k } \bigg ( \frac { 1 } { 2 } \eta _ { k } ^ { 2 } \bigg ( \mathbb { E } _ { k } \| \hat { \zeta } _ { k } \| ^ { 2 } \eta _ { k } ^ { 2 } \bigg ( \mathbb { E } _ { k } \| \hat { \zeta } _ { k } \| ^ { 2 } \eta _ { k } ^ { 2 } \bigg ) \bigg ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{array}
+$$
+
+Therefore,
+
+$$
+\begin{array} { l } { \displaystyle \hat { \theta } _ { 1 } - \theta _ { 1 } = ( \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } \xi _ { 1 } - \lambda \hat { \Lambda } _ { 1 } ^ { - 1 } \theta _ { 1 } ) + \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \overline { { \Phi } } _ { 2 } ( \theta _ { 2 } - \hat { \theta } _ { 2 } ) } \\ { \displaystyle = ( \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } \xi _ { 1 } - \lambda \hat { \Lambda } _ { 1 } ^ { - 1 } \theta _ { 1 } ) + \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \overline { { \Phi } } _ { 2 } ( \hat { \Lambda } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \xi _ { 2 } - \lambda \hat { \Lambda } _ { 2 } ^ { - 1 } \theta _ { 2 } ) } \\ { \displaystyle \quad \quad + \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \overline { { \Phi } } _ { 2 } \hat { \Lambda } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \overline { { \Phi } } _ { 3 } ( \theta _ { 3 } - \hat { \theta } _ { 3 } ) } \\ { \displaystyle \quad = \ldots } \\ { \displaystyle \quad = \sum _ { h = 1 } ^ { H } \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \overline { { \Phi } } _ { 2 } \hat { \Lambda } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \overline { { \Phi } } _ { 3 } \ldots ( \hat { \Lambda } _ { h } ^ { - 1 } \Phi _ { h } ^ { \top } \xi _ { h } - \lambda \hat { \Lambda } _ { h } ^ { - 1 } \theta _ { h } ) . } \end{array}
+$$
+
+Also note that
+
+$$
+\begin{array} { r } { ( Q ^ { \pi } ( s _ { 1 } , \pi ( s _ { 1 } ) ) - \hat { Q } ( s _ { 1 } , \pi ( s _ { 1 } ) ) ) ^ { 2 } = \| \theta _ { 1 } - \hat { \theta } _ { 1 } \| _ { \bar { \Lambda } _ { 1 } } ^ { 2 } . } \end{array}
+$$
+
+# D.2 PROOF OF THEOREM 5.2
+
+By matrix concentration inequality (Tropp, 2015), we have the following lemma.
+
+Lemma D.1. For each $h \in [ H ]$ , with probability $1 - \delta / ( 4 H )$ , for some universal constant $C$ , we have
+
+$$
+\left\| \frac { 1 } { N } \Phi _ { h } ^ { \top } \Phi _ { h } - \Lambda _ { h } \right\| _ { 2 } \leq C \sqrt { d \log ( d H / \delta ) / N } .
+$$
+
+and
+
+$$
+\left\| \frac { 1 } { N } \overline { { \Phi } } _ { h + 1 } \overline { { \Phi } } _ { h + 1 } - \overline { { \Lambda } } _ { h + 1 } \right\| _ { 2 } \leq C \sqrt { d \log ( d H / \delta ) / N } .
+$$
+
+Therefore, since $\lambda = C H \sqrt { d \log ( d H / \delta ) N }$ , with probability $1 - \delta / ( 4 H )$ , we have
+
+$$
+\hat { \Lambda } _ { h } = \Phi _ { h } ^ { \top } \Phi _ { h } + \lambda I \succeq N \Lambda _ { h } .
+$$
+
+Note that
+
+$$
+\begin{array} { l } { \displaystyle ( { \mathcal Q } ^ { \pi } ( s _ { 1 } , \pi ( s _ { 1 } ) ) - \hat { Q } ( s _ { 1 } , \pi ( s _ { 1 } ) ) ) ^ { 2 } } \\ { \displaystyle \leq H \cdot \left( \sum _ { h = 1 } ^ { H } \left\| \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \overline { { \Phi } } _ { 2 } \hat { \Lambda } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \overline { { \Phi } } _ { 3 } \cdot \cdot \cdot ( \hat { \Lambda } _ { h } ^ { - 1 } \Phi _ { h } ^ { \top } \xi _ { h } - \lambda \hat { \Lambda } _ { h } ^ { - 1 } \theta _ { h } ) \right\| _ { \overline { { \Lambda } } _ { 1 } } ^ { 2 } \right) } \\ { \displaystyle \leq 2 H \cdot \left( \sum _ { h = 1 } ^ { H } \left\| \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \overline { { \Phi } } _ { 2 } \hat { \Lambda } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \overline { { \Phi } } _ { 3 } \cdot \cdot \cdot \hat { \Lambda } _ { h } ^ { - 1 } \Phi _ { h } ^ { \top } \xi _ { h } \right\| _ { \overline { { \Lambda } } _ { 1 } } ^ { 2 } + \sum _ { h = 1 } ^ { H } \left\| \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \overline { { \Phi } } _ { 2 } \hat { \Lambda } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \overline { { \Phi } } _ { 3 } \cdot \cdot \cdot \cdot \lambda \hat { \Lambda } _ { h } ^ { - 1 } \theta _ { h } \right\| _ { \overline { { \Lambda } } _ { 1 } } ^ { 2 } \right) } \end{array}
+$$
+
+For each $h \in [ H ]$ ,
+
+$$
+\begin{array} { r l } & { \quad \| \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \overline { { \Phi } } _ { 2 } \hat { \Lambda } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \overline { { \Phi } } _ { 3 } \cdots \hat { \Lambda } _ { h } ^ { - 1 } \Phi _ { h } ^ { \top } \xi _ { h } \| _ { \overline { { \Lambda } } _ { 1 } } ^ { 2 } } \\ & { \leq \| \Phi _ { 1 } \hat { \Lambda } _ { 1 } ^ { - 1 } \overline { { \Lambda } } _ { 1 } \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \| _ { 2 } \cdot \| \overline { { \Phi } } _ { 2 } \hat { \Lambda } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \overline { { \Phi } } _ { 3 } \cdots \hat { \Lambda } _ { h } ^ { - 1 } \Phi _ { h } ^ { \top } \xi _ { h } \| _ { 2 } ^ { 2 } } \\ & { \leq \| \hat { \Lambda } _ { 1 } ^ { - 1 / 2 } \overline { { \Lambda } } _ { 1 } \hat { \Lambda } _ { 1 } ^ { - 1 / 2 } \| _ { 2 } \cdot \| \Phi _ { 1 } \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \| _ { 2 } \cdot \| \overline { { \Phi } } _ { 2 } \hat { \Lambda } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \overline { { \Phi } } _ { 2 } \cdot \cdot \hat { \Lambda } _ { h } ^ { - 1 } \Phi _ { h } ^ { \top } \xi _ { h } \| _ { 2 } ^ { 2 } } \\ & \leq \| \hat { \Lambda } _ { 1 } ^ { - 1 / 2 } \overline { { \Lambda } } _ { 1 } \hat { \Lambda } _ { 1 } ^ { - 1 / 2 } \| _ { 2 } \cdot \displaystyle \prod _ { h ^ { \prime } = 1 } ^ { h - 1 } \Big ( \| \Phi _ { h ^ { \prime } } \hat { \Lambda } _ { h ^ { \prime } } ^ { - 1 } \Phi _ { h ^ { \prime } } ^ { \top } \| _ { 2 } \cdot \| \hat { \Phi } _ { h ^ { \prime } + 1 } ^ { - 1 / 2 } ( \overline { { \Phi } } _ { h ^ { \prime } + 1 } ^ { \top } \overline { { \Phi } } _ { h ^ { \prime } + 1 } ) \hat { \Lambda } _ h ^ { \prime } \end{array}
+$$
+
+Similarly,
+
+$$
+\begin{array} { r l } & { \quad \| \hat { \boldsymbol { \Lambda } } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \overline { { \Phi } } _ { 2 } \hat { \boldsymbol { \Lambda } } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \overline { { \Phi } } _ { 3 } \cdot \cdot \lambda \hat { \boldsymbol { \Lambda } } _ { h } ^ { - 1 } \theta _ { h } \| _ { \overline { { \boldsymbol { \Lambda } } } _ { 1 } } ^ { 2 } } \\ & { \leq \| \hat { \boldsymbol { \Lambda } } _ { 1 } ^ { - 1 / 2 } \overline { { \boldsymbol { \Lambda } } } _ { 1 } \hat { \boldsymbol { \Lambda } } _ { 1 } ^ { - 1 / 2 } \| _ { 2 } \cdot \displaystyle \prod _ { h ^ { \prime } = 1 } ^ { h - 1 } \Big ( \| \Phi _ { h ^ { \prime } } \hat { \boldsymbol { \Lambda } } _ { h ^ { \prime } } ^ { - 1 } \Phi _ { h ^ { \prime } } ^ { \top } \| _ { 2 } \cdot \| \hat { \boldsymbol { \Lambda } } _ { h ^ { \prime } + 1 } ^ { - 1 / 2 } ( \overline { { \Phi } } _ { h ^ { \prime } + 1 } ^ { \top } \overline { { \Phi } } _ { h ^ { \prime } + 1 } ) \hat { \boldsymbol { \Lambda } } _ { h + 1 } ^ { - 1 / 2 } \| _ { 2 } \Big ) \cdot \lambda ^ { 2 } \cdot \| \theta _ { h } \| _ { \hat { \boldsymbol { \Lambda } } _ { h } ^ { - 1 } } ^ { 2 } } \\ & { \leq \| \hat { \boldsymbol { \Lambda } } _ { 1 } ^ { - 1 / 2 } \overline { { \boldsymbol { \Lambda } } } _ { 1 } \hat { \boldsymbol { \Lambda } } _ { 1 } ^ { - 1 / 2 } \| _ { 2 } \cdot \displaystyle \prod _ { h ^ { \prime } = 1 } ^ { h - 1 } \Big ( \| \Phi _ { h ^ { \prime } } \hat { \boldsymbol { \Lambda } } _ { h ^ { \prime } } ^ { - 1 } \Phi _ { h ^ { \prime } } ^ { \top } \| _ { 2 } \cdot \| \hat { \boldsymbol { \Lambda } } _ { h ^ { \prime } + 1 } ^ { - 1 / 2 } ( \overline { { \Phi } } _ { h ^ { \prime } + 1 } ^ { \top } \overline { { \Phi } } _ { h ^ { \prime } + 1 } ) \hat { \boldsymbol { \Lambda } } _ { h ^ { \prime } + 1 } ^ { - 1 / 2 } \| _ { 2 } \cdot \lambda \cdot H ^ { 2 } d . } \end{array}
+$$
+
+For all $h \in [ H ]$ , we have
+
+$$
+\| \Phi _ { h } \hat { \Lambda } _ { h } ^ { - 1 } \Phi _ { h } ^ { \top } \| _ { 2 } \leq 1
+$$
+
+and
+
+$$
+\begin{array} { r } { \lVert \hat { \Lambda } _ { h } ^ { - 1 / 2 } ( \overline { { \Phi } } _ { h } ^ { \top } \overline { { \Phi } } _ { h } ) \hat { \Lambda } _ { h } ^ { - 1 / 2 } \rVert _ { 2 } \leq \lVert N \hat { \Lambda } _ { h } ^ { - 1 / 2 } \overline { { \Lambda } } _ { h } \hat { \Lambda } _ { h } ^ { - 1 / 2 } \rVert _ { 2 } + \lVert \hat { \Lambda } _ { h } ^ { - 1 / 2 } ( \overline { { \Phi } } _ { h } ^ { \top } \overline { { \Phi } } _ { h } - N \overline { { \Lambda } } _ { h } ) \hat { \Lambda } _ { h } ^ { - 1 / 2 } \rVert _ { 2 } . } \end{array}
+$$
+
+Conditioned on the event in Lemma D.1,
+
+$$
+\hat { \Lambda } _ { h } \succeq N \Lambda _ { h } \succeq \frac { N } { C _ { h } } \overline { { \Lambda } } _ { h } ,
+$$
+
+which implies $\| N \hat { \Lambda } _ { h } ^ { - 1 / 2 } \overline { { \Lambda } } _ { h } \hat { \Lambda } _ { h } ^ { - 1 / 2 } \| \le C _ { h }$ . Moreover, conditioned on the event in Lemma D.1,
+
+$$
+\begin{array} { r } { \| \hat { \boldsymbol { \Lambda } } _ { h } ^ { - 1 / 2 } ( \overline { { \Phi } } _ { h } ^ { \top } \overline { { \Phi } } _ { h } - N \overline { { \boldsymbol { \Lambda } } } _ { h } ) \hat { \boldsymbol { \Lambda } } _ { h } ^ { - 1 / 2 } \| _ { 2 } \leq C \sqrt { d \log ( d H / \delta ) N } / \lambda . } \end{array}
+$$
+
+Thus,
+
+$$
+\| \hat { \boldsymbol { \Lambda } } _ { 1 } ^ { - 1 / 2 } \overline { { \boldsymbol { \Lambda } } } _ { 1 } \hat { \boldsymbol { \Lambda } } _ { 1 } ^ { - 1 / 2 } \| _ { 2 } \leq C _ { 1 } / N .
+$$
+
+and
+
+$$
+\begin{array} { r } { \| \hat { \boldsymbol \Lambda } _ { h } ^ { - 1 / 2 } ( \overline { { \Phi } } _ { h } ^ { \top } \overline { { \Phi } } _ { h } ) \hat { \boldsymbol \Lambda } _ { h } ^ { - 1 / 2 } \| _ { 2 } \leq C _ { h } + C \sqrt { d \log ( d H / \delta ) N } / \lambda . } \end{array}
+$$
+
+Finally, by Theorem 1.2 in (Hsu et al., 2012a), with probability $1 - \delta / ( 4 H )$ , for some constant $C ^ { \prime }$ , we have
+
+$$
+\begin{array} { r } { \| \xi _ { h } \| _ { \Phi _ { h } \hat { \Lambda } _ { h } ^ { - 1 } \Phi _ { h } ^ { \top } } ^ { 2 } \le C ^ { \prime } H ^ { 2 } d \log ( H / \delta ) . } \end{array}
+$$
+
+Therefore,
+
+$$
+\begin{array} { r l } & { \quad \quad \left\| \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \overline { { \Phi } } _ { 2 } \hat { \Lambda } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \overline { { \Phi } } _ { 3 } \cdots \hat { \Lambda } _ { h } ^ { - 1 } \Phi _ { h } ^ { \top } \xi _ { h } \right\| _ { \overline { { \Lambda } } _ { 1 } } ^ { 2 } + \left\| \hat { \Lambda } _ { 1 } ^ { - 1 } \Phi _ { 1 } ^ { \top } \overline { { \Phi } } _ { 2 } \hat { \Lambda } _ { 2 } ^ { - 1 } \Phi _ { 2 } ^ { \top } \overline { { \Phi } } _ { 3 } \cdots \lambda \hat { \Lambda } _ { h } ^ { - 1 } \theta _ { h } \right\| _ { \overline { { \Lambda } } _ { 1 } } ^ { 2 } } \\ & { \leq \frac { C _ { 1 } } { N } ( C _ { 2 } + C \sqrt { d \log ( d / \delta ) N } / \lambda ) \times \cdots \times ( C _ { h } + C \sqrt { d \log ( d / \delta ) N } / \lambda ) \times ( C ^ { \prime } H ^ { 2 } d \log ( H / \delta ) + \lambda H } \\ & { \leq \frac { C _ { 1 } } { N } ( C _ { 2 } + 1 / H ) \times \cdots \times ( C _ { h } + 1 / H ) \times ( C ^ { \prime } H ^ { 2 } d \log ( H / \delta ) + \lambda H ^ { 2 } d ) } \\ & { \leq \frac { e } { N } C _ { 1 } \times C _ { 2 } \times \cdots \times C _ { h } \times ( C ^ { \prime } H ^ { 2 } d \log ( H / \delta ) + C d H ^ { 3 } \sqrt { d \log ( d H / \delta ) N } ) . } \end{array}
+$$
+
+Let $c > 0$ be a large enough constant. We now have
+
+$$
+\mathbb { E } _ { s _ { 1 } } [ ( Q _ { 1 } ^ { \pi } ( s _ { 1 } , \pi ( s _ { 1 } ) ) - \hat { Q } _ { 1 } ( s _ { 1 } , \pi ( s _ { 1 } ) ) ) ^ { 2 } ] \leq c \cdot \left( \prod _ { h = 1 } ^ { H } C _ { h } \right) \cdot d H ^ { 5 } \cdot \sqrt { \frac { d \log ( d H / \delta ) } { N } } .
+$$
\ No newline at end of file
diff --git a/parse/train/30EvkP2aQLD/30EvkP2aQLD_content_list.json b/parse/train/30EvkP2aQLD/30EvkP2aQLD_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..4059f1648204f808d1f05aa505a7ce688165baa8
--- /dev/null
+++ b/parse/train/30EvkP2aQLD/30EvkP2aQLD_content_list.json
@@ -0,0 +1,3002 @@
+[
+ {
+ "type": "text",
+ "text": "WHAT ARE THE STATISTICAL LIMITS OF OFFLINE RL WITH LINEAR FUNCTION APPROXIMATION? ",
+ "text_level": 1,
+ "bbox": [
+ 176,
+ 99,
+ 813,
+ 146
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Ruosong Wang Carnegie Mellon University ruosongw@andrew.cmu.edu ",
+ "bbox": [
+ 184,
+ 170,
+ 411,
+ 212
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Dean P. Foster \nUniversity of Pennsylvania and Amazon \ndean@foster.net \nSham M. Kakade \nUniversity of Washington, Seattle and Microsoft Research \nsham@cs.washington.edu ",
+ "bbox": [
+ 549,
+ 170,
+ 813,
+ 212
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "",
+ "bbox": [
+ 183,
+ 233,
+ 565,
+ 275
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "ABSTRACT ",
+ "text_level": 1,
+ "bbox": [
+ 454,
+ 311,
+ 544,
+ 327
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Offline reinforcement learning seeks to utilize offline (observational) data to guide the learning of (causal) sequential decision making strategies. The hope is that offline reinforcement learning coupled with function approximation methods (to deal with the curse of dimensionality) can provide a means to help alleviate the excessive sample complexity burden in modern sequential decision making problems. However, the extent to which this broader approach can be effective is not well understood, where the literature largely consists of sufficient conditions. ",
+ "bbox": [
+ 233,
+ 343,
+ 764,
+ 440
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "This work focuses on the basic question of what are necessary representational and distributional conditions that permit provable sample-efficient offline reinforcement learning. Perhaps surprisingly, our main result shows that even if: i) we have realizability in that the true value function of every policy is linear in a given set of features and 2) our off-policy data has good coverage over all features (under a strong spectral condition), any algorithm still (information-theoretically) requires a number of offline samples that is exponential in the problem horizon to nontrivially estimate the value of any given policy. Our results highlight that sampleefficient offline policy evaluation is not possible unless significantly stronger conditions hold; such conditions include either having low distribution shift (where the offline data distribution is close to the distribution of the policy to be evaluated) or significantly stronger representational conditions (beyond realizability). ",
+ "bbox": [
+ 233,
+ 443,
+ 764,
+ 608
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "1 INTRODUCTION ",
+ "text_level": 1,
+ "bbox": [
+ 176,
+ 635,
+ 336,
+ 651
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Offline methods (also known as off-policy methods or batch methods) are a promising methodology to alleviate the sample complexity burden in challenging reinforcement learning (RL) settings, particularly those where sample efficiency is paramount (Mandel et al., 2014; Gottesman et al., 2018; Wang et al., 2018; Yu et al., 2019). Off-policy methods are often applied together with function approximation schemes; such methods take sample transition data and reward values as inputs, and approximate the value of a target policy or the value function of the optimal policy. Indeed, many practical deep RL algorithms find their prototypes in the literature of offline RL. For example, when running on off-policy data (sometimes termed as “experience replay”), deep $Q$ -networks (DQN) (Mnih et al., 2015) can be viewed as an analog of Fitted $Q$ -Iteration (Gordon, 1999) with neural networks being the function approximators. More recently, there are an increasing number of both model-free (Laroche et al., 2019; Fujimoto et al., 2019; Jaques et al., 2020; Kumar et al., 2019; Agarwal et al., 2020) and model-based (Ross & Bagnell, 2012; Kidambi et al., 2020) offline RL methods, with steady improvements in performance (Fujimoto et al., 2019; Kumar et al., 2019; Wu et al., 2020; Kidambi et al., 2020). ",
+ "bbox": [
+ 174,
+ 666,
+ 825,
+ 861
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "However, despite the importance of these methods, the extent to which data reuse is possible, especially when off-policy methods are combined with function approximation, is not well understood. For example, deep $Q$ -network requires millions of samples to solve certain Atari games (Mnih et al., 2015). Also important is that in some safety-critical settings, we seek guarantees when offlinetrained policies can be effective (Thomas, 2014; Thomas et al., 2019). A basic question here is that if there are fundamental statistical limits on such methods, where sample-efficient offline RL is simply not possible without further restrictions on the problem. ",
+ "bbox": [
+ 174,
+ 868,
+ 823,
+ 924
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "",
+ "bbox": [
+ 174,
+ 103,
+ 823,
+ 145
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "In the context of supervised learning, it is well-known that empirical risk minimization is sampleefficient if the hypothesis class has bounded complexity. For example, suppose the agent is given a $d$ -dimensional feature extractor, and the ground truth labeling function is a (realizable) linear function with respect to the feature mapping. Here, it is well-known that a polynomial number of samples in $d$ suffice for a given target accuracy. Furthermore, in this realizable case, provided the training data has a good feature coverage, then we will have good accuracy against any test distribution.1 ",
+ "bbox": [
+ 174,
+ 152,
+ 825,
+ 250
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "In the more challenging offline RL setting, it is unclear if sample-efficient methods are possible, even under analogous assumptions. This is our motivation to consider the following question: ",
+ "bbox": [
+ 171,
+ 257,
+ 823,
+ 285
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "What are the statistical limits for offline RL with linear function approximation? ",
+ "text_level": 1,
+ "bbox": [
+ 217,
+ 297,
+ 779,
+ 313
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "Here, one may hope that value estimation for a given policy is possible in the offline RL setting under the analogous set of assumptions that enable sample-efficient supervised learning, i.e., 1) (realizability) the features can perfectly represent the value functions and 2) (good coverage) the feature covariance matrix of our off-policy data has lower bounded eigenvalues. ",
+ "bbox": [
+ 174,
+ 324,
+ 825,
+ 380
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "The extant body of provable methods on offline RL either make representational assumptions that are far stronger than realizability or assume distribution shift conditions that are far stronger than having coverage with regards to the spectrum of the feature covariance matrix of the data distribution. For example, Szepesvari & Munos (2005) analyze offline RL methods by assuming a representational ´ condition where the features satisfy (approximate) closedness under Bellman updates, which is a far stronger representation condition than realizability. Recently, Xie & Jiang (2020a) propose a offline RL algorithm that only requires realizability as the representation condition. However, the algorithm in (Xie & Jiang, 2020a) requires a more stringent data distribution condition. Whether it is possible to design a sample-efficient offline RL method under the realizability assumption and a reasonable data coverage assumption — an open problem in (Chen & Jiang, 2019) — is the focus of this work. ",
+ "bbox": [
+ 174,
+ 387,
+ 825,
+ 526
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "Our Contributions. Perhaps surprisingly, our main result shows that, under only the above two assumptions, it is information-theoretically not possible to design a sample-efficient algorithm to non-trivially estimate the value of a given policy. The following theorem is an informal version of the result in Section 4. ",
+ "bbox": [
+ 174,
+ 542,
+ 825,
+ 598
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "Theorem 1.1 (Informal). In the offline RL setting, suppose the data distributions have (polynomially) lower bounded eigenvalues, and the $Q$ -functions of every policy are linear with respect to a given feature mapping. Any algorithm requires an exponential number of samples in the horizon $H$ to output a non-trivially accurate estimate of the value of any given policy $\\pi$ , with constant probability. ",
+ "bbox": [
+ 174,
+ 603,
+ 825,
+ 672
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "This hardness result states that even if the $Q$ -functions of all polices are linear with respect to the given feature mapping, we still require an exponential number of samples to evaluate any given policy. Note that this representation condition is significantly stronger than assuming realizability with regards to only a single target policy; it assumes realizability for all policies. Regardless, even under this stronger representation condition, it is hard to evaluate any policy, as specified in our hardness result. ",
+ "bbox": [
+ 174,
+ 685,
+ 825,
+ 768
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "This result also formalizes a key issue in offline reinforcement learning with function approximation: geometric error amplification. To better illustrate the issue, in Section 5, we analyze the classical Least-Squares Policy Evaluation (LSPE) algorithm under the realizability assumption, which demonstrates how the error propagates as the algorithm proceeds. Here, our analysis shows that, if we only rely on the realizability assumption, then a far more stringent condition is required for sample-efficient offline policy evaluation: the off-policy data distribution must be quite close to the distribution induced by the policy to be evaluated. ",
+ "bbox": [
+ 173,
+ 775,
+ 825,
+ 873
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "Our results highlight that sample-efficient offline RL is simply not possible unless either the distribution shift condition is sufficiently mild or we have stronger representation conditions that go well beyond realizability. See Section 5 for more details. ",
+ "bbox": [
+ 174,
+ 103,
+ 823,
+ 145
+ ],
+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Furthermore, our hardness result implies an exponential separation on the sample complexity between offline RL and supervised learning, since supervised learning (which is equivalent to offline RL with $H = 1$ ) is possible with polynomial number of samples under the same set of assumptions. ",
+ "bbox": [
+ 176,
+ 152,
+ 820,
+ 195
+ ],
+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "A few additional points are worth emphasizing with regards to our lower bound construction: ",
+ "bbox": [
+ 176,
+ 202,
+ 782,
+ 217
+ ],
+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "• Our results imply that Least-Squares Policy Evaluation (LSPE, i.e., using Bellman backups with linear regression) will fail. Interestingly, while LSPE will provide an unbiased estimator, our results imply that it will have exponential variance in the problem horizon. \nOur construction is simple and does not rely on having a large state or action space: the size of the state space is only $O ( d \\cdot H )$ where $d$ is the feature dimension and $H$ is the planning horizon, and the size of the action space is only is 2. This stands in contrast to other RL lower bounds, which typically require state spaces that are exponential in the problem horizon (e.g. see (Du et al., 2020)). \n• We provide two hard instances, one with a sparse reward (and stochastic transitions) and another with deterministic dynamics (and stochastic rewards). These two hard instances jointly imply that both the estimation error on reward values and the estimation error on the transition probabilities could be geometrically amplified in offline RL. Of possibly broader interest is that our hard instances are, to our knowledge, the first concrete examples showing that geometric error amplification is real in RL problems (even with realizability). While this is a known concern in the analysis of RL algorithms, there have been no concrete examples exhibiting such behavior under only a realizability assumption. ",
+ "bbox": [
+ 173,
+ 223,
+ 826,
+ 445
+ ],
+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "2 RELATED WORK ",
+ "text_level": 1,
+ "bbox": [
+ 176,
+ 477,
+ 344,
+ 492
+ ],
+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "We now survey prior work on offline RL, largely focusing on theoretical results. We also discuss results on the error amplification issue in RL. Concurrent to this work, Xie & Jiang (2020a) propose a offline RL algorithm under the realizability assumption, which requires stronger distribution shift conditions. We will discuss this work shortly. ",
+ "bbox": [
+ 174,
+ 515,
+ 825,
+ 570
+ ],
+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Existing Algorithms and Analysis. Offline RL with value function approximation is closely related to Approximate Dynamic Programming (Bertsekas & Tsitsiklis, 1995). Existing work (Munos, 2003; Szepesvari & Munos, 2005; Antos et al., 2008; Munos & Szepesv ´ ari, 2008; Tosatto et al., ´ 2017; Xie & Jiang, 2020b; Duan & Wang, 2020) that analyze the sample complexity of approximate dynamic programming-based approaches usually make the following two categories of assumptions: (i) representation conditions that assume the function class approximates the value functions well and (ii) distribution shift conditions that assume the given data distribution has sufficient coverage over the state-action space. As mentioned in the introduction, the desired representation condition would be realizability, which only assumes the value function of the policy to be evaluated lies in the function class (for the case of offline policy evaluation) or the optimal value function lies in the function class (for the case of finding near-optimal policies), and existing works usually make stronger assumptions. For example, Szepesvari & Munos (2005); Duan & Wang (2020) assume (ap- ´ proximate) closedness under Bellman updates, which is much stronger than realizability. Whether it is possible to design a sample-efficient offline RL method under the realizability assumption and reasonable data coverage assumption, is left as an open problem in (Chen & Jiang, 2019). ",
+ "bbox": [
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+ "type": "text",
+ "text": "To measure the coverage over the state-action space of the given data distribution, existing works assume the concentrability coefficient (introduced by Munos (2003)) to be bounded. The concentrability coefficient, informally speaking, is the largest possible ratio between the probability for a state-action pair $( s , a )$ to be visited by a policy, and the probability that $( s , a )$ appears on the data distribution. Since we work with linear function approximation in this work, we measure the distribution shift in terms of the spectrum of the feature covariance matrices (see Assumption 2), which is a well-known sufficient condition in the context of supervised learning and is much more natural for the case of linear function approximation. ",
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+ "type": "text",
+ "text": "Concurrent to this work, Xie & Jiang (2020a) propose an algorithm that works under the realizability assumption instead of other stronger representation conditions used in prior work. However, the algorithm in (Xie & Jiang, 2020a) requires a much stronger data distribution condition which assumes a stringent version of concentrability coefficient introduced by (Munos, 2003) to be bounded. In contrast, in this work we measure the distribution shift in terms of the spectrum of the feature covariance matrix of the data distribution, which is more natural than the concentrability coefficient for the case of linear function approximation. ",
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+ "type": "text",
+ "text": "Recently, there has been great interest in approaching offline policy evaluation (Precup, 2000) via importance sampling. For recent work on this topic, see (Dud´ık et al., 2011; Mandel et al., 2014; Thomas et al., 2015; Li et al., 2015; Jiang & Li, 2016; Thomas & Brunskill, 2016; Guo et al., 2017; Wang et al., 2017; Liu et al., 2018; Farajtabar et al., 2018; Xie et al., 2019; Kallus & Uehara, 2019; Liu et al., 2019; Uehara & Jiang, 2019; Kallus & Uehara, 2020; Jiang & Huang, 2020; Feng et al., 2020). Offline policy evaluation with importance sampling incurs exponential variance in the planning horizon when the behavior policy is significantly different from the policy to be evaluated. Bypassing such exponential dependency requires non-trivial function approximation assumptions (Jiang & Huang, 2020; Feng et al., 2020; Liu et al., 2018). Finally, Kidambi et al. (2020) provide a model-based offline RL algorithm, with a theoretical analysis based on hitting times. ",
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+ "type": "text",
+ "text": "Hardness Results. Algorithm-specific hardness results have been known for a long time in the literature of Approximate Dynamic Programming. See Chapter 4 in (Van Roy, 1994) and also (Gordon, 1995; Tsitsiklis & Van Roy, 1996). These works demonstrate that certain approximate dynamic programming-based methods will diverge on hard cases. However, such hardness results only hold for a restricted class of algorithms, and to demonstrate the fundamental difficulty of offline RL, it is more desirable to obtain information-theoretic lower bounds as initiated by Chen & Jiang (2019). ",
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+ "type": "text",
+ "text": "Existing (information-theoretic) exponential lower bounds (Krishnamurthy et al., 2016; Sun et al., 2017; Chen & Jiang, 2019) usually construct unstructured MDPs with an exponentially large state space. Du et al. (2020) prove an exponential lower bound under the assumption that the optimal $Q$ - function is approximately linear. The condition that the optimal $Q$ -function is only approximately linear is crucial for the hardness result in Du et al. (2020). The techniques in (Du et al., 2020) are later generalized to other settings (Kumar et al., 2020; Wang et al., 2020; Mou et al., 2020). ",
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+ "type": "text",
+ "text": "Error Amplification In RL. Error amplification induced by distribution shift and long planning horizon is a known issue in the theoretical analysis of RL algorithms. See (Gordon, 1995; 1996; Munos & Moore, 1999; Ormoneit & Sen, 2002; Kakade, 2003; Zanette et al., 2019) for papers on this topic and additional assumptions that mitigate this issue. Error amplification in offline RL is also observed in empirical works (see e.g. (Fujimoto et al., 2019)). In this work, we provide the first information-theoretic lower bound showing that geometric error amplification is real in offline RL. ",
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+ "type": "text",
+ "text": "3 THE OFFLINE POLICY EVALUATION PROBLEM ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Throughout this paper, for a given integer $H$ , we use $[ H ]$ to denote the set $\\{ 1 , 2 , \\ldots , H \\}$ ",
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+ "type": "text",
+ "text": "Episodic Reinforcement Learning. Let $M = ( \\mathcal { S } , \\mathcal { A } , P , R , H )$ be a Markov Decision Process (MDP) where $s$ is the state space, $\\mathcal { A }$ is the action space, $P : \\mathcal { S } \\times \\mathcal { A } \\Delta ( \\mathcal { S } )$ is the transition operator which takes a state-action pair and returns a distribution over states, $\\Dot { R : } \\Dot { S } \\times \\mathcal { A } \\Delta ( \\mathbb { R } )$ is the reward distribution, $H \\in \\mathbb { Z } _ { + }$ is the planning horizon. For simplicity, we assume a fixed initial state $s _ { 1 } \\in { \\mathcal { S } }$ . A (stochastic) policy $\\pi : S \\Delta ( { \\mathcal { A } } )$ chooses an action $a$ randomly based on the current state $s$ . The policy $\\pi$ induces a (random) trajectory $s _ { 1 } , a _ { 1 } , r _ { 1 } , s _ { 2 } , a _ { 2 } , r _ { 2 } , \\ldots , s _ { H } , a _ { H } , r _ { H }$ , where $a _ { 1 } \\sim \\pi _ { 1 } ( s _ { 1 } )$ , $r _ { 1 } \\sim R ( s _ { 1 } , a _ { 1 } )$ , $s _ { 2 } \\sim P ( s _ { 1 } , a _ { 1 } )$ , $a _ { 2 } \\sim \\pi _ { 2 } ( s _ { 2 } )$ , etc. To streamline our analysis, for each $h \\in [ H ]$ , we use $S _ { h } \\subseteq S$ to denote the set of states at level $h$ , and we assume $S _ { h }$ do not intersect with each other. We assume, almost surely, that $r _ { h } \\in [ - 1 , 1 ]$ for all $h \\in [ H ]$ . ",
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+ "text": "Value Functions. Given a policy $\\begin{array} { r } { \\mathbb { E } \\left[ \\sum _ { h ^ { \\prime } = h } ^ { H } r _ { h ^ { \\prime } } \\mid s _ { h } = s , a _ { h } = a , \\pi \\right] } \\end{array}$ an d , $\\textit { h } \\in \\ [ H ]$ $\\begin{array} { r } { V _ { h } ^ { \\pi } ( s ) = \\mathbb { E } \\left[ \\sum _ { h ^ { \\prime } = h } ^ { H } r _ { h ^ { \\prime } } \\mid s _ { h } = s , \\pi \\right] } \\end{array}$ and $( s , a ) \\in S _ { h } \\times \\mathcal { A }$ , define h. For a policy $Q _ { h } ^ { \\pi } ( s , a ) \\ =$ $\\pi$ , we define $V ^ { \\pi } = V _ { 1 } ^ { \\pi } ( s _ { 1 } )$ to be the value of $\\pi$ from the fixed initial state $s _ { 1 }$ . ",
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+ "type": "text",
+ "text": "Linear Function Approximation. When applying linear function approximation schemes, it is commonly assumed that the agent is given a feature extractor $\\phi : \\mathcal { S } \\times \\bar { \\mathcal { A } } \\mathbb { R } ^ { d }$ which can either be hand-crafted or a pre-trained neural network that transforms a state-action pair to a $d$ -dimensional embedding, and the $Q$ -functions can be predicted by linear functions of the features. In this paper, we are interested in the following realizability assumption. ",
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+ "type": "text",
+ "text": "Assumption 1 (Realizable Linear Function Approximation). For every policy $\\pi : S \\Delta ( { \\mathcal { A } } )$ , there exists $\\theta _ { 1 } ^ { \\pi } , \\ldots \\theta _ { H } ^ { \\pi } \\in \\mathbb { R } ^ { d }$ such that for all $( s , a ) \\in S \\times A$ and $h \\in [ H ] , Q _ { h } ^ { \\pi } ( s , a ) = \\left( \\theta _ { h } ^ { \\pi } \\right) ^ { \\top } \\phi ( s , a ) .$ . ",
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+ "type": "text",
+ "text": "Note that our assumption is much stronger than assuming realizability with regards to a single policy $\\pi$ (say the policy that we wish to evaluate); our assumption imposes realizability for all policies. ",
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+ "text": "Offline Reinforcement Learning. This paper is concerned with the offline RL setting. In this setting, thtributions $\\{ \\mu _ { h } \\} _ { h = 1 } ^ { H }$ oes not have dirwhere for each $h \\in [ H ]$ , $\\mu _ { h } \\in \\Delta ( S _ { h } \\times \\mathcal { A } )$ stead is given access to data d. The inputs of the agent are $H$ datasets $\\{ D _ { h } \\} _ { h = 1 } ^ { H }$ 1 , and for each $h \\in [ H ]$ , $D _ { h }$ consists i.i.d. samples of the form $( s , a , r , s ^ { \\prime } ) \\in$ $S _ { h } \\times \\mathcal { A } \\times \\mathbb { R } \\times \\mathcal { S } _ { h + 1 }$ tuples, where $( s , \\dot { a } ) \\stackrel { . } { \\sim } \\mu _ { h } , r \\sim r ( s , a ) , s ^ { \\prime } \\sim \\bar { P ( } s , a ) $ . ",
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+ "type": "text",
+ "text": "In this paper, we focus on the offline policy evaluation problem with linear function approximation: given a policy accurate estim $\\pi : S \\Delta ( { \\mathcal { A } } )$ f $\\pi$ d a fe(i.e., $V ^ { \\pi }$ re extractor ) approxima $\\phi : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R } ^ { d }$ , the goal is toected datasets $\\{ D _ { h } \\} _ { h = 1 } ^ { H }$ with as few samples as possible. ",
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+ "type": "text",
+ "text": "Notation. For a vector $x \\in \\mathbb { R } ^ { d }$ , we use $\\lVert x \\rVert _ { 2 }$ to denote its $\\ell _ { 2 }$ norm. For a positive semidefinite matrix $A$ , we use $\\| A \\| _ { 2 }$ to denote its operator norm, and $\\sigma _ { \\mathrm { m i n } } ( A )$ to denote its smallest eigenvalue. For two positive semidefinite matrices $A$ and $B$ , we write $A \\succeq B$ to denote the Lowner partial ¨ ordering of matrices, i.e, $A \\succeq B$ if and only if $A - B$ is positive semidefinite. For a policy $\\pi : { \\cal { S } } $ $\\Delta \\left( \\mathcal { A } \\right)$ , we use $\\mu _ { h } ^ { \\pi }$ to denote the marginal distribution of $s _ { h }$ under $\\pi$ , i.e., $\\mu _ { h } ^ { \\pi } ( s ) = \\operatorname* { P r } [ s _ { h } = s \\mid \\pi ]$ . For a vector $x \\in \\mathbb { R } ^ { d }$ and a positive semidefinite matrix $A \\in \\mathbb { R } ^ { d \\times d }$ , we use $\\| x \\| _ { A }$ to denote $\\sqrt { x ^ { \\top } A x }$ . ",
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+ "type": "text",
+ "text": "4 THE LOWER BOUND: REALIZABILITY AND COVERAGE ARE INSUFFICIENT ",
+ "text_level": 1,
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+ "text": "We now present our main hardness result for offline policy evaluation with linear function approximation. It should be evident that without feature coverage in our dataset, realizability alone is clearly not sufficient for sample-efficient estimation. Here, we will make the strongest possible assumption, with regards to the conditioning of the feature covariance matrix. ",
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+ "text": "Assumption 2 (Feature Coverage). For all $( s , a ) \\in S \\times A ,$ , assume our feature map is bounded such that $\\| \\phi ( s , a ) \\| _ { 2 } \\leq 1$ . Furthermore, suppose for each $h \\in [ H ]$ , the data distributions $\\mu _ { h }$ satisfy the following minimum eigenvalue condition: $\\sigma _ { \\mathrm { m i n } } \\left( \\mathbb { E } _ { ( s , a ) \\sim \\mu _ { h } } [ \\dot { \\phi } ( \\dot { s } , a ) \\phi ( s , a ) ^ { \\top } ] \\right) = 1 / d .$ . 2 ",
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+ "type": "text",
+ "text": "Clearly, for the case where $H = 1$ , the realizability assumption (Assumption 1), and feature coverage assumption (Assumption 2) imply that the ordinary least squares estimator will accurately estimate $\\theta _ { 1 }$ .3 Our main result now shows that these assumptions are not sufficient for offline policy evaluation for long horizon problems. ",
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+ "type": "text",
+ "text": "Theorem 4.1. Suppose Assumption 2 holds. Fix an algorithm that takes as input both a policy and $a$ feature mapping. There exists a (deterministic) MDP satisfying Assumption $^ { l }$ , such that for any policy $\\pi : S \\Delta ( { \\mathcal { A } } )$ , the algorithm requires $\\Omega ( ( d / 2 ) ^ { H } )$ samples to output the value of $\\pi$ up to constant additive approximation error with probability at least 0.9. ",
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+ "text": "Although we focus on offline policy evaluation in this work, our hardness result also holds for finding near-optimal policies under Assumption 1 in the offline RL setting. Below we give a simple reduction. At the initial state, if the agent chooses action $a _ { 1 }$ , then the agent receives a fixed reward value (say 0.5) and terminates. If the agent chooses action $a _ { 2 }$ , then the agent transits to our hard instance. In order to find a policy with suboptimality at most 0.5, the agent must evaluate the value of the optimal policy in our hard instance up to an error of 0.5, and hence the hardness result holds. ",
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+ "text": "Remark 1 (The sparse reward case). As stated, the theorem uses a deterministic MDP (with stochastic rewards). See Appendix C for another hard case where the transition is stochastic and the reward is deterministic and sparse (only occurring at two states at $h = H$ ). ",
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+ "text": "Remark 2 (Least-Squares Policy Evaluation (LSPE) has exponential variance). For offline policy evaluation with linear function approximation, the most na¨ıve algorithm here would be LSPE, i.e., using ordinary least squares (OLS) to estimate $\\theta ^ { \\pi }$ , starting at level $h = H$ and then proceeding backwards to level $h = 1$ , using the plug-in estimator from the previous level. Here, LSPE will provide an unbiased estimate (provided the feature covariance matrices are full rank, which will occur with high probability). As a direct corollary, the above theorem implies that LSPE has exponential variance in $H$ . See Section 5 for a more detailed discussion on LSPE. More generally, our theorem implies that there is no estimator that can avoid such exponential dependence in the offline setting. ",
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+ "text": "Remark 3 (Least-Squares Value Iteration (LSVI) versus Least-Squares Policy Iteration (LSPI)). In the offline setting, under Assumptions 1 and 2, in order to find a near-optimal policy, the most na¨ıve algorithm would be LSVI, i.e., using ordinary least squares (OLS) to estimate $\\theta ^ { * }$ , starting at level $h = H$ and then proceeding backwards to level $h = 1$ , using the plug-in estimator from the previous level and the bellman operator. The above theorem implies that LSVI will require an exponential number of samples to find a near-optimal policy. On the other hand, if the regression targets are collected by using rollouts (i.e. on-policy sampling) as in LSPI (Lagoudakis & Parr, 2003), then a polynomial number of samples suffice. See Section D in (Du et al., 2020) for an analysis. Thus, Theorem 4.1 implies an exponential separation on the sample complexity between LSVI and LSPI. Of course, LSPI requires adaptive data samples and thus does not work in the offline setting. ",
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+ "text": "One may wonder if Theorem 4.1 still holds when the data distributions $\\{ \\mu _ { h } \\} _ { h = 1 } ^ { H }$ are induced by a policy. In Appendix C, we prove another exponential sample complexity lower bound under the additional assumption that the data distributions are induced by a fixed policy $\\pi$ . However, under such an assumption, it is impossible to prove a hardness result as strong as Theorem 4.1 (which shows that evaluating any policy is hard), since one can at least evaluate the policy $\\pi$ that induces the data distributions. Nevertheless, we are able to prove the hardness of offline policy evaluation, under a weaker version of Assumption 1. See Appendix $\\textrm { C }$ for more details. ",
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+ "text": "In the remaining part of this section, we give the hard instance construction and the proof of Theorem 4.1. We use $d$ the denote the feature dimension, and we assume $d$ is even for simplicity. We use $\\hat { d }$ to denote $d / 2$ for convenience. We also provide an illustration of the construction in Figure 1. ",
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+ "type": "text",
+ "text": "ac, Action Spcontains d Transtates or. and action space . For each $\\mathcal { A } = \\{ a _ { 1 } , a _ { 2 } \\}$ . For each, for each $h \\in [ H ]$ $S _ { h }$ $\\hat { d } + 1$ $s _ { h } ^ { 1 } , s _ { h } ^ { 2 } , \\ldots , s _ { h } ^ { \\hat { d } }$ $s _ { h } ^ { \\hat { d } + 1 }$ $h \\in [ H - 1 ]$ $c \\in \\{ 1 , 2 , \\ldots , \\hat { d } + 1 \\}$ , we have $P ( s _ { h } ^ { c } , a _ { 1 } ) = s _ { h + 1 } ^ { \\hat { d } + 1 }$ and $P ( s _ { h } ^ { c } , a _ { 1 } ) = s _ { h + 1 } ^ { c }$ . ",
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+ "type": "text",
+ "text": "Reward Distributions. Let $0 \\leq r _ { 0 } \\leq \\hat { d } ^ { - H / 2 }$ be a parameter to be determined. For each $( h , c ) \\in$ $[ H - 1 ] \\times [ \\hat { d } ]$ and $a \\in { \\mathcal { A } }$ , we set $R ( s _ { h } ^ { c } , a ) = 0$ and $R ( s _ { h } ^ { \\hat { d } + 1 } , a ) = r _ { 0 } \\cdot ( \\hat { d } ^ { 1 / 2 } - 1 ) \\cdot \\hat { d } ^ { ( H - h ) / 2 }$ . For the last level, for each $c \\in [ \\hat { d } ]$ and $a \\in { \\mathcal { A } }$ , we set $R ( s _ { H } ^ { c } , a ) = \\left\\{ \\begin{array} { l l } { { 1 } } \\\\ { { - 1 } } \\end{array} \\right.$ with probability with probability $( 1 + r _ { 0 } ) / 2$ $( 1 - r _ { 0 } ) / 2$ so that $\\mathbb { E } [ R ( s _ { H } ^ { c } , a ) ] = r _ { 0 }$ . Moreover, for all actions $a \\in \\mathcal { A } , R ( s _ { H } ^ { \\hat { d } + 1 } , a ) = r _ { 0 } \\cdot \\hat { d } ^ { 1 / 2 }$ . ",
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+ "text": "Feature Mapping. Let $e _ { 1 } , e _ { 2 } , \\ldots , e _ { d }$ be a set of orthonormal vectors in $\\mathbb { R } ^ { d }$ . Here, one possible choice is to set $e _ { 1 } , e _ { 2 } , \\ldots , e _ { d }$ to be the standard basis vectors. For each $( h , c ) \\in [ H ] \\times [ \\hat { d } ]$ , we set $\\phi ( s _ { h } ^ { c } , a _ { 1 } ) = e _ { c } , \\phi ( s _ { h } ^ { c } , a _ { 2 } ) = e _ { c + { \\hat { d } } } $ , and $\\begin{array} { r } { \\phi ( s _ { h } ^ { \\hat { d } + 1 } , a ) = \\sum _ { c \\in \\hat { d } } e _ { c } / \\hat { d } ^ { 1 / 2 } } \\end{array}$ for all $a \\in { \\mathcal { A } }$ . ",
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+ "type": "text",
+ "text": "Verifying Assumption 1. The following lemma shows that Assumption 1 holds for our construction. The formal proof can be found in Appendix A. ",
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+ "text": "Lemma 4.2. For every policy $\\pi : S \\Delta ( { \\mathcal { A } } )$ , for each $h \\in [ H ]$ , for all $( s , a ) \\in S _ { h } \\times \\mathcal { A } ,$ , we have $Q _ { h } ^ { \\pi } ( s , a ) = \\left( \\theta _ { h } ^ { \\pi } \\right) ^ { \\top } \\phi ( s , a )$ for some $\\theta _ { h } ^ { \\pi } \\in \\mathbb { R } ^ { d }$ . ",
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+ "text": "Figure 1: An illustration of the hard instance. Recall that $\\hat { d } = d / 2$ . States on the top are those in the first level $( h = 1 )$ ), while states at the bottom are those in the last level $h = H$ ). Solid line (with arrow) corresponds to transitions associated with action $a _ { 1 }$ , while dotted line (with arrow) corresponds to transitions associated with action $a _ { 2 }$ . For each level $h \\in [ H ]$ , reward values and $Q$ -values associated with $s _ { h } ^ { 1 } , s _ { h } ^ { 2 } , \\ldots , s _ { h } ^ { \\hat { d } }$ are marked on the left, while reward values and $Q$ -values associated with $s _ { h } ^ { \\hat { d } + 1 }$ are mark on the right. Rewards and transitions are all deterministic, except for the reward distributions associated with $s _ { H } ^ { 1 } , s _ { H } ^ { 2 } , \\ldots , s _ { H } ^ { \\hat { d } }$ . We mark the ex tation of the reward uniformly at random from those states in the dashed rectangle, i.e., $h \\in [ H ]$ $\\{ s _ { h } ^ { 1 } , s _ { h } ^ { 2 } , \\ldots , s _ { h } ^ { \\hat { d } } \\}$ $\\mu _ { h }$ , while the action is chosen uniformly at random from $\\{ a _ { 1 } , a _ { 2 } \\}$ . Suppose the initial state is $s _ { 1 } ^ { \\hat { d } + 1 }$ . When $r _ { 0 } = 0$ , the value of the policy is 0. When $r _ { 0 } = \\hat { d } ^ { - H / 2 }$ , the value of the policy is $r _ { 0 } \\cdot \\hat { d } ^ { H / 2 } = 1$ . ",
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+ "text": "The Data Distributions. For each level $h \\in [ H ]$ , the data distribution $\\mu _ { h }$ is a uniform distribution over $\\{ ( s _ { h } ^ { 1 } , a _ { 1 } ) , ( s _ { h } ^ { 1 } , a _ { 2 } ) , ( s _ { h } ^ { 2 } , a _ { 1 } ) , ( s _ { h } ^ { 2 } , a _ { 2 } ) , \\ldots , ( s _ { h } ^ { \\hat { d } } , a _ { 1 } ) , ( s _ { h } ^ { \\hat { d } } , a _ { 2 } ) \\}$ . Notice that $( s _ { h } ^ { \\hat { d } + 1 } , a )$ is not in the hsupport of $\\mu _ { h }$ hfor all $a \\in { \\mathcal { A } }$ h h. It can be seen that, $\\begin{array} { r } { \\mathbb { E } _ { ( s , a ) \\sim \\mu _ { h } } \\left[ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } \\right] = \\frac { 1 } { d } \\sum _ { c = 1 } ^ { d } e _ { c } e _ { c } ^ { \\top } = \\frac { 1 } { d } I . } \\end{array}$ ",
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+ "type": "text",
+ "text": "The Lower Bound. We show that it is information-theoretically hard for any algorithm to distinguish the case $r _ { 0 } = 0$ and $r _ { 0 } = \\hat { d } ^ { - H / 2 }$ . We fix the initial state to be $s _ { 1 } ^ { \\hat { d } + 1 }$ , and consider any policy $\\pi$ . When $r _ { 0 } = 0$ , all reward values will be zero, and thus the value of $\\pi$ is zero. On the other hand, when $r _ { 0 } = \\hat { d } ^ { - H / 2 }$ , the value of $\\pi$ would be $r _ { 0 } \\cdot \\hat { d } ^ { H / 2 } = 1$ . Thus, if the algorithm approximates the value of the policy up to an error of $1 / 2$ , then it must distinguish the case that $r _ { 0 } = 0$ and $r _ { 0 } = \\hat { d } ^ { - H / 2 }$ . ",
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+ "type": "text",
+ "text": "We first notice that for the case $r _ { 0 } = 0$ and $r _ { 0 } = \\hat { d } ^ { - H / 2 }$ , the data distributions $\\{ \\mu _ { h } \\} _ { h = 1 } ^ { H }$ , the feature mapping $\\phi : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R } ^ { d }$ , the policy $\\pi$ to be evaluated and the transition operator $P$ are the same. Thus, in order to distinguish the case $r _ { 0 } = 0$ and $r _ { 0 } = \\hat { d } ^ { - H / 2 }$ , the only way is to query the reward distribution by using sampling taken from the data distributions. For all state-action pairs $( s , a )$ in the support of the data distributions of the first $H - 1$ levels, the reward distributions will be identical. This is because for all $s \\in \\mathcal { S } _ { h } \\setminus \\{ s _ { h } ^ { \\hat { d } + 1 } \\}$ and $a \\in { \\mathcal { A } }$ , we have $R ( s , a ) = 0$ . For the case $r _ { 0 } = 0$ and $r _ { 0 } = \\hat { d } ^ { - H / 2 }$ , for all state-action pairs $( s , a )$ in the support of the data distribution of the last level, $R ( s , a ) = \\left\\{ { 1 \\atop - 1 } \\right.$ with probability with probability $( 1 + r _ { 0 } ) / 2$ $\\dot { ( 1 - r _ { 0 } ) / 2 }$ Therefore, to distinguish the case that $r _ { 0 } ~ = ~ 0$ and $r _ { 0 } ~ = ~ \\hat { d } ^ { - H / 2 }$ , the agent needs to distinguish two reward distributions $r _ { 1 } = \\left\\{ { \\begin{array} { l l } { 1 } & { { \\mathrm { w i t h ~ p r o b a b i l i t y ~ } } 1 / 2 } \\\\ { - 1 } & { { \\mathrm { w i t h ~ p r o b a b i l i t y ~ } } 1 / 2 } \\end{array} } \\right.$ and $r _ { 2 } = \\left\\{ \\begin{array} { l l } { { \\stackrel { - } { 1 } } } \\\\ { { - 1 } } \\end{array} \\right.$ with probability with probability $\\begin{array} { c } { { ( 1 + \\hat { d } ^ { - H / 2 } ) / 2 } } \\\\ { { ( 1 - \\hat { d } ^ { - H / 2 } ) / 2 } } \\end{array}$ . Now we invoke Lemma B.1 in Section B by setting $\\varepsilon = \\hat { d } ^ { - H / 2 } / 2$ and $\\delta = 0 . 9$ . By Lemma B.1, in order to distinguish $r _ { 1 }$ and $r _ { 2 }$ with probability at least 0.9, any algorithm requires $\\Omega ( \\hat { d } ^ { H } )$ samples. ",
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+ "table_caption": [
+ "Algorithm 1 Least-Squares Policy Evaluation "
+ ],
+ "table_footnote": [],
+ "table_body": " | 1: Input: policy π to be evaluated, number of samples N, regularization parameter 入 > 0 |
| 2:Let QH+1(·,·)=O andVH+1(-)= 0 |
| forh=H,H-1,...,1do |
| 4: | Take samples (s,a)~ μh,rh ~ r(s,a) and sh ~P(s,a) for eachi∈ [N] |
| 5: | LetAh=∑i∈[N](sn,an）(s,a）T+λI |
| 6: 7: | LetOn=An1（∑N（s,）·(nh+V+1(3)） |
| Let Qn(-,) = (.,·)TOn and Vh(.)=Q(·,π()) |
",
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+ "text": "",
+ "bbox": [
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+ "text": "Remark 4. The key in our construction is the state sdˆ+h in each level, whose feature vector is defined to be $\\textstyle \\sum _ { c \\in { \\hat { d } } } e _ { c } / { \\hat { d } } ^ { 1 / 2 }$ . In each level, $s _ { h } ^ { \\hat { d } + 1 }$ amplifies the $Q$ -value by a $\\hat { d } ^ { 1 / 2 }$ factor, due to the linearity of the $Q$ -function. After all the $H$ levels, the value will be amplified by a $\\hat { d } ^ { H / 2 }$ factor. Since $s _ { h } ^ { \\hat { d } + 1 }$ is not in the support of the data distribution, the only way to estimate the value of the policy istimate the expected reward value in the last level. Our construction forces the estimation error of the last level to be amplified exponentially and thus implies an exponential lower bound. ",
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+ "type": "text",
+ "text": "5 UPPER BOUNDS: LOW DISTRIBUTION SHIFT OR POLICY COMPLETENESS ARE SUFFICIENT ",
+ "text_level": 1,
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+ "text": "In order to illustrate the error amplification issue and discuss conditions that permit sample-efficient offline RL, in this section, we analyze Least-Squares Policy Evaluation when applied to the offline policy evaluation problem under the realizability assumption. The algorithm is presented in Algorithm 1. For simplicity here we assume the policy $\\pi$ to be evaluated is deterministic. ",
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+ "type": "text",
+ "text": "Notation. For each $h \\in [ H ]$ , define $\\Lambda _ { h } = \\mathbb { E } _ { ( s , a ) \\sim \\mu _ { h } } \\left[ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } \\right]$ to be the feature covariance matrix at level $h$ . For each $h \\in [ H - 1 ]$ , define $\\overline { { \\Lambda } } _ { h + 1 } = \\mathbb { E } _ { ( s , a ) \\sim \\mu _ { h } , \\overline { { s } } \\sim P ( \\cdot \\vert s , a ) } \\left[ \\phi ( \\overline { { s } } , \\pi ( \\overline { { s } } ) ) \\phi ( \\overline { { s } } , \\pi ( \\overline { { s } } ) ) ^ { \\top } \\right]$ to be the feature covariance matrix of the one-step lookahead distribution at level $h$ . Moreover, define $\\overline { { \\Lambda } } _ { 1 } = \\phi ( s _ { 1 } , \\pi ( s _ { 1 } ) ) \\phi ( s _ { 1 } , \\pi ( s _ { 1 } ) ) ^ { \\top }$ . We define $\\Phi _ { h }$ to be a $N \\times d$ matrix, whose $i$ -th row is $\\phi ( s _ { h } ^ { i } , a _ { h } ^ { i } )$ , and define $\\overline { { \\Phi } } _ { h + 1 }$ to be another $N \\times d$ matrix whose $i$ -th row is $\\phi ( \\overline { { s } } _ { h } ^ { i } , \\pi ( \\overline { { s } } _ { h } ^ { i } ) )$ . For each $h \\in [ H ]$ and $i \\in [ N ]$ , define $\\xi _ { h } ^ { i } = r _ { h } ^ { i } + V ( \\overline { { s } } _ { h } ^ { i } ) - Q ( s _ { h } ^ { i } , a _ { h } ^ { i } )$ . We use $\\xi _ { h }$ to denote a vector whose $i$ -th entry is $\\xi _ { h } ^ { i }$ . ",
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+ "type": "text",
+ "text": "Now we present a general lemma that characterizes the estimation error of Algorithm 1 by an equality. The proof can be found in Appendix D. Later, we apply this general lemma to special cases. ",
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+ "text": "Lemma 5.1. Suppose $\\lambda > 0$ in Algorithm $^ { l }$ , and for the given policy $\\pi$ , there exists $\\theta _ { 1 } , \\theta _ { 2 } , \\ldots , \\theta _ { d } \\in$ $\\mathbb { R } ^ { d }$ such that for each $h \\in [ H ]$ , $Q _ { h } ^ { \\bar { \\pi } } ( s , a ) = \\phi ( s , \\bar { a } ) ^ { \\top } \\theta _ { h }$ for all $( s , a ) \\in S _ { h } \\times \\mathcal { A }$ . Then we have ",
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+ "type": "equation",
+ "img_path": "images/ae3c942c91242009b562629605d1f25f0be72411e576b0506d5890cac0fed046.jpg",
+ "text": "$$\n( Q ^ { \\pi } ( s _ { 1 } , \\pi ( s _ { 1 } ) ) - \\hat { Q } ( s _ { 1 } , \\pi ( s _ { 1 } ) ) ) ^ { 2 } = \\displaystyle \\left\\| \\sum _ { h = 1 } ^ { H } \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\cdot \\cdot \\cdot ( \\hat { \\Lambda } _ { h } ^ { - 1 } \\Phi _ { h } ^ { \\top } \\xi _ { h } - \\lambda \\hat { \\Lambda } _ { h } ^ { - 1 } \\theta _ { h } ) \\right\\| _ { \\overline { { \\Lambda } } _ { 1 } } ^ { 2 } .\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "Now we consider two special cases where the estimation error in Equation (1) can be upper bounded. ",
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+ "type": "text",
+ "text": "Low Distribution Shift. The first special we focus on is the case where the distribution shift between the data distributions and the distribution induced by the policy to be evaluated is low. To measure the distribution shift formally, our main assumption is as follows. ",
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+ "text": "Assumption 3. We assume that for each $h \\in [ H ]$ , there exists $C _ { h } \\geq 1$ such that $\\overline { { \\Lambda _ { h } } } \\preceq C _ { h } \\Lambda _ { h }$ . ",
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+ "type": "text",
+ "text": "Remark 5. For each Therefore, Assumptio $h \\in [ H ]$ , if be r $\\sigma _ { \\mathrm { m i n } } ( \\Lambda _ { h } ) \\succeq \\frac { 1 } { C _ { h } } I$ for some ssumption $C _ { h } \\geq 1$ e . $\\overline { { { \\Lambda } } } _ { h } \\preceq I \\preceq C _ { h } \\Lambda _ { h }$ $C _ { h } \\Lambda _ { h } \\succeq I$ to the original version of Assumption 3 as it gives a tighter characterization of the distribution shift. ",
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+ "type": "text",
+ "text": "Now we state the theoretical guarantee of Algorithm 1. The proof can be found in Appendix D. ",
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+ "type": "text",
+ "text": "Theorem 5.2. Suppose for the given policy $\\pi$ , there exists $\\theta _ { 1 } , \\theta _ { 2 } , \\ldots , \\theta _ { d } \\ \\in \\ \\mathbb { R } ^ { d }$ such that for√ each $h \\in [ H ] , Q _ { h } ^ { \\pi } ( s , a ) \\ : = \\ : \\phi ( s , a ) ^ { \\top } \\theta _ { h }$ for all $( s , a ) \\ \\in \\ S _ { h } \\times { \\mathcal A }$ and $\\| \\theta _ { h } \\| _ { 2 } ~ \\le ~ H \\sqrt { d } .$ 4 Let $\\lambda = C H \\sqrt { d \\log ( d H / \\delta ) N }$ for some $C > 0$ . With probability at least $1 - \\delta$ , for some $c > 0$ , $\\begin{array} { r } { ( Q _ { 1 } ^ { \\pi } ( s _ { 1 } , \\pi ( s _ { 1 } ) ) - \\hat { Q } _ { 1 } ( s _ { 1 } , \\pi ( s _ { 1 } ) ) ) ^ { 2 } \\leq c \\cdot \\prod _ { h = 1 } ^ { H } C _ { h } \\cdot d H ^ { 5 } \\cdot \\sqrt { d \\log ( d H / \\delta ) / N } } \\end{array}$ . ",
+ "bbox": [
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+ },
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+ "type": "text",
+ "text": "Remark 6. The factor $\\textstyle \\prod _ { h = 1 } ^ { H } C _ { h }$ in Theorem 5.2 implies that the estimation error will be amplified geometrically. Now we discuss how the error is amplified when running Algorithm 1 on the instance in Section 4 to better illustrate the issue. If we run Algorithm 1 on the hard instance in Section 4, when $h = H$ , the estimation error on $V ( s _ { H } ^ { c } )$ would be roughly $N ^ { - 1 / 2 }$ for each $c \\in [ \\hat { d } ]$ . When using the linear predictor at level $H$ to predict the value of $s _ { H } ^ { * }$ , the error will be amplified by $\\hat { d } ^ { 1 / 2 }$ . When $h = H - 1$ , the dataset contains only $s _ { H - 1 } ^ { c }$ for $c \\in [ \\hat { d } ]$ , and the estimation error on the value of $s _ { H - 1 } ^ { c }$ will be the same as that of $s _ { H } ^ { * }$ , which is roughly $( \\hat { d } / N ) ^ { 1 / 2 }$ . Again, the estimation error on the value of $s _ { H - 1 } ^ { * }$ will be $( \\hat { d } ^ { 2 } / N ) ^ { 1 / 2 }$ when using the linear predictor at level $H - 1$ . The error will eventually be amplified by a factor of $\\hat { d } ^ { H / 2 }$ , which corresponds to the factor $\\textstyle \\prod _ { h = 1 } ^ { H } C _ { h }$ in Theorem 5.2. ",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "Policy Completeness. In offline RL, another representation condition is closedness under Bellman update (Szepesvari & Munos, 2005; Duan & Wang, 2020), which is stronger than realizability. In ´ the context of offline policy evaluation, we have the following policy completeness assumption. ",
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "Assumption 4. For the given policy $\\pi$ , for any $h > 1$ and $\\theta _ { h } \\in \\mathbb { R } ^ { d }$ , there exists $\\theta ^ { \\prime } \\in \\mathbb { R } ^ { d }$ such that for any $\\begin{array} { r } { ( s , a ) \\in \\mathcal S _ { h - 1 } \\times \\mathcal A , \\mathbb E [ R ( s , a ) ] + \\sum _ { s ^ { \\prime } \\in { \\cal S } _ { h } } P ( s ^ { \\prime } \\mid s , a ) \\phi ( s ^ { \\prime } , \\pi ( s ^ { \\prime } ) ) ^ { \\top } \\theta _ { h } = \\phi ( s , a ) ^ { \\top } \\theta ^ { \\prime } . } \\end{array}$ . ",
+ "bbox": [
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "Under Assumption 4 and the assumption that $\\sigma _ { \\operatorname* { m i n } } ( \\Lambda _ { h } ) \\geq \\lambda _ { 0 }$ for all $h \\in [ H ]$ for some $\\lambda _ { 0 } > 0$ , Duan & Wang (2020) have shown that for Algorithm 1, by taking $N = \\mathrm { p o l } \\bar { \\mathrm { y } } ( \\bar { H } , d , 1 / \\varepsilon , 1 / \\lambda _ { 0 } )$ , we have $( Q _ { 1 } ^ { \\pi } ( s _ { 1 } , \\pi ( s _ { 1 } ) ) - \\hat { Q } _ { 1 } ( s _ { 1 } , \\pi ( s _ { 1 } ) ) ) ^ { 2 } \\leq \\varepsilon$ . We refer interested readers to (Duan & Wang, 2020). ",
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "We remark that the above analysis again implies that geometric error amplification is a real issue in offline RL, and sample-efficient offline RL is impossible unless the distribution shift is sufficiently low, i.e., $\\textstyle \\prod _ { h = 1 } ^ { H } C _ { h }$ is bounded, or strong representation condition (e.g. policy completeness) holds. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "6 CONCLUSION ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "While the extant body of provable results in the literature largely focus on sufficient conditions for sample-efficient offline RL, this work focuses on obtaining a better understanding of the necessary conditions, where we seek to understand to what extent mild assumptions can imply sample-efficient offline RL. This work shows that for off-policy evaluation, even if we are given a representation that can perfectly represent the value function of the given policy and the data distribution has good coverage over the features, any provable algorithm still requires an exponential number of samples to non-trivially approximate the value of the given policy. These results highlight that provable sample-efficient offline RL is simply not possible unless either the distribution shift condition is sufficiently mild or we have stronger representation conditions that go well beyond realizability. ",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "ACKNOWLEDGMENTS ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "The authors would like to thank Akshay Krishnamurthy, Alekh Agarwal, Wen Sun, and Nan Jiang for numerous helpful discussion. Sham M. Kakade gratefully acknowledges funding from the ONR award N00014-18-1-2247, and NSF Awards CCF-1703574 and CCF-1740551. Ruosong Wang was supported in part by the NSF IIS1763562, US Army W911NF1920104, and ONR Grant N000141812861. Research performed while Ruosong Wang was an intern at Microsoft Research. ",
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+ },
+ {
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+ "text": "A PROOF OF LEMMA 4.2 ",
+ "text_level": 1,
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+ {
+ "type": "text",
+ "text": "Proof. We first verify $Q ^ { \\pi }$ is linear for the first $H - 1$ levels. For each $( h , c ) \\in [ H - 1 ] \\times [ \\hat { d } ]$ , we have ",
+ "bbox": [
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+ {
+ "type": "equation",
+ "img_path": "images/20e2f82f083315c34a6372cfc8601eda3bdec13111334ef8f8e4d716070da026.jpg",
+ "text": "$$\nQ _ { h } ^ { \\pi } ( s _ { h } ^ { c } , a _ { 1 } ) = R ( s _ { h } ^ { c } , a _ { 1 } ) + R ( s _ { h + 1 } ^ { \\hat { d } + 1 } , a _ { 1 } ) + R ( s _ { h + 2 } ^ { \\hat { d } + 1 } , a _ { 1 } ) + \\ldots + R ( s _ { H } ^ { \\hat { d } + 1 } , a _ { 1 } ) = r _ { 0 } \\cdot \\hat { d } ^ { ( H - h ) / 2 } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "type": "text",
+ "text": "Moreover, for all $a \\in { \\mathcal { A } }$ , ",
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "equation",
+ "img_path": "images/26a6565e194fb4e7d4ad7047fd0657ab1f85ed87efc1665e0a2724854b375ea3.jpg",
+ "text": "$$\n\\begin{array} { r } { \\mathcal { Q } _ { h } ^ { \\pi } ( s _ { h } ^ { \\hat { d } + 1 } , a ) = R ( s _ { h } ^ { \\hat { d } + 1 } , a ) + R ( s _ { h + 1 } ^ { \\hat { d } + 1 } , a _ { 1 } ) + R ( s _ { h + 2 } ^ { \\hat { d } + 1 } , a _ { 1 } ) + . . . + R ( s _ { H } ^ { \\hat { d } + 1 } , a _ { 1 } ) = r _ { 0 } \\cdot \\hat { d } ^ { ( H - h + 1 ) / 2 } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "Therefore, if we define ",
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "equation",
+ "img_path": "images/54fcd2dd6f55180ccd68f147b6e964775504c997fdce54bc2f950525a174eb3e.jpg",
+ "text": "$$\n\\boldsymbol { \\theta } _ { h } ^ { \\pi } = \\sum _ { c = 1 } ^ { \\hat { d } } \\boldsymbol { r } _ { 0 } \\cdot \\hat { d } ^ { ( H - h ) / 2 } \\cdot \\boldsymbol { e } _ { c } + \\sum _ { c = 1 } ^ { \\hat { d } } Q _ { h } ^ { \\pi } ( s _ { h } ^ { c } , a _ { 2 } ) \\cdot \\boldsymbol { e } _ { c + \\hat { d } } ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "then $Q _ { h } ^ { \\pi } ( s , a ) = \\left( \\theta _ { h } ^ { \\pi } \\right) ^ { \\top } \\phi ( s , a )$ for all $( s , a ) \\in S _ { h } \\times \\mathcal { A }$ ",
+ "bbox": [
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+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "Now we verify that the $Q$ -function is linear for the last level. Clearly, for all $c \\in [ \\hat { d } ]$ and $a \\in { \\mathcal { A } }$ , $Q _ { H } ^ { \\pi } ( s _ { H } ^ { c } , a ) = r _ { 0 }$ and $Q _ { H } ^ { \\pi } ( s _ { H } ^ { \\hat { d } + 1 } , a ) = r _ { 0 } \\cdot \\sqrt { \\hat { d } }$ . Thus by defining $\\begin{array} { r } { \\theta _ { H } ^ { \\pi } = \\sum _ { c = 1 } ^ { d } r _ { 0 } \\cdot e _ { c } } \\end{array}$ , we have $Q _ { H } ^ { \\pi } ( s , a ) = \\left( \\theta _ { H } ^ { \\pi } \\right) ^ { \\top } \\phi ( s , a )$ for all $( s , a ) \\in { \\cal S } _ { H } \\times \\mathcal { A }$ . ",
+ "bbox": [
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+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "B A TECHNICAL LEMMA ",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "We need the following lemma in the proof of our hardness results. ",
+ "bbox": [
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+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "Lemma B.1. Let α be a random variable uniformly distributed on $\\{ \\alpha _ { + } , \\alpha _ { - } \\}$ , where $\\alpha _ { - } = 1 / 2$ and $\\alpha _ { + } = 1 / 2 + \\varepsilon$ with $0 < \\varepsilon < 1$ . Suppose that $\\xi _ { 1 } , \\xi _ { 2 } , \\ldots , \\xi _ { m }$ are i.i.d. $\\{ + 1 , - 1 \\}$ -valued random variables with $\\operatorname* { P r } [ \\xi _ { i } = + 1 ] = \\alpha$ for all $i \\in [ m ]$ . Let $f$ be a function from $\\bar { \\{ + 1 , - 1 \\} } ^ { m }$ to $\\{ \\alpha _ { + } , \\alpha _ { - } \\}$ . Suppose $m \\le C / \\varepsilon ^ { 2 } \\log ( 1 / \\delta )$ for some fixed constant $C$ . Then ",
+ "bbox": [
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+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "equation",
+ "img_path": "images/8baf597341492de00e54fb80d5764f5c6ea5b5799a08174e53a081bb1892c63c.jpg",
+ "text": "$$\n\\operatorname* { P r } [ f ( \\xi _ { 1 } , \\xi _ { 2 } , \\ldots , \\xi _ { m } ) \\neq \\alpha ] > \\delta .\n$$",
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+ "text": "To our best knowledge, Lemma B.1 was first proved in (Chernoff, 1972) and has enormous applications in statistical learning theory (see, e.g., Chapter 5 in (Anthony & Bartlett, 2009)) and bandits (Mannor & Tsitsiklis, 2004). ",
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+ "text": "To prove Lemma B.1, one can first prove that the maximum likelihood estimator (MLE) is optimal and then show that MLE requires $\\dot { \\Omega ( 1 / \\varepsilon ^ { 2 } \\log ( 1 / \\delta ) ) }$ samples to correctly output $\\alpha$ with probability $1 - \\delta$ by anti-concentration. Lemma B.1 can also be proved by using information theory. See, e.g., (Kaufmann et al., 2016) for such a proof. ",
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+ "type": "text",
+ "text": "C ANOTHER HARD INSTANCE ",
+ "text_level": 1,
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+ "text": "In this section, we present another hard case under a weaker version of Assumption 1. Here the transition operator is stochastic and the reward is deterministic and sparse, meaning that the reward value is non-zero only for the last level. Moreover, the data distributions $\\{ \\mu _ { h } \\} _ { h = 1 } ^ { H }$ are induced by a fixed policy $\\pi _ { \\mathrm { d a t a } }$ . We also illustrate the construction in Figure 2. Throughout this section, we use $d$ the denote the feature dimension, and we assume $d$ is an even integer for simplicity. We use $\\hat { d }$ to denote $d / 2 - 1$ . ",
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+ "text": "In this section, we adopt the following realizability assumption, which is a weaker version of Assumption 1. ",
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+ "text": "Assumption 5 (Realizability). For the policy $\\pi : { \\mathcal { S } } \\to \\Delta ( { \\mathcal { A } } )$ to be evaluated, there exists $\\theta _ { 1 } , \\ldots \\theta _ { H } \\in \\mathbb { R } ^ { d }$ such that for all $( s , a ) \\in S \\times A$ and $h \\in [ H ]$ , ",
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+ "text": "$$\nQ _ { h } ^ { \\pi } ( s , a ) = \\theta _ { h } ^ { \\top } \\phi ( s , a ) .\n$$",
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+ "Figure 2: An illustration of the hard instance. Recall that $\\hat { d } = d / 2 - 1$ . States on the top are those in the first level $( h = 1 )$ ), while states at the bottom are those in the last level ( $h = H$ ). Dotted line (with arrow) corresponds to transitions associated with actions $a _ { 1 } , a _ { 2 } , \\ldots , a _ { \\hat { d } }$ , while solid line (with arrow) corresponds to transitions associated with actions $a _ { \\hat { d } + 1 } , a _ { \\hat { d } + 2 } , \\dotsc , a _ { d }$ . We omit the transition associated with $a _ { 1 } , a _ { 2 } , \\ldots , a _ { \\hat { d } }$ in the figure if all actions give the same transition. For each level $h \\in [ H ]$ , $Q$ -values associated with $s _ { h } ^ { 1 } , s _ { h } ^ { 2 } , \\ldots , s _ { h } ^ { \\hat { d } } , s _ { h } ^ { + } , s _ { h } ^ { - }$ , sdˆh, s+h , s−h are marked on the left, while transition distributions and $Q$ -values associated with + − $s _ { h } ^ { \\hat { d } + 1 }$ are marked on the right. Rewards are all deterministic, and the only two states ) with non-zero reward values are marked in black and grey. Consider the fixed policy that returns $\\mathit { a } _ { d }$ for all input states. When $r _ { 0 } = 0$ , the value of the policy is 0. When $r _ { 0 } = \\hat { d } ^ { - ( \\bar { H } - 2 ) / 2 }$ , the value of the policy $s = r _ { 0 } \\hat { d } ^ { ( H - 2 ) / 2 } = 1$ . "
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+ "text": "State Space, Action Space and Transition Operator. In this hard case, the action space ${ \\mathcal { A } } =$ $\\{ a _ { 1 } , a _ { 2 } , \\ldots , a _ { d } \\}$ contains $d$ elements. $S _ { 1 }$ contains a single state $s _ { 1 }$ . For each $h \\geq 2$ , $S _ { h }$ contains ${ \\hat { d } } + 3$ states $s _ { h } ^ { 1 } , s _ { h } ^ { 2 } , \\ldots , s _ { h } ^ { \\hat { d } } , s _ { h } ^ { \\hat { d } + 1 } , s _ { h } ^ { + }$ and $s _ { h } ^ { - }$ . ",
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+ "text": "Let $0 \\leq r _ { 0 } \\leq \\hat { d } ^ { - ( H - 2 ) / 2 }$ be a parameter to be determined. We first define the transition operator for the first level. We have ",
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+ "text": "$$\nP ( s _ { 1 } , a ) = \\left\\{ \\begin{array} { l l } { s _ { 2 } ^ { c } } & { a = a _ { c } , c \\in [ \\hat { d } ] } \\\\ { s _ { 2 } ^ { + } } & { a = a _ { \\hat { d } + 1 } } \\\\ { s _ { 2 } ^ { - } } & { a = a _ { \\hat { d } + 2 } } \\\\ { s _ { 2 } ^ { \\hat { d } + 1 } } & { a \\in \\{ a _ { \\hat { d } + 3 } , a _ { \\hat { d } + 4 } , \\dots , a _ { d } \\} } \\end{array} \\right. .\n$$",
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+ "text": "Now we define the transition operator when $h \\in \\{ 2 , 3 , \\ldots , H - 2 \\}$ . For each $h \\in \\{ 2 , 3 , \\ldots , H - 2 \\}$ , $a \\in { \\mathcal { A } }$ and $c \\in [ \\hat { d } ]$ , we have $P ( s _ { h } ^ { c } , a ) = s _ { h + 1 } ^ { \\hat { d } + 1 }$ , $P ( s _ { h } ^ { + } , a ) = s _ { h + 1 } ^ { + }$ and $P ( s _ { h } ^ { - } , a ) = s _ { h + 1 } ^ { - }$ . For each $h \\in \\{ 2 , 3 , \\ldots , H - 2 \\}$ and $c \\in [ \\hat { d } ]$ , we have $P ( s _ { h } ^ { \\hat { d } + 1 } , a _ { c } ) = s _ { h + 1 } ^ { c }$ . For all $a \\in \\left\\{ a _ { \\hat { d } + 1 } , a _ { \\hat { d } + 2 } , \\dotsc , a _ { d } \\right\\}$ , ",
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+ "type": "text",
+ "text": "we have ",
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+ "text": "$$\nP \\left( s _ { h } ^ { \\hat { d } + 1 } , a \\right) = \\left\\{ \\begin{array} { l l } { s _ { h + 1 } ^ { + } } & { \\mathrm { w i t h ~ p r o b a b i l i t y ~ } ( 1 + r _ { 0 } \\cdot \\hat { d } ^ { ( H - h ) / 2 } ) / 2 } \\\\ { s _ { h + 1 } ^ { - } } & { \\mathrm { w i t h ~ p r o b a b i l i t y ~ } ( 1 - r _ { 0 } \\cdot \\hat { d } ^ { ( H - h ) / 2 } ) / 2 } \\end{array} \\right. .\n$$",
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+ "type": "text",
+ "text": "Now we define the transition operator for the second last level. For all $c \\in [ \\hat { d } ]$ and $a \\in { \\mathcal { A } }$ , we have ",
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+ "text": "$$\nP ( s _ { H - 1 } ^ { c } , a ) = \\left\\{ \\begin{array} { l l } { s _ { H } ^ { + } } & { \\mathrm { w i t h ~ p r o b a b i l i t y ~ } ( 1 + r _ { 0 } ) / 2 } \\\\ { s _ { H } ^ { - } } & { \\mathrm { w i t h ~ p r o b a b i l i t y ~ } ( 1 - r _ { 0 } ) / 2 } \\end{array} \\right. .\n$$",
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+ "text": "For all $a \\in { \\mathcal { A } }$ , we have $P ( s _ { H - 1 } ^ { + } , a ) = s _ { H } ^ { + }$ and $P ( s _ { H - 1 } ^ { - } , a ) \\ = \\ s _ { H } ^ { - }$ . For each $c \\in [ \\hat { d } ]$ , we have $P ( s _ { H - 1 } ^ { \\hat { d } + 1 } , a _ { c } ) = s _ { H } ^ { c }$ . For all $a \\in \\left\\{ a _ { \\hat { d } + 1 } , a _ { \\hat { d } + 2 } , \\dotsc , a _ { d } \\right\\}$ , we have ",
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+ "text": "$$\nP ( s _ { H - 1 } ^ { \\hat { d } + 1 } , a ) = \\left\\{ \\begin{array} { l l } { s _ { H } ^ { + } } & { \\mathrm { w i t h } \\mathrm { p r o b a b i l i t y } \\left( 1 + r _ { 0 } \\cdot \\sqrt { \\hat { d } } \\right) / 2 } \\\\ { s _ { H } ^ { - } } & { \\mathrm { w i t h } \\mathrm { p r o b a b i l i t y } \\left( 1 - r _ { 0 } \\cdot \\sqrt { \\hat { d } } \\right) / 2 } \\end{array} \\right. .\n$$",
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+ "type": "text",
+ "text": "Reward Values. In this hard case, all reward values are deterministic, and reward values can be non-zero only for the last level. Formally, we have ",
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+ "text": "$$\nR ( s , a ) = \\left\\{ { \\begin{array} { l l } { 1 } & { s = s _ { H } ^ { + } } \\\\ { - 1 } & { s = s _ { H } ^ { - } } \\\\ { 0 } & { { \\mathrm { o t h e r w i s e } } } \\end{array} } \\right. .\n$$",
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+ "text": "Feature Mapping. As in the in hard instance in Section 4, let $e _ { 1 } , e _ { 2 } , \\ldots , e _ { d }$ be a set of of orthonormal vectors in $\\mathbb { R } ^ { d }$ . For the initial state, for each $c \\in [ d ]$ , we have $\\phi ( s _ { 1 } , a _ { c } ) = e _ { c }$ . ",
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+ "type": "text",
+ "text": "Now we define the feature mapping when $h \\in \\{ 2 , 3 , \\ldots , H \\}$ . For each $h \\in \\{ 2 , 3 , \\ldots , H \\}$ , $a \\in { \\mathcal { A } }$ and $c \\in [ \\hat { d } ] , \\phi ( s _ { h } ^ { c } , a ) = e _ { c } , \\phi ( s _ { h } ^ { + } , a ) = e _ { \\hat { d } + 1 }$ and $\\phi ( s _ { h } ^ { - } , a ) = e _ { \\hat { d } + 2 }$ . Moreover, for all actions $a \\in { \\mathcal { A } }$ , ",
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+ "text": "$$\n\\phi ( s _ { h } ^ { \\hat { d } + 1 } , a ) = \\left\\{ { \\begin{array} { l l } { e _ { \\hat { d } + 2 + c } } & { a = a _ { c } , c \\in [ { \\hat { d } } ] } \\\\ { { \\frac { 1 } { d ^ { 1 / 2 } } } \\left( e _ { 1 } + e _ { 2 } + \\ldots + e _ { \\hat { d } } \\right) } & { a \\in \\left\\{ a _ { \\hat { d } + 1 } , a _ { \\hat { d } + 2 } , \\ldots , a _ { d } \\right\\} } \\end{array} } \\right. .\n$$",
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+ "type": "text",
+ "text": "Clearly, for all (s, a) ∈ S × A, $\\| \\phi ( s , a ) \\| _ { 2 } \\leq 1$ . ",
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+ "text": "Verifying Assumption 5. Now we consider the deterministic policy $\\pi : { \\mathcal { S } } A$ , which is defined to be $\\pi ( s ) = a _ { d }$ for all $s \\in S$ . We show that Assumption 5 holds. ",
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+ "type": "text",
+ "text": "When $h = 1$ , define ",
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+ "text": "$$\n\\theta _ { 1 } = \\sum _ { c = 1 } ^ { \\hat { d } } r _ { 0 } \\cdot \\hat { d } ^ { ( H - 3 ) / 2 } \\cdot e _ { c } + e _ { \\hat { d } + 1 } - e _ { \\hat { d } + 2 } + \\sum _ { c = 1 } ^ { \\hat { d } } r _ { 0 } \\cdot \\hat { d } ^ { ( H - 2 ) / 2 } \\cdot e _ { \\hat { d } + 2 + c } .\n$$",
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+ "text": "For each $h \\in \\{ 2 , 3 , \\ldots , H - 2 \\}$ , define ",
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+ "text": "$$\n\\theta _ { h } = \\sum _ { c = 1 } ^ { \\hat { d } } r _ { 0 } \\cdot \\hat { d } ^ { ( H - h - 1 ) / 2 } \\cdot e _ { c } + e _ { \\hat { d } + 1 } - e _ { \\hat { d } + 2 } + \\sum _ { c = 1 } ^ { \\hat { d } } r _ { 0 } \\cdot \\hat { d } ^ { ( H - h - 2 ) / 2 } \\cdot e _ { \\hat { d } + 2 + c } .\n$$",
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+ "text": "For the second last level $h = H - 1$ , define ",
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+ "text": "$$\n\\theta _ { H - 1 } = \\sum _ { c = 1 } ^ { \\hat { d } } r _ { 0 } \\cdot e _ { c } + e _ { \\hat { d } + 1 } - e _ { \\hat { d } + 2 } .\n$$",
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+ "text": "Finally, for the last level $h = H$ , define ",
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+ "text": "$$\n\\theta _ { H } = e _ { \\hat { d } + 1 } - e _ { \\hat { d } + 2 } .\n$$",
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+ "type": "text",
+ "text": "It can be verified that for each $h \\in [ H ]$ , $Q _ { h } ^ { \\pi } ( s , a ) = \\theta _ { h } ^ { \\top } \\phi ( s , a )$ for all $( s , a ) \\in S _ { h } \\times \\mathcal { A }$ . ",
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+ "type": "text",
+ "text": "The Data Distributions. For the first level, the data distribution $\\mu _ { 1 }$ is defined to be the uniform distribution over $\\{ ( s _ { 1 } , a _ { c } ) ~ | ~ c ~ \\in ~ [ d ] \\}$ . For each $h \\geq 2$ , the data distribution $\\mu _ { h }$ is a uniform distribution over ",
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+ "img_path": "images/98895987379f06852e232c87b7cac8b00890955f81df350fcbcb45bf6a0654bc.jpg",
+ "text": "$$\n\\{ ( s _ { h } ^ { 1 } , a _ { 1 } ) , ( s _ { h } ^ { 2 } , a _ { 1 } ) , \\ldots , ( s _ { h } ^ { \\hat { d } } , a _ { 1 } ) , ( s _ { h } ^ { + } , a _ { 1 } ) , ( s _ { h } ^ { - } , a _ { 1 } ) , ( s _ { h } ^ { \\hat { d } + 1 } , a _ { 1 } ) , ( s _ { h } ^ { \\hat { d } + 1 } , a _ { 2 } ) , \\ldots , ( s _ { h } ^ { \\hat { d } + 1 } , a _ { \\hat { d } } ) \\} .\n$$",
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+ "text": "Notice that again $( s _ { h } ^ { \\hat { d } + 1 } , a )$ is not in the support of $\\mu _ { h }$ for all actions $a \\in \\left\\{ a _ { \\hat { d } + 1 } , a _ { \\hat { d } + 2 } , \\dotsc , a _ { d } \\right\\}$ . It can be seen that for all $h \\in [ H ]$ , ",
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+ "text": "$$\n\\mathbb { E } _ { ( s , a ) \\sim \\mu _ { h } } [ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } ] = \\frac { 1 } { d } \\sum _ { c = 1 } ^ { d } e _ { c } e _ { c } ^ { \\top } = \\frac { 1 } { d } I .\n$$",
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+ "type": "text",
+ "text": "Moreover, by defining ",
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+ "text": "$$\n\\pi _ { \\operatorname* { d a t a } } ( s ) = \\left\\{ \\begin{array} { l l } { \\operatorname { U n i f o r m } ( A ) } & { s = s _ { 1 } } \\\\ { a _ { 1 } } & { s \\in \\{ s _ { h } ^ { c } \\mid h \\in \\{ 2 , 3 , \\ldots , H \\} , c \\in [ \\hat { d } ] \\} } \\\\ { a _ { 1 } } & { s \\in \\{ s _ { h } ^ { + } \\mid h \\in \\{ 2 , 3 , \\ldots , H \\} \\} } \\\\ { a _ { 1 } } & { s \\in \\{ s _ { h } ^ { - } \\mid h \\in \\{ 2 , 3 , \\ldots , H \\} \\} } \\\\ { \\operatorname { U n i f o r m } ( \\{ a _ { 1 } , a _ { 2 } , \\ldots , a _ { \\hat { d } } \\} ) } & { s \\in \\{ s _ { h } ^ { \\hat { d } + 1 } \\mid h \\in \\{ 2 , 3 , \\ldots , H \\} \\} } \\end{array} \\right. ,\n$$",
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+ "type": "text",
+ "text": "we have $\\mu _ { h } = \\mu _ { h } ^ { \\pi _ { \\mathrm { d a t a } } }$ for all $h \\in [ H ]$ ",
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+ "type": "text",
+ "text": "The Lower Bound. Now we show that it is information-theoretically hard for any algorithm to distinguish the case $r _ { 0 } = 0$ and $r _ { 0 } = \\hat { d } ^ { - ( H - 2 ) / 2 }$ in the offline setting by taking samples from the data distributions $\\{ \\mu _ { h } \\} _ { h = 1 } ^ { H }$ . Here we consider the above policy $\\pi$ defined above which returns action $a _ { d }$ for all input states. Notice that when $r _ { 0 } = 0$ , the value of the policy would be zero. On the other hand, when $\\stackrel { \\cdot } { r } _ { 0 } = \\hat { d } ^ { - ( H - 2 ) / 2 }$ , the value of the policy would be $r _ { 0 } \\cdot \\hat { d } ^ { ( \\overleftarrow { H } - 2 ) / 2 } = 1 .$ . Therefore, if the algorithm approximates the value of the policy up to an approximation error of $1 / 2$ , then it must distinguish the case that $r _ { 0 } = 0$ and $r _ { 0 } = \\hat { d } ^ { - ( H - 2 ) / 2 }$ . ",
+ "bbox": [
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+ "text": "We first notice that for the case $r _ { 0 } = 0$ and $r _ { 0 } = \\hat { d } ^ { - ( H - 2 ) / 2 }$ , the data distributions $\\{ \\mu _ { h } \\} _ { h = 1 } ^ { H }$ , the feature mapping $\\phi : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R } ^ { d }$ , the policy $\\pi$ to be evaluated and the reward distributions $R$ are the same. Thus, in order to distinguish the case $r _ { 0 } = 0$ and $r _ { 0 } = \\hat { d } ^ { - ( H - 2 ) / 2 }$ , the only way is to query the transition operator $P$ by using sampling taken from the data distributions. ",
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+ "type": "text",
+ "text": "Now, for all state-action pairs $( s , a )$ in the support of the data distributions of the first $H - 2$ lev \nelsing amely only $\\mu _ { 1 } , \\mu _ { 2 } , \\ldots , \\mu _ { H - 2 } )$ , the transition opeion distributions of ang-, and $r _ { 0 }$ $( s _ { h } ^ { \\hat { d } + 1 } , a _ { \\hat { d } + 1 } ) , ( s _ { h } ^ { \\hat { d } + 1 } , a _ { \\hat { d } + 2 } ) , \\dots , ( s _ { h } ^ { \\hat { d } + 1 } , a _ { d } )$ support of in the supp f of $h ~ \\in ~ [ H \\mathrm { ~ - ~ } 2 ]$ . Moreover, for anyso be identical no mat \n$( s , a ) \\in \\{ s _ { H - 1 } ^ { + } , s _ { H - 1 } ^ { - } , s _ { H - 1 } ^ { \\hat { d } + 1 } \\} \\times \\mathcal { A }$ $\\mu _ { H - 1 }$ $P ( s , a )$ \nter $r _ { 0 } = 0$ or $r _ { 0 } = \\hat { d } ^ { - ( H - 2 ) / 2 }$ . For those state-action pairs $( s , a )$ in the support of $\\mu _ { H - 1 }$ with \n$s \\not \\in \\{ s _ { H - 1 } ^ { + } , s _ { H - 1 } ^ { - } , s _ { H - 1 } ^ { \\hat { d } + 1 } \\}$ , we have ",
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+ },
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+ "type": "equation",
+ "img_path": "images/3923d7417aca0ce99b714732257f17509ea4cdfc956725d9ef4dd19654d62acb.jpg",
+ "text": "$$\nP ( s , a ) = \\left\\{ \\begin{array} { l l } { s _ { H } ^ { + } } & { \\mathrm { w i t h } \\ p r o b a b i l i t y \\ ( 1 + r _ { 0 } ) / 2 } \\\\ { s _ { H } ^ { - } } & { \\mathrm { w i t h } \\ p r o b a b i l i t y \\ ( 1 - r _ { 0 } ) / 2 } \\end{array} \\right. .\n$$",
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+ },
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+ "type": "text",
+ "text": "Again, this is because $( s _ { H - 1 } ^ { \\hat { d } + 1 } , a )$ is not in the support of $\\mu _ { H - 1 }$ for all $a \\in \\left\\{ a _ { \\hat { d } + 1 } , a _ { \\hat { d } + 2 } , \\dotsc , a _ { d } \\right\\}$ ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Therefore, in order to distinguish the case $r _ { 0 } = 0$ and $r _ { 0 } = \\hat { d } ^ { - ( H - 2 ) / 2 }$ , the agent needs distinguish two transition distributions ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/0fe8bae359e8250df180debadc1391756ffe35c0d55a1e67825fa124ecc85b2f.jpg",
+ "text": "$$\np _ { 1 } = { \\left\\{ \\begin{array} { l l } { s _ { H } ^ { + } } & { { \\mathrm { w i t h ~ p r o b a b i l i t y ~ } } 1 / 2 } \\\\ { s _ { H } ^ { - } } & { { \\mathrm { w i t h ~ p r o b a b i l i t y ~ } } 1 / 2 } \\end{array} \\right. }\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "and ",
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "equation",
+ "img_path": "images/213c2d1f6f8006f0ad57680ab50cc19ad912456c73418c5c8a550f318dc82780.jpg",
+ "text": "$$\np _ { 2 } = \\left\\{ s _ { H } ^ { + } \\quad \\mathrm { w i t h } \\mathrm { p r o b a b i l i t y } ( 1 + \\hat { d } ^ { - ( H - 2 ) / 2 } ) / 2 \\right.\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "Again, by Lemma B.1, in order to distinguish $p _ { 1 }$ and $p _ { 2 }$ with probability at least 0.9, one needs $\\Omega ( \\hat { d } ^ { H - 2 } )$ samples. Formally, we have the following theorem. ",
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+ },
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+ "type": "text",
+ "text": "Theorem C.1. Suppose Assumption 2 holds, and rewards are deterministic and could be nonezero only for state-action pairs in the last level. Fix an algorithm that takes as input both a policy and a feature mapping. There exists an MDP satisfying Assumption 5, such that for a fixed policy $\\pi : { \\mathcal { S } } A$ , the algorithm requires $\\Omega ( ( d / 2 - 1 ) ^ { H / 2 } )$ samples to output the value of π up to constant additive approximation error with probability at least 0.9. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "D ANALYSIS OF ALGORITHM 1 ",
+ "text_level": 1,
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+ },
+ {
+ "type": "text",
+ "text": "D.1 PROOF OF LEMMA 5.1 ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Clearly, ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "For the first te ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/a794e2984d07f734f3b1f7eee9882b84ec8c3c8980c3b8a66b48b1ae2f524bab.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\frac { 1 } { \\sqrt { 3 } } \\sum _ { k \\geq 0 , k \\leq 1 } \\bar { \\zeta } _ { k } \\bigg ( \\frac { 1 } { 2 } \\eta _ { k } ^ { 2 } \\bigg ( \\mathbb { E } _ { k } \\| \\hat { \\zeta } _ { k } \\| ^ { 2 } \\eta _ { k } ^ { 2 } \\bigg ( \\mathbb { E } _ { k } \\| \\hat { \\zeta } _ { k } \\| ^ { 2 } \\eta _ { k } ^ { 2 } \\bigg ) \\bigg ) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\end{array}\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "Therefore, ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/2a3f5d3a13ff2478cb320974abe1bd48a8543ddb6d99844d86ae3f2995a70a01.jpg",
+ "text": "$$\n\\begin{array} { l } { \\displaystyle \\hat { \\theta } _ { 1 } - \\theta _ { 1 } = ( \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } \\xi _ { 1 } - \\lambda \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\theta _ { 1 } ) + \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } ( \\theta _ { 2 } - \\hat { \\theta } _ { 2 } ) } \\\\ { \\displaystyle = ( \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } \\xi _ { 1 } - \\lambda \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\theta _ { 1 } ) + \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } ( \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\xi _ { 2 } - \\lambda \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\theta _ { 2 } ) } \\\\ { \\displaystyle \\quad \\quad + \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\overline { { \\Phi } } _ { 3 } ( \\theta _ { 3 } - \\hat { \\theta } _ { 3 } ) } \\\\ { \\displaystyle \\quad = \\ldots } \\\\ { \\displaystyle \\quad = \\sum _ { h = 1 } ^ { H } \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\overline { { \\Phi } } _ { 3 } \\ldots ( \\hat { \\Lambda } _ { h } ^ { - 1 } \\Phi _ { h } ^ { \\top } \\xi _ { h } - \\lambda \\hat { \\Lambda } _ { h } ^ { - 1 } \\theta _ { h } ) . } \\end{array}\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "Also note that ",
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+ "type": "equation",
+ "img_path": "images/80cd714ffe24921aa94bd54a18fc45075e46a89d468f6ac7919e07b9b06ed96a.jpg",
+ "text": "$$\n\\begin{array} { r } { ( Q ^ { \\pi } ( s _ { 1 } , \\pi ( s _ { 1 } ) ) - \\hat { Q } ( s _ { 1 } , \\pi ( s _ { 1 } ) ) ) ^ { 2 } = \\| \\theta _ { 1 } - \\hat { \\theta } _ { 1 } \\| _ { \\bar { \\Lambda } _ { 1 } } ^ { 2 } . } \\end{array}\n$$",
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+ {
+ "type": "text",
+ "text": "D.2 PROOF OF THEOREM 5.2 ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "By matrix concentration inequality (Tropp, 2015), we have the following lemma. ",
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+ "type": "text",
+ "text": "Lemma D.1. For each $h \\in [ H ]$ , with probability $1 - \\delta / ( 4 H )$ , for some universal constant $C$ , we have ",
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+ "type": "equation",
+ "img_path": "images/c7c64ef9476b39f93cff38824d7eb84c7c88ae51fb7bf4cd1c2c2b697de684d0.jpg",
+ "text": "$$\n\\left\\| \\frac { 1 } { N } \\Phi _ { h } ^ { \\top } \\Phi _ { h } - \\Lambda _ { h } \\right\\| _ { 2 } \\leq C \\sqrt { d \\log ( d H / \\delta ) / N } .\n$$",
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+ "type": "text",
+ "text": "and ",
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+ {
+ "type": "equation",
+ "img_path": "images/d18992896bb27ede910b917fc278c6f358083295a4c8af0afb592cf1f1efa217.jpg",
+ "text": "$$\n\\left\\| \\frac { 1 } { N } \\overline { { \\Phi } } _ { h + 1 } \\overline { { \\Phi } } _ { h + 1 } - \\overline { { \\Lambda } } _ { h + 1 } \\right\\| _ { 2 } \\leq C \\sqrt { d \\log ( d H / \\delta ) / N } .\n$$",
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+ "type": "text",
+ "text": "Therefore, since $\\lambda = C H \\sqrt { d \\log ( d H / \\delta ) N }$ , with probability $1 - \\delta / ( 4 H )$ , we have ",
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+ "type": "equation",
+ "img_path": "images/3c7c5d913557d3e6cd3122de5023e285a00f1ee050cf0b46e2d9c9532373ac7a.jpg",
+ "text": "$$\n\\hat { \\Lambda } _ { h } = \\Phi _ { h } ^ { \\top } \\Phi _ { h } + \\lambda I \\succeq N \\Lambda _ { h } .\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "Note that ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/3f640e4d0c72d9b55eaa1cc73875fd439ff8e57122c01f2a179ebeee99f83023.jpg",
+ "text": "$$\n\\begin{array} { l } { \\displaystyle ( { \\mathcal Q } ^ { \\pi } ( s _ { 1 } , \\pi ( s _ { 1 } ) ) - \\hat { Q } ( s _ { 1 } , \\pi ( s _ { 1 } ) ) ) ^ { 2 } } \\\\ { \\displaystyle \\leq H \\cdot \\left( \\sum _ { h = 1 } ^ { H } \\left\\| \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\overline { { \\Phi } } _ { 3 } \\cdot \\cdot \\cdot ( \\hat { \\Lambda } _ { h } ^ { - 1 } \\Phi _ { h } ^ { \\top } \\xi _ { h } - \\lambda \\hat { \\Lambda } _ { h } ^ { - 1 } \\theta _ { h } ) \\right\\| _ { \\overline { { \\Lambda } } _ { 1 } } ^ { 2 } \\right) } \\\\ { \\displaystyle \\leq 2 H \\cdot \\left( \\sum _ { h = 1 } ^ { H } \\left\\| \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\overline { { \\Phi } } _ { 3 } \\cdot \\cdot \\cdot \\hat { \\Lambda } _ { h } ^ { - 1 } \\Phi _ { h } ^ { \\top } \\xi _ { h } \\right\\| _ { \\overline { { \\Lambda } } _ { 1 } } ^ { 2 } + \\sum _ { h = 1 } ^ { H } \\left\\| \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\overline { { \\Phi } } _ { 3 } \\cdot \\cdot \\cdot \\cdot \\lambda \\hat { \\Lambda } _ { h } ^ { - 1 } \\theta _ { h } \\right\\| _ { \\overline { { \\Lambda } } _ { 1 } } ^ { 2 } \\right) } \\end{array}\n$$",
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+ {
+ "type": "text",
+ "text": "For each $h \\in [ H ]$ , ",
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+ },
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+ "type": "equation",
+ "img_path": "images/9ec79d4746f8905d6df4d5b1764ac53c1187875809618c8fcca355596bf01604.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\quad \\| \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\overline { { \\Phi } } _ { 3 } \\cdots \\hat { \\Lambda } _ { h } ^ { - 1 } \\Phi _ { h } ^ { \\top } \\xi _ { h } \\| _ { \\overline { { \\Lambda } } _ { 1 } } ^ { 2 } } \\\\ & { \\leq \\| \\Phi _ { 1 } \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\overline { { \\Lambda } } _ { 1 } \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\| _ { 2 } \\cdot \\| \\overline { { \\Phi } } _ { 2 } \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\overline { { \\Phi } } _ { 3 } \\cdots \\hat { \\Lambda } _ { h } ^ { - 1 } \\Phi _ { h } ^ { \\top } \\xi _ { h } \\| _ { 2 } ^ { 2 } } \\\\ & { \\leq \\| \\hat { \\Lambda } _ { 1 } ^ { - 1 / 2 } \\overline { { \\Lambda } } _ { 1 } \\hat { \\Lambda } _ { 1 } ^ { - 1 / 2 } \\| _ { 2 } \\cdot \\| \\Phi _ { 1 } \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\| _ { 2 } \\cdot \\| \\overline { { \\Phi } } _ { 2 } \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } \\cdot \\cdot \\hat { \\Lambda } _ { h } ^ { - 1 } \\Phi _ { h } ^ { \\top } \\xi _ { h } \\| _ { 2 } ^ { 2 } } \\\\ & \\leq \\| \\hat { \\Lambda } _ { 1 } ^ { - 1 / 2 } \\overline { { \\Lambda } } _ { 1 } \\hat { \\Lambda } _ { 1 } ^ { - 1 / 2 } \\| _ { 2 } \\cdot \\displaystyle \\prod _ { h ^ { \\prime } = 1 } ^ { h - 1 } \\Big ( \\| \\Phi _ { h ^ { \\prime } } \\hat { \\Lambda } _ { h ^ { \\prime } } ^ { - 1 } \\Phi _ { h ^ { \\prime } } ^ { \\top } \\| _ { 2 } \\cdot \\| \\hat { \\Phi } _ { h ^ { \\prime } + 1 } ^ { - 1 / 2 } ( \\overline { { \\Phi } } _ { h ^ { \\prime } + 1 } ^ { \\top } \\overline { { \\Phi } } _ { h ^ { \\prime } + 1 } ) \\hat { \\Lambda } _ h ^ { \\prime } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Similarly, ",
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+ "text": "$$\n\\begin{array} { r l } & { \\quad \\| \\hat { \\boldsymbol { \\Lambda } } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } \\hat { \\boldsymbol { \\Lambda } } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\overline { { \\Phi } } _ { 3 } \\cdot \\cdot \\lambda \\hat { \\boldsymbol { \\Lambda } } _ { h } ^ { - 1 } \\theta _ { h } \\| _ { \\overline { { \\boldsymbol { \\Lambda } } } _ { 1 } } ^ { 2 } } \\\\ & { \\leq \\| \\hat { \\boldsymbol { \\Lambda } } _ { 1 } ^ { - 1 / 2 } \\overline { { \\boldsymbol { \\Lambda } } } _ { 1 } \\hat { \\boldsymbol { \\Lambda } } _ { 1 } ^ { - 1 / 2 } \\| _ { 2 } \\cdot \\displaystyle \\prod _ { h ^ { \\prime } = 1 } ^ { h - 1 } \\Big ( \\| \\Phi _ { h ^ { \\prime } } \\hat { \\boldsymbol { \\Lambda } } _ { h ^ { \\prime } } ^ { - 1 } \\Phi _ { h ^ { \\prime } } ^ { \\top } \\| _ { 2 } \\cdot \\| \\hat { \\boldsymbol { \\Lambda } } _ { h ^ { \\prime } + 1 } ^ { - 1 / 2 } ( \\overline { { \\Phi } } _ { h ^ { \\prime } + 1 } ^ { \\top } \\overline { { \\Phi } } _ { h ^ { \\prime } + 1 } ) \\hat { \\boldsymbol { \\Lambda } } _ { h + 1 } ^ { - 1 / 2 } \\| _ { 2 } \\Big ) \\cdot \\lambda ^ { 2 } \\cdot \\| \\theta _ { h } \\| _ { \\hat { \\boldsymbol { \\Lambda } } _ { h } ^ { - 1 } } ^ { 2 } } \\\\ & { \\leq \\| \\hat { \\boldsymbol { \\Lambda } } _ { 1 } ^ { - 1 / 2 } \\overline { { \\boldsymbol { \\Lambda } } } _ { 1 } \\hat { \\boldsymbol { \\Lambda } } _ { 1 } ^ { - 1 / 2 } \\| _ { 2 } \\cdot \\displaystyle \\prod _ { h ^ { \\prime } = 1 } ^ { h - 1 } \\Big ( \\| \\Phi _ { h ^ { \\prime } } \\hat { \\boldsymbol { \\Lambda } } _ { h ^ { \\prime } } ^ { - 1 } \\Phi _ { h ^ { \\prime } } ^ { \\top } \\| _ { 2 } \\cdot \\| \\hat { \\boldsymbol { \\Lambda } } _ { h ^ { \\prime } + 1 } ^ { - 1 / 2 } ( \\overline { { \\Phi } } _ { h ^ { \\prime } + 1 } ^ { \\top } \\overline { { \\Phi } } _ { h ^ { \\prime } + 1 } ) \\hat { \\boldsymbol { \\Lambda } } _ { h ^ { \\prime } + 1 } ^ { - 1 / 2 } \\| _ { 2 } \\cdot \\lambda \\cdot H ^ { 2 } d . } \\end{array}\n$$",
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+ "text": "$$\n\\| \\Phi _ { h } \\hat { \\Lambda } _ { h } ^ { - 1 } \\Phi _ { h } ^ { \\top } \\| _ { 2 } \\leq 1\n$$",
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+ "text": "$$\n\\begin{array} { r } { \\lVert \\hat { \\Lambda } _ { h } ^ { - 1 / 2 } ( \\overline { { \\Phi } } _ { h } ^ { \\top } \\overline { { \\Phi } } _ { h } ) \\hat { \\Lambda } _ { h } ^ { - 1 / 2 } \\rVert _ { 2 } \\leq \\lVert N \\hat { \\Lambda } _ { h } ^ { - 1 / 2 } \\overline { { \\Lambda } } _ { h } \\hat { \\Lambda } _ { h } ^ { - 1 / 2 } \\rVert _ { 2 } + \\lVert \\hat { \\Lambda } _ { h } ^ { - 1 / 2 } ( \\overline { { \\Phi } } _ { h } ^ { \\top } \\overline { { \\Phi } } _ { h } - N \\overline { { \\Lambda } } _ { h } ) \\hat { \\Lambda } _ { h } ^ { - 1 / 2 } \\rVert _ { 2 } . } \\end{array}\n$$",
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+ "text": "$$\n\\hat { \\Lambda } _ { h } \\succeq N \\Lambda _ { h } \\succeq \\frac { N } { C _ { h } } \\overline { { \\Lambda } } _ { h } ,\n$$",
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+ "text": "which implies $\\| N \\hat { \\Lambda } _ { h } ^ { - 1 / 2 } \\overline { { \\Lambda } } _ { h } \\hat { \\Lambda } _ { h } ^ { - 1 / 2 } \\| \\le C _ { h }$ . Moreover, conditioned on the event in Lemma D.1, ",
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+ "text": "$$\n\\begin{array} { r } { \\| \\hat { \\boldsymbol { \\Lambda } } _ { h } ^ { - 1 / 2 } ( \\overline { { \\Phi } } _ { h } ^ { \\top } \\overline { { \\Phi } } _ { h } - N \\overline { { \\boldsymbol { \\Lambda } } } _ { h } ) \\hat { \\boldsymbol { \\Lambda } } _ { h } ^ { - 1 / 2 } \\| _ { 2 } \\leq C \\sqrt { d \\log ( d H / \\delta ) N } / \\lambda . } \\end{array}\n$$",
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+ "text": "$$\n\\| \\hat { \\boldsymbol { \\Lambda } } _ { 1 } ^ { - 1 / 2 } \\overline { { \\boldsymbol { \\Lambda } } } _ { 1 } \\hat { \\boldsymbol { \\Lambda } } _ { 1 } ^ { - 1 / 2 } \\| _ { 2 } \\leq C _ { 1 } / N .\n$$",
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+ "type": "equation",
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+ "text": "$$\n\\begin{array} { r } { \\| \\hat { \\boldsymbol \\Lambda } _ { h } ^ { - 1 / 2 } ( \\overline { { \\Phi } } _ { h } ^ { \\top } \\overline { { \\Phi } } _ { h } ) \\hat { \\boldsymbol \\Lambda } _ { h } ^ { - 1 / 2 } \\| _ { 2 } \\leq C _ { h } + C \\sqrt { d \\log ( d H / \\delta ) N } / \\lambda . } \\end{array}\n$$",
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+ "text": "Finally, by Theorem 1.2 in (Hsu et al., 2012a), with probability $1 - \\delta / ( 4 H )$ , for some constant $C ^ { \\prime }$ , we have ",
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+ "type": "equation",
+ "img_path": "images/338970a0fdadb8f420c39e132f02752370880618d5a28946bca55245ccfbe467.jpg",
+ "text": "$$\n\\begin{array} { r } { \\| \\xi _ { h } \\| _ { \\Phi _ { h } \\hat { \\Lambda } _ { h } ^ { - 1 } \\Phi _ { h } ^ { \\top } } ^ { 2 } \\le C ^ { \\prime } H ^ { 2 } d \\log ( H / \\delta ) . } \\end{array}\n$$",
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+ "text": "$$\n\\begin{array} { r l } & { \\quad \\quad \\left\\| \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\overline { { \\Phi } } _ { 3 } \\cdots \\hat { \\Lambda } _ { h } ^ { - 1 } \\Phi _ { h } ^ { \\top } \\xi _ { h } \\right\\| _ { \\overline { { \\Lambda } } _ { 1 } } ^ { 2 } + \\left\\| \\hat { \\Lambda } _ { 1 } ^ { - 1 } \\Phi _ { 1 } ^ { \\top } \\overline { { \\Phi } } _ { 2 } \\hat { \\Lambda } _ { 2 } ^ { - 1 } \\Phi _ { 2 } ^ { \\top } \\overline { { \\Phi } } _ { 3 } \\cdots \\lambda \\hat { \\Lambda } _ { h } ^ { - 1 } \\theta _ { h } \\right\\| _ { \\overline { { \\Lambda } } _ { 1 } } ^ { 2 } } \\\\ & { \\leq \\frac { C _ { 1 } } { N } ( C _ { 2 } + C \\sqrt { d \\log ( d / \\delta ) N } / \\lambda ) \\times \\cdots \\times ( C _ { h } + C \\sqrt { d \\log ( d / \\delta ) N } / \\lambda ) \\times ( C ^ { \\prime } H ^ { 2 } d \\log ( H / \\delta ) + \\lambda H } \\\\ & { \\leq \\frac { C _ { 1 } } { N } ( C _ { 2 } + 1 / H ) \\times \\cdots \\times ( C _ { h } + 1 / H ) \\times ( C ^ { \\prime } H ^ { 2 } d \\log ( H / \\delta ) + \\lambda H ^ { 2 } d ) } \\\\ & { \\leq \\frac { e } { N } C _ { 1 } \\times C _ { 2 } \\times \\cdots \\times C _ { h } \\times ( C ^ { \\prime } H ^ { 2 } d \\log ( H / \\delta ) + C d H ^ { 3 } \\sqrt { d \\log ( d H / \\delta ) N } ) . } \\end{array}\n$$",
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+ "text": "$$\n\\mathbb { E } _ { s _ { 1 } } [ ( Q _ { 1 } ^ { \\pi } ( s _ { 1 } , \\pi ( s _ { 1 } ) ) - \\hat { Q } _ { 1 } ( s _ { 1 } , \\pi ( s _ { 1 } ) ) ) ^ { 2 } ] \\leq c \\cdot \\left( \\prod _ { h = 1 } ^ { H } C _ { h } \\right) \\cdot d H ^ { 5 } \\cdot \\sqrt { \\frac { d \\log ( d H / \\delta ) } { N } } .\n$$",
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| 2:Let QH+1(·,·)=O andVH+1(-)= 0 |
| forh=H,H-1,...,1do |
| 4: | Take samples (s,a)~ μh,rh ~ r(s,a) and sh ~P(s,a) for eachi∈ [N] |
| 5: | LetAh=∑i∈[N](sn,an）(s,a）T+λI |
| 6: 7: | LetOn=An1（∑N（s,）·(nh+V+1(3)） |
| Let Qn(-,) = (.,·)TOn and Vh(.)=Q(·,π()) |
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| 2:Let QH+1(·,·)=O andVH+1(-)= 0 |
| forh=H,H-1,...,1do |
| 4: | Take samples (s,a)~ μh,rh ~ r(s,a) and sh ~P(s,a) for eachi∈ [N] |
| 5: | LetAh=∑i∈[N](sn,an）(s,a）T+λI |
| 6: 7: | LetOn=An1（∑N（s,）·(nh+V+1(3)） |
| Let Qn(-,) = (.,·)TOn and Vh(.)=Q(·,π()) |
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| 2:Let QH+1(·,·)=O andVH+1(-)= 0 |
| forh=H,H-1,...,1do |
| 4: | Take samples (s,a)~ μh,rh ~ r(s,a) and sh ~P(s,a) for eachi∈ [N] |
| 5: | LetAh=∑i∈[N](sn,an）(s,a）T+λI |
| 6: 7: | LetOn=An1（∑N（s,）·(nh+V+1(3)） |
| Let Qn(-,) = (.,·)TOn and Vh(.)=Q(·,π()) |
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+# PITFALLS OF IN-DOMAIN UNCERTAINTY ESTIMATION AND ENSEMBLING IN DEEP LEARNING
+
+Arsenii Ashukha ∗ Samsung AI Center Moscow, HSE‡ aashukha@bayesgroup.ru
+
+Alexander Lyzhov ∗ Samsung AI Center Moscow, Skoltech†, HSE§ alyzhov@bayesgroup.ru
+
+Dmitry Molchanov ∗ Samsung AI Center Moscow, HSE‡ dmolch@bayesgroup.ru
+
+Dmitry Vetrov HSE‡, Samsung AI Center Moscow dvetrov@bayesgroup.ru
+
+# ABSTRACT
+
+Uncertainty estimation and ensembling methods go hand-in-hand. Uncertainty estimation is one of the main benchmarks for assessment of ensembling performance. At the same time, deep learning ensembles have provided state-of-the-art results in uncertainty estimation. In this work, we focus on in-domain uncertainty for image classification. We explore the standards for its quantification and point out pitfalls of existing metrics. Avoiding these pitfalls, we perform a broad study of different ensembling techniques. To provide more insight in this study, we introduce the deep ensemble equivalent score (DEE) and show that many sophisticated ensembling techniques are equivalent to an ensemble of only few independently trained networks in terms of test performance.
+
+# video / code / blog post
+
+# 1 INTRODUCTION
+
+Deep neural networks (DNNs) have become one of the most popular families of machine learning models. The predictive performance of DNNs for classification is often measured in terms of accuracy. However, DNNs have been shown to yield inaccurate and unreliable probability estimates, or predictive uncertainty (Guo et al., 2017). This has brought considerable attention to the problem of uncertainty estimation with deep neural networks.
+
+There are many faces to uncertainty estimation. Different desirable uncertainty estimation properties of a model require different settings and metrics to capture them. Out-of-domain uncertainty of the model is measured on the data that does not follow the same distribution as the training dataset (out-of-domain data). Out-of-domain data can include images corrupted with rotations or blurring, adversarial attacks (Szegedy et al., 2013) or data points from a completely different dataset. The model is expected to be resistant to data corruptions and to be more uncertain on out-of-domain data than on in-domain data. On the contrary, in-domain uncertainty of the model is measured on data taken from the training data distribution, i.e. data from the same domain. In this setting the model is expected to produce reliable probability estimates, e.g. the model shouldn’t be too overconfident in its wrong predictions.
+
+Pitfalls of metrics We show that many common metrics of in-domain uncertainty estimation (e.g. log-likelihood, Brier score, calibration metrics, etc.) are either not comparable across different models or fail to provide a reliable ranking. We address some of the stated pitfalls and point out more reasonable evaluation schemes. For instance, although temperature scaling is not a standard for ensembling techniques, it is a must for a fair evaluation. With this in mind, the calibrated log-likelihood avoids most of the stated pitfalls and generally is a reasonable metric for in-domain uncertainty estimation task.
+
+Pitfalls of ensembles Equipped with the proposed evaluation framework, we are revisiting the evaluation of ensembles of DNNs—one of the major tools for uncertainty estimation. We introduce the deep ensemble equivalent (DEE) score that measures the number of independently trained models that, when ensembled, achieve the same performance as the ensembling technique of interest. The DEE score allows us to compare ensembling techniques across different datasets and architectures using a unified scale. Our study shows that most of the popular ensembling techniques require averaging predictions across dozens of samples (members of an ensemble), yet are essentially equivalent to an ensemble of only few independently trained models.
+
+Missing part of ensembling In our study, test-time data augmentation (TTA) turned out to be a surprisingly strong baseline for uncertainty estimation and a simple way to improve ensembles. Despite being a popular technique in large-scale classification, TTA seems to be overlooked in the community of uncertainty estimation and ensembling.
+
+# 2 SCOPE OF THE PAPER
+
+We use standard benchmark problems of image classification which comprise a common setting in research on learning ensembles of neural networks. There are other relevant settings where the correctness of probability estimates can be a priority, and ensembling techniques are used to improve it. These settings include, but are not limited to, regression, language modeling (Gal, 2016), image segmentation (Gustafsson et al., 2019), active learning (Settles, 2012) and reinforcement learning (Buckman et al., 2018; Chua et al., 2018).
+
+We focus on in-domain uncertainty, as opposed to out-of-domain uncertainty. Out-of-domain uncertainty includes detection of inputs that come from a completely different domain or have been corrupted by noise or adversarial attacks. This setting has been thoroughly explored by (Ovadia et al., 2019).
+
+We only consider methods that are trained on clean data with simple data augmentation. Some other methods use out-of-domain data (Malinin & Gales, 2018) or more elaborate data augmentation, e.g. mixup (Zhang et al., 2017) or adversarial training (Lakshminarayanan et al., 2017) to improve accuracy, robustness and uncertainty.
+
+We use conventional training procedures. We use the stochastic gradient descent (SGD) and use batch normalization (Ioffe & Szegedy, 2015), both being the de-facto standards in modern deep learning. We refrain from using more elaborate optimization techniques including works on superconvergence (Smith & Topin, 2019) and stochastic weight averaging (SWA) (Izmailov et al., 2018). These techniques can be used to drastically accelerate training and to improve the predictive performance. Thus, we do not comment on the training time of different ensembling methods since the use of these and other more efficient training techniques would render such a comparison obsolete.
+
+A number of related works study ways of approximating and accelerating prediction in ensembles. The distillation mechanism allows to approximate the prediction of an ensemble by a single neural network (Hinton et al., 2015; Balan et al., 2015; Tran et al., 2020), whereas fast dropout (Wang & Manning, 2013) and deterministic variational inference (Wu et al., 2018) allow to approximate the predictive distribution of specific stochastic computation graphs. We measure the raw power of ensembling techniques without these approximations.
+
+All of the aforementioned alternative settings are orthogonal to the scope of this paper and are promising points of interest for further research.
+
+# 3 PITFALLS OF IN-DOMAIN UNCERTAINTY ESTIMATION
+
+No single metric measures all the desirable properties of uncertainty estimates obtained by a model of interest. Because of this, the community is using many different metrics in an attempt to capture the quality of uncertainty estimation, such as the Brier score (Brier, 1950), log-likelihood (Quinonero-Candela et al., 2005), metrics of calibration (Guo et al., 2017; Nixon et al., 2019), performance of misclassification detection (Malinin & Gales, 2018), and threshold–accuracy curves (Lakshminarayanan et al., 2017). In the section we highlight the pitfalls of the aforementioned metrics, and demonstrate that these pitfalls can significantly affect evaluation, changing the ranking of the methods.
+
+
+Figure 1: The average log-likelihood of two different ensembling techniques for ResNet50 on ImageNet dataset before (solid) and after (dashed) temperature scaling. Without the temperature scaling, test-time data augmentation decreases the log-likelihood of plain deep ensembles. However, when the temperature scaling is enabled, deep ensembles with test-time data augmentation outperform plain deep ensembles.
+
+
+Figure 2: Thresholded adaptive calibration error (TACE) is highly sensitive to the threshold and the number of bins. It does not provide a consistent ranking of different ensembling techniques. Here TACE is reported for VGG16BN model on CIFAR-100 dataset and is evaluated at the optimal temperature.
+
+Notation We consider a classification problem with a dataset that consists of $N$ training and $n$ testing pairs $( x _ { i } , y _ { i } ^ { * } ) \sim p ( x , y )$ , where $x _ { i }$ is an object and $y _ { i } ^ { * } \in \{ 1 , \ldots , C \}$ is a discrete class label. A probabilistic classifier maps an object $x _ { i }$ into a predictive distribution ${ \dot { p } } ( y \mid x _ { i } )$ . The predictive distribution ${ \hat { p } } ( y \mid x _ { i } )$ of a deep neural network is typically defined by the softmax function ${ \hat { p } } ( y \mid x ) =$ $\mathrm { S o f t m a x } ( z ( x ) / T )$ , where $z ( x )$ is a vector of logits and $T$ is a scalar parameter standing for the temperature of the predictive distribution. This scalar parameter is usually set to $T = 1$ or is tuned on a validation set (Guo et al., 2017). The maximum probability $\operatorname* { m a x } _ { c } \dot { \hat { p } } ( y = c | x _ { i } )$ is called the confidence of a classifier $\hat { p }$ on an object $x _ { i }$ . $\mathbb { I } [ \cdot ]$ denotes the indicator function throughout the text.
+
+# 3.1 LOG-LIKELIHOOD AND BRIER SCORE
+
+The average test log-likelihood $\begin{array} { r } { \mathbf { L } \mathbf { L } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \log \hat { p } ( y = y _ { i } ^ { * } \mid x _ { i } ) } \end{array}$ is a popular metric for measuring the quality of in-domain uncertainty of deep learning models. It directly penalizes high probability scores assigned to incorrect labels and low probability scores assigned to the correct labels $y _ { i } ^ { * }$ .
+
+LL is sensitive to the softmax temperature $T$ . The temperature that has been implicitly learned during training can be far from optimal for the test data. However, a nearly optimal temperature can be found post-hoc by maximizing the log-likelihood on validation data. This approach is called temperature scaling or calibration (Guo et al., 2017). Despite its simplicity, the temperature scaling results in a notable improvement in the LL.
+
+While ensembling techniques tend to have better temperature than single models, the default choice of $T = 1$ is still suboptimal. Comparing the LL with suboptimal temperatures—that is often the case in practice—can potentially produce an arbitrary ranking of different methods.
+
+Comparison of the log-likelihood should only be performed at the optimal temperature.
+
+Empirically, we demonstrate that the overall ordering of methods and also the best ensembling method according to the LL can vary depending on temperature $T$ . While this applies to most ensembling techniques (see Figure 10), this effect is most noticeable on experiments with data augmentation on ImageNet (Figure 1).
+
+We introduce a new metric called the calibrated log-likelihood that is the log-likelihood at the optimal temperature.
+
+The calibrated log-likelihood considers a model and a post-training calibration as a unified system, targeting to measure all models in the equal conditions of optimal temperature. That allows to avoid measuring calibration error that can be eliminated by a simple temperature scaling. The metric significantly affects the results of the comparison. For example, in Figure 10 the differences between Bayesian (VI, K-FAC, SWAG, dropout) and conventional non-Bayesian networks become much less pronounced, and in most cases making conventional non-Bayesian networks match the performance of Bayesian ones (VI, K-FAC, Dropout) on ResNet110, ResNet164, and WideResNet.
+
+We show how to obtain an unbiased estimate of the calibrated log-likelihood without a held-out validation set in Section 3.5.
+
+LL also demonstrates a high correlation with accuracy $( \rho > 0 . 8 6 )$ , that in case of calibrated LL becomes even stronger $( \rho > 0 . 9 5 )$ . That suggests that while (calibrated) LL measures the uncertainty of the model, it still has a significant dependence on the accuracy and vice versa. A model with higher accuracy would likely have a higher log-likelihood. See Figure 9 in Appendix C for more details.
+
+Brier score $\begin{array} { r } { \mathrm { { B S } } = \frac { 1 } { n } \frac { 1 } { C } \sum _ { i = 1 } ^ { n } \sum _ { c = 1 } ^ { C } ( \mathbb { I } [ y _ { i } ^ { * } = c ] - \hat { p } ( y = c | x _ { i } ) ) ^ { 2 } } \end{array}$ has also been known for a long time as a metric for verification of predicted probabilities (Brier, 1950). Similarly to the log-likelihood, the Brier score penalizes low probabilities assigned to correct predictions and high probabilities assigned to wrong ones. It is also sensitive to the temperature of the softmax distribution and behaves similarly to the log-likelihood. While these metrics are not strictly equivalent, they show a high empirical correlation for a wide range of models on CIFAR-10, CIFAR-100 and ImageNet datasets (see Figure 8 in Appendix C ).
+
+# 3.2 MISCLASSIFICATION DETECTION
+
+Detection of wrong predictions of the model, or misclassifications, is a popular downstream problem relevant to the problem of in-domain uncertainty estimation. Since misclassification detection is essentially a binary classification problem, some papers measure its quality using conventional metrics for binary classification such as AUC-ROC and AUC-PR (Malinin & Gales, 2018; Cui et al., 2019; Mozejko et al., 2018). These papers use an uncertainty criterion like confidence or predictive ˙ entropy $\mathcal { H } [ \bar { p } ( \boldsymbol { y } | \boldsymbol { x } _ { i } ) ]$ as a prediction score.
+
+While these metrics can be used to assess the misclassification detection performance of a single model, they cannot be used to directly compare misclassification performance across different models. Correct and incorrect predictions are specific for every model, therefore, every model induces its own binary classification problem. The induced problems can differ significantly, since different models produce different confidences and misclassify different objects. In other words, comparing such metrics implies a comparison of performance of classifiers that solve different classification problems. Such metrics are therefore incomparable.
+
+AUCs for misclassification detection cannot be directly compared between different models.
+
+While comparing AUCs is incorrect in the setting of misclassification detection, it is correct to compare these metrics in many out-of-domain data detection problems. In that case, both objects and targets of the induced binary classification problems remain the same for all models. All outof-domain objects have a positive label and all in-domain objects have a negative label. Note that this condition does not necessarily hold in the problem of detection of adversarial attacks. Different models generally have different inputs after an adversarial attack, so such AUC-based metrics might still be flawed.
+
+# 3.3 CLASSIFICATION WITH REJECTION
+
+Accuracy-confidence curves are another way to measure the performance of misclassification detection. These curves measure the accuracy on the set of objects with confidence $\operatorname* { m a x } _ { c } { \hat { p } } ( y = c | x _ { i } )$ above a certain threshold $\tau$ (Lakshminarayanan et al., 2017) and ignoring or rejecting the others.
+
+The main problem with accuracy-confidence curves is that they rely too much on calibration and the actual values of confidence. Models with different temperatures have different numbers of objects at each confidence level which does not allow for a meaningful comparison. To overcome this problem, one can swhich from thresholding by the confidence level to thresholding by the number of rejected objects. The corresponding curves are then less sensitive to temperature scaling and thus allow to compare the rejection ability in a more meaningful way. Such curves have been known as accuracyrejection curves (Nadeem et al., 2009). In order to obtain a scalar metric for easy comparisons, one can compute the area under this curve, resulting in AU-ARC (Nadeem et al., 2009).
+
+# 3.4 CALIBRATION METRICS
+
+Informally speaking, a probabilistic classifier is calibrated if any predicted class probability is equal to the true class probability according to the underlying data distribution (see (Vaicenavicius et al., 2019) for formal definitions). Any deviation from the perfect calibration is called miscalibration. For brevity, we will use $\hat { p } _ { i , c }$ to denote ${ \hat { p } } ( y = c | x _ { i } )$ in the current section.
+
+Expected calibration error (ECE) (Naeini et al., 2015) is a metric that estimates model miscalibration by binning the assigned probability scores and comparing them to average accuracies inside these bins. Assuming ${ B } _ { m } ^ { \bar { } }$ denotes the $m$ -th bin and $M$ is overall number of bins, the ECE is defined as follows:
+
+$$
+\mathrm { E C E } = \sum _ { m = 1 } ^ { M } \frac { | B _ { m } | } { n } \left| \operatorname { a c c } ( B _ { m } ) - \operatorname { c o n f } ( B _ { m } ) \right| ,
+$$
+
+where $\begin{array} { r } { \operatorname { a c c } ( B ) = | B | ^ { - 1 } \sum _ { i \in B } { \mathbb { I } } [ \operatorname { a r g m a x } _ { c } \hat { p } _ { i , c } = y _ { i } ^ { * } ] \mathrm { ~ a n d ~ c o n f } ( B ) = | B | ^ { - 1 } \sum _ { i \in B } \hat { p } _ { i , y _ { i } ^ { * } } . } \end{array}$
+
+A recent line of works on measuring calibration in deep learning (Vaicenavicius et al., 2019; Kumar et al., 2019; Nixon et al., 2019) outlines several problems of the ECE score. Firstly, ECE is a biased estimate of the true calibration. Secondly, ECE-like scores cannot be optimized directly since they are minimized by a model with constant uniform predictions, making the infinite temperature $T = + \infty$ its global optimum. Thirdly, ECE only estimates miscalibration in terms of the maximum assigned probability whereas practical applications may require the full predicted probability vector to be calibrated. Finally, biases of ECE on different models may not be equal, rendering the miscalibration estimates incompatible. Similar concerns are also discussed by Ding et al. (2019).
+
+Thresholded adaptive calibration error (TACE) was proposed as a step towards solving some of these problems (Nixon et al., 2019). TACE disregards all predicted probabilities that are less than a certain threshold (hence thresholded), chooses the bin locations adaptively so that each bin has the same number of objects (hence adaptive), and estimates miscalibration of probabilties across all classes in the prediction (not just the top-1 predicted class as in ECE). Assuming that $B _ { m } ^ { \mathrm { { T A } } }$ denotes the $m$ -th thresholded adaptive bin and $\bar { M }$ is the overall number of bins, TACE is defined as follows:
+
+$$
+\begin{array} { r } { \mathrm { \ o b j s } ( B ^ { \mathtt { T A } } , c ) = | B ^ { \mathtt { T A } } | ^ { - 1 } \sum _ { i \in B ^ { \mathtt { T A } } } \mathbb { I } [ y _ { i } ^ { * } = c ] \mathrm { \ a n d \ c o n f } ( B ^ { \mathtt { T A } } , c ) = | B ^ { \mathtt { T A } } | ^ { - 1 } \sum _ { i \in B ^ { \mathtt { T A } } } \hat { p } _ { i } } \end{array}
+$$
+
+$$
+\mathrm { T A C E } = \frac { 1 } { C M } \sum _ { c = 1 } ^ { C } \sum _ { m = 1 } ^ { M } \frac { | B _ { m } ^ { \mathrm { T A } } | } { n } \left| \mathrm { o b j s } ( B _ { m } ^ { \mathrm { T A } } , c ) - \mathrm { c o n f } ( B _ { m } ^ { \mathrm { T A } } , c ) \right| ,
+$$
+
+Although TACE does solve several problems of ECE and is useful for measuring calibration of a specific model, it still cannot be used as a reliable criterion for comparing different models. Theory suggests that it is still a biased estimate of true calibration with different bias for each model (Vaicenavicius et al., 2019). In practice, we find that TACE is sensitive to its two parameters, the number of bins and the threshold, and does not provide a consistent ranking of different models, as shown in Figure 2.
+
+# 3.5 CALIBRATED LOG-LIKELIHOOD AND TEST-TIME CROSS-VALIDATION
+
+There are two common ways to perform temperature scaling using a validation set when training on datasets that only feature public training and test sets (e.g. CIFARs). The public training set might be divided into a smaller training set and validation set, or the public test set can be split into test and validation parts (Guo et al., 2017; Nixon et al., 2019). The problem with the first method is that the resulting models cannot be directly compared with all the other models that have been trained on the full training set. The second approach, however, provides an unbiased estimate of metrics such as log-likelihood and Brier score but introduces more variance.
+
+In order to reduce the variance of the second approach, we perform a “test-time cross-validation”. We randomly divide the test set into two equal parts, then compute metrics for each half of the test set using the temperature optimized on another half. We repeat this procedure five times and average the results across different random partitions to reduce the variance of the computed metrics.
+
+# 4 A STUDY OF ENSEMBLING & DEEP ENSEMBLE EQUIVALENT
+
+Ensembles of deep neural networks have become a de-facto standard for uncertainty estimation and improving the quality of deep learning models (Hansen & Salamon, 1990; Krizhevsky et al., 2009; Lakshminarayanan et al., 2017). There are two main directions of training ensembles of DNNs: training stochastic computation graphs and obtaining separate snapshots of neural network parameters.
+
+Methods based on the paradigm of stochastic computation graphs introduce some kind of random noise over the weights or activations of deep learning models. When the model is trained, each sample of the noise corresponds to a member of the ensemble. During test time, the predictions are averaged across the noise samples. These methods include (test-time) data augmentation, dropout (Srivastava et al., 2014; Gal & Ghahramani, 2016), variational inference (Blundell et al., 2015; Kingma et al., 2015; Louizos & Welling, 2017), batch normalization (Ioffe & Szegedy, 2015; Teye et al., 2018; Atanov et al., 2019), Laplace approximation (Ritter et al., 2018) and many more.
+
+Snapshot-based methods aim to obtain sets of weights for deep learning models and then to average the predictions across these weights. The weights can be trained independently (e.g. deep ensembles (Lakshminarayanan et al., 2017)), collected on different stages of a training trajectory (e.g. snapshot ensembles (Huang et al., 2017) and fast geometric ensembles (Garipov et al., 2018)), or obtained from a sampling process (e.g. MCMC-based methods (Welling & Teh, 2011; Zhang et al., 2019)). These two paradigms can be combined. Some works suggest construction of ensembles of stochastic computation graphs (Tomczak et al., 2018), while others make use of the collected snapshots to construct a stochastic computation graph (Wang et al., 2018; Maddox et al., 2019).
+
+In this paper we consider the following ensembling techniques: deep ensembles (Lakshminarayanan et al., 2017), snapshot ensembles (SSE by Huang et al. (2017)), fast geometric ensembling (FGE by Garipov et al. (2018)), SWA-Gaussian (SWAG by Maddox et al. (2019)), cyclical SGLD (cSGLD by Zhang et al. (2019)), variational inference (VI by Blundell et al. (2015)), K-FAC Laplace approximation (Ritter et al., 2018), dropout (Srivastava et al., 2014) and test-time data augmentation (Krizhevsky et al., 2009). These techniques were chosen to cover a diverse set of approaches keeping their predictive performance in mind.
+
+All these techniques can be summarized as distributions $q _ { m } ( \omega )$ over parameters $\omega$ of computation graphs, where $m$ stands for the technique. During testing, one can average the predictions across parameters $\omega \sim q _ { m } ( \omega )$ to approximate the predictive distribution
+
+$$
+\hat { p } ( y _ { i } | x _ { i } ) \approx \int p ( y _ { i } | x _ { i } , \omega ) q _ { m } ( \omega ) d \omega \simeq \frac { 1 } { K } \sum _ { k = 1 } ^ { K } p ( y _ { i } | x _ { i } , \omega _ { k } ) , \quad \omega _ { k } \sim q _ { m } ( \omega ) \omega _ { k } ^ { \quad \mathrm { ~ d ~ } } ,
+$$
+
+For example,Dirac’s deltas $S$ rks can be represented in this form as a mix, centered at independently trained snapshots re of . Sim $S$ $\begin{array} { r } { q _ { \mathrm { { D E } } } ( \omega ) = \frac { 1 } { S } \sum _ { s = 1 } ^ { S } \delta ( \omega - \omega _ { s } ) } \end{array}$ $\omega _ { s }$ over the weight matrices and convolutional kernels $\omega$ is represented as $q _ { \mathrm { v I } } ( \omega ) = \mathcal { N } ( \omega | \mu , \mathrm { d i a g } ( \sigma ^ { 2 } ) )$ , $\mu$ and $\sigma ^ { 2 }$ being the optimal variational means and variances respectively.
+
+If one considers data augmentation as a part of the computational graph, it can be parameterized by the coordinates of the random crop and the flag for whether to flip the image horizontally or not. Sampling from the corresponding $q _ { \mathrm { a u g } } ( \omega )$ would generate different ways to augment the data during inference. However, as data augmentation is present by default during the training of all othe mentioned ensembling techniques, it is suitable to study it in combination with these methods and not as a separate ensembling technique. We perform such an evaluation in Section 4.3.
+
+
+Figure 3: The deep ensemble equivalent score (DEE) for different numbers of samples on CIFAR10, CIFAR-100, and ImageNet datasets averaged across different deep convolutional architectures. A deep ensemble equivalent score (DEE) of a model is equal to the minimum size of a deep ensemble (an ensemble of independently train networks) that achieves the same performance as the model under consideration. The score is measured in the number of models (higher is better). The area between average lower and upper bounds of DEE is shaded. The plot demonstrates that all of the ensembling techniques are far less efficient than deep ensembles during inference and fail to produce the same level of performance as deep ensembles. The comparison that is normalized on training time is presented in Appendix A.
+
+Typically, the approximation (equation 3) requires $K$ independent forward passes through a neural network, making the test-time budget directly comparable across all methods.
+
+# 4.1 DEEP ENSEMBLE EQUIVALENT
+
+Most ensembling techniques under consideration are either bounded to a single mode, or provide positively correlated samples. Deep ensemble, on the other hand, is a simple technique that provides independent samples from different modes of the loss landscape, which, intuitively, should result in a better ensemble. Therefore deep ensembles can be considered a strong baseline for the performance of other ensembling techniques given a fixed test-time computation budget.
+
+Comparing the performance of ensembling techniques is, however, a challenging problem. Different models on different datasets achieve different values of metrics; their dependence on the number of samples is non-trivial, and varies depending on a specific model and dataset. Values of the metrics are thus lacking in interpretability as the gain in performance has to be compared against a modeland dataset-specific baseline.
+
+Aiming to introduce perspective and interpretability in our study, we introduce the deep ensemble equivalent score that employs deep ensembles to measure the performance of other ensembling techniques. Specifically, the deep ensemble equivalent score answers the following question:
+
+What size of deep ensemble yields the same performance as a particular ensembling method?
+
+Following the insights from the previous sections, we base the deep ensemble equivalent on the calibrated log-likelihood (CLL). Formally speaking, we define the deep ensemble equivalent (DEE) for an ensembling method $m$ and its upper and lower bounds as follows:
+
+$$
+\begin{array} { r l r } & { } & { \mathrm { D E E } _ { m } ( k ) = \operatorname* { m i n } \Big \{ l \in \mathbb { R } , l \geq 1 \Big | \mathrm { C L L } _ { D E } ^ { \mathrm { m e a n } } ( l ) \geq \mathrm { C L L } _ { m } ^ { \mathrm { m e a n } } ( k ) \Big \} , \quad } \\ & { } & { \mathrm { D E E } _ { m } ^ { \mathrm { ~ u p p e r ~ } } ( k ) = \operatorname* { m i n } \Big \{ l \in \mathbb { R } , l \geq 1 \Big | \mathrm { C L L } _ { D E } ^ { \mathrm { m e a n } } ( l ) \mp \mathrm { C L L } _ { D E } ^ { \mathrm { s t d } } ( l ) \geq \mathrm { C L L } _ { m } ^ { \mathrm { m e a n } } ( k ) \Big \} , } \end{array}
+$$
+
+
+Figure 4: An illustration of test-time augmentation (TTA) for an ensemble. We apply every member of an ensemble to a separate random augmentation of an image. The predictions of all members are averaged to produce a final prediction of an ensemble. In our experiments, TTA leads to a significant boost of the performance for most of the ensembling techniques on ImageNet with a sufficient computational budget (see Figure 5).
+
+where $\mathrm { C L L } _ { m } ^ { \mathrm { m e a n / s t d } } ( l )$ are the mean and the standard deviation of the calibrated log-likelihood achieved by an ensembling method $m$ with $l$ samples. We compute $\mathrm { C L L } _ { \mathrm { D E } } ^ { \mathrm { m e a n } } ( l )$ and $\mathrm { C L L } _ { \mathrm { D E } } ^ { \mathrm { s t d } } ( l )$ for natural numbers $l \in { \mathbb { N } } _ { > 0 }$ and use linear interpolation to define them for real values $l \geq 1$ . In the following plots we report $\mathrm { D E E } _ { m } ( k )$ for different methods $m$ with different numbers of samples $k$ , and shade the area between the respective lower and upper bounds $\mathrm { D E E } _ { m } ^ { \mathrm { l o w e r } } ( k )$ and $\mathrm { D E E } _ { m } ^ { \mathrm { u p p e r } } ( k )$ .
+
+# 4.2 EXPERIMENTS
+
+We compute the deep ensemble equivalent (DEE) of various ensembling techniques for four popular deep architectures: VGG16 (Simonyan & Zisserman, 2014), PreResNet110/164 (He et al., 2016), and WideResNet28x10 (Zagoruyko & Komodakis, 2016) on CIFAR-10/100 datasets (Krizhevsky et al., 2009), and ResNet50 (He et al., 2016) on ImageNet dataset (Russakovsky et al., 2015). We use PyTorch (Paszke et al., 2017) for implementation of these models, building upon available public implementations. Our implementation closely matches the quality of methods that has been reported in original works. Technical details on training, hyperparameters and implementations can be found in Appendix B. The source code and all computed metrics are available on GitHub1.
+
+As one can see on Figure 3, ensembling methods clearly fall into three categories. SSE and cSGLD outperform all other techniques except deep ensembles and enjoy a near-linear scaling of DEE with the number of samples on CIFAR datasets. The investigation of weight-space trajectories of cSGLD and SSE (Huang et al., 2017; Zhang et al., 2019) suggests that these methods can efficiently explore different modes of the loss landscape. In terms of the deep ensemble equivalent, these methods do not saturate unlike other methods that are bound to a single mode. We found SSE to still saturate on ImageNet. This is likely due to suboptimal hyperparameters of the cyclic learning rate schedule. More verbose results are presented in Figures 11–13 and in Table 5 and Table 8 in Appendix C.
+
+In our experiments SSE typically outperforms cSGLD. This is mostly due to the fact that SSE has a much larger training budget. The cycle lengths and learning rates of SSE and cSGLD are comparable, however, SSE collects one snapshot per cycle while cSGLD collects three snapshots. This makes samples from SSE less correlated with each other while increasing the training budget threefold. Both SSE and cSGLD can be adjusted to obtain a different trade-off between the training budget and the DEE-to-samples ratio. We reused the schedules provided in the original papers (Huang et al., 2017; Zhang et al., 2019).
+
+
+Figure 5: How to read results: $\times \ \frac { \mathrm { \ t e s t { - } t i m e \ a u g } } { 1 0 \ \mathrm { s a m p l e s } } \quad \Phi , \times \ \frac { \mathrm { \ t e s t { - } t i m e \ a u g } } { 5 0 \ \mathrm { s a m p l e s } } \quad \cdot$ . The negative calibrated loglikelihood (lower is better) for different ensembling techniques on ImageNet. We report performance for two regimes. Central-crop evaluation $( \times \times )$ means every member of an ensemble is applied to a central crop of an image, and test-time data augmentation ( ) means each member of the ensemble is applied to a separate random augmentation of the image. Test-time data augmentation significantly improves ensembles with no additional computational cost. Interestingly, a single model with TTA performs competitively with methods that require significantly larger parameters budget, and training complexity, e.g., a single model with TTA performs closely to pure deep ensembles (DE) of the same size.
+
+Being more “local” methods, FGE and SWAG perform worse than SSE and cSGLD, but still significantly outperform “single-snapshot” methods like dropout, K-FAC Laplace approximation and variational inference. We hypothesize that by covering a single mode with a set of snapshots, FGE and SWAG provide a better fit for the local geometry than models trained as stochastic computation graphs. This implies that the performance of FGE and SWAG should be achievable by singlesnapshot methods. However, one might need more elaborate posterior approximations and better inference techniques in order to match the performance of FGE and SWAG by training a stochastic computation graph end-to-end (as opposed to SWAG that constructs a stochastic computation graph post-hoc).
+
+The deep ensemble equivalent curves allow us to notice the common behaviour of different methods, e.g. the relation between deep ensembles, snapshot methods, advanced local methods and singlesnapshot local methods. They also allow us to notice inconsistencies that may indicate a suboptimal choice of hyperparameters. For example, we find that SSE on ImageNet quickly saturates, unlike SSE on CIFAR datasets (Figure 3). This may indicate that the hyperparameters used on ImageNet are not good enough for efficient coverage of different modes of the loss landscape. We also find that SSE on WideResNet on CIFAR-10 achieves a DEE score of 100 on approx. 70 samples (Figure 12). This may indicate that the members of the deep ensemble for this dataset-architecture pair are underfitted and may benefit from longer training or a different learning rate schedule. Such inconsistencies might be more difficult to spot using plain calibrated log-likelihood plots.
+
+# 4.3 TEST-TIME DATA AUGMENTATION IMPROVES ENSEMBLES FOR FREE
+
+Data augmentation is a time-honored technique that is widely used in deep learning, and is a crucial component for training modern DNNs. Test-time data augmentation has been used for a long time to improve the performance of convolutional networks. For example, multi-crop evaluation has been a standard procedure for the ImageNet challenge (Simonyan & Zisserman, 2014; Szegedy et al., 2015; He et al., 2016). It is, however, often overlooked in the literature on ensembling techniques in deep learning. In this section, we study the effect of test-time data augmentation on the aforementioned ensembling techniques. To keep the test-time computation budget the same, we sample one random augmentation for each member of an ensemble. Figure 5 reports the calibrated log-likelihood on combination of ensembles and test-time data augmentation for ImageNet. Other metrics and results on CIFAR-10/100 datasets are reported in Appendix C. We have used the standard data augmentation: random horizontal flips and random padded crops for CIFAR-10/100 datasets, and random horizontal flips and random resized crops for ImageNet (see more details in Appendix B).
+
+Test-time data augmentation (Figure 4) consistently improves most ensembling methods, especially on ImageNet, where we see a clear improvement across all methods (Figure 5 and Table 7). The performance gain for powerful ensembles (deep ensembles, SSE and cSGLD) on CIFAR datasets is not as dramatic (Figures 14–15 and Table 4). This is likely due to the fact that CIFAR images are small, making data augmentation limited, whereas images from ImageNet allow for a large number of diverse samples of augmented images. On the other hand, while the performance of “single-snapshot” methods (e.g. variational inference, K-FAC Laplace and dropout) is improved significantly, they perform approximately as good as an augmented version of a single model across all datasets.w
+
+Interestingly, test-time data augmentation on ImageNet improves accuracy but decreases the uncalibrated log-likelihood of deep ensembles (Table 7 in Appendix C). Test-time data augmentation breaks the nearly optimal temperature of deep ensembles and requires temperature scaling to reveal the actual performance of the method, as discussed in Section 3.1. The experiment demonstrates that ensembles may be highly miscalibrated by default while still providing superior predictive performance after calibration.
+
+We would like to note that test-time data augmentation does not always break the calibration of an ensemble, and, on the contrary, test-time data augmentation often improves the calibration of an ensemble. In our experiments, decalibration was caused by the extreme magnitude of a random crop, that is conventionally used for ImageNet augmentation. Using less extreme magnitude of the random crop fixes decalibration, that makes test-time data augmentation a more practical method that provides out-of-the-box calibration. Although, as we demonstrated earlier, there is no guarantee that any ensemble is calibrated out-of-the-box. If we are willing to apply post-hoc calibration, the final performance can be much better with more severe augmentations.
+
+# 5 DISCUSSION & CONCLUSION
+
+We have explored the field of in-domain uncertainty estimation and performed an extensive evaluation of modern ensembling techniques. Our main findings can be summarized as follows:
+
+• Temperature scaling is a must even for ensembles. While ensembles generally have better calibration out-of-the-box, they are not calibrated perfectly and can benefit from the procedure. A comparison of log-likelihoods of different ensembling methods without temperature scaling might not provide a fair ranking, especially if some models happen to be miscalibrated. Many common metrics for measuring in-domain uncertainty are either unreliable (ECE and analogues) or cannot be used to compare different methods (AUC-ROC, AUC-PR for misclassification detection; accuracy-confidence curves). In order to perform a fair comparison of different methods, one needs to be cautious of these pitfalls. Many popular ensembling techniques require dozens of samples for test-time averaging, yet are essentially equivalent to a handful of independently trained models. Deep ensembles dominate other methods given a fixed test-time budget. The results indicate, in particular, that exploration of different modes in the loss landscape is crucial for good predictive performance.
+• Methods that are stuck in a single mode are unable to compete with methods that are designed to explore different modes of the loss landscape. Would more elaborate posterior approximations and better inference techniques shorten this gap? Test-time data augmentation is a surprisingly strong baseline for in-domain uncertainty estimation. It can significantly improve other methods without increasing training time or model size since data augmentation is usually already present during training.
+
+Our takeaways are aligned with the take-home messages of Ovadia et al. (2019) that relate to indomain uncertainty estimation. We also observe a stable ordering of different methods in our experiments, and observe that deep ensembles with few members outperform methods based on stochastic computation graphs.
+
+A large number of unreliable metrics inhibits a fair comparison of different methods. Because of this, we urge the community to aim for more reliable benchmarks in the numerous setups of uncertainty estimation.
+
+# ACKNOWLEDGMENTS
+
+Dmitry Vetrov and Dmitry Molchanov were supported by the Russian Science Foundation grant no. 19-71-30020. This research was supported in part through computational resources of HPC facilities at NRU HSE.
+
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+
+Figure 6: The mean cost of training of an ensemble vs. the mean deep ensemble equivalent score. Each marker on the plot denotes one snapshot of weights.
+
+
+Figure 7: Cost of training of an ensemble vs. its quality (DEE). Each marker on a plot denotes one snapshot of weights.
+
+# A IS ”EFFICIENT” TRAINING OF ENSEMBLES EFFICIENT AT ALL?
+
+Yes! If you care most about training time efficiency. All snapshot based methods (SSE, cSGLD and FGE) on average (Figure 6) tend to achieve the same performance as deep ensembles using $2 { - } 5 \times$ less training epochs on CIFAR datasets.
+
+The gain comes at a cost of inference efficiency and memory consumption. ”Efficient” snapshotbased methods require to store much more samples of weights compared to deep ensembles making inference significantly more expensive (up to $\times 2 5$ ) given the same predictive performance.
+
+You need to get lucky with hyperparameter choice. While on average ”efficient” snapshot-based methods require less training resources, they might be completely inefficient if your hyperparameter choice is sub-optimal (see Figure 7). Such hyperparameters as the maximum learning rate, the length of learning rate decay cycle, the snapshot saving schedule can all impact the performance significantly.
+
+This analysis assumes that we use only conventional training procedure. Many models can most likely be trained, stored and executed much more efficiently with methods like super-convergence, stochastic weight averaging, compression, distillation and others. These methods are out of the scope of the paper but are interesting topics for future research. The current choice of hyperparameters may also not be optimal. We reuse the hyperparameters used in the original papers.
+
+# B EXPERIMENTAL DETAILS
+
+Implementations of deep ensembles, SWAG, FGE and K-FAC Laplace are heavily based on the original PyTorch implementations of stochastic weight averaging (SWA) 2 and SWAG 3. Implementations of cyclical MCMC and snapshot ensembles are based on the original implementation of cyclical MCMC 4. We hypothesize that the optimal hyperparameters of ensembling methods may vary widely depending on the computational budget and the number of samples in the ensemble. Searching for the optimal values for each configuration is outside the scope of this paper so we stick to the originally proposed hyperparameters whenever possible.
+
+Implied probabilistic model Conventional neural networks for classification are usually trained using the average cross-entropy loss function with weight decay regularization hidden inside an optimizer in a deep learning framework like PyTorch. The underlying optimization problem can be written as follows:
+
+$$
+L ( w ) = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \hat { p } ( y _ { i } ^ { * } \mid x _ { i } , w ) + \frac { \lambda } { 2 } \| w \| ^ { 2 } \operatorname* { m i n } _ { w } ,
+$$
+
+where $\{ ( x _ { i } , y _ { i } ^ { * } ) \} _ { i = 1 } ^ { N }$ is the training dataset of $N$ objects $x _ { i }$ with corresponding labels $y _ { i } ^ { * } , \lambda$ is the weight decay scale and $\hat { p } ( j | x _ { i } , w )$ denotes the probability that a neural network with parameters $w$ assigns to class $j$ when evaluated on object $x _ { i }$ .
+
+The cross-entropy loss defines a likelihood function $p ( y ^ { * } \mid x , w )$ and weight decay regularization, or $L _ { 2 }$ regularization corresponding to a certain Gaussian prior distribution $p ( w )$ . The whole optimization objective then corresponds to the maximum a posteriori inference in the following probabilistic model:
+
+$$
+\begin{array} { c } { { p ( y ^ { * } , w \mid x ) = p ( y ^ { * } \mid x , w ) p ( w ) , } } \\ { { \displaystyle \log p ( y ^ { * } \mid x , w ) = \log \prod _ { i = 1 } ^ { N } p ( y _ { i } ^ { * } \mid x _ { i } , w ) = \sum _ { i = 1 } ^ { N } \log \hat { p } ( y _ { i } ^ { * } \mid x _ { i } , w ) , } } \\ { { \displaystyle \log p ( w ) = \frac { - N \lambda } { 2 } \| w \| ^ { 2 } + \mathrm { c o n s t } \iff p ( w ) = \mathcal { N } \left( w \mid 0 , ( N \lambda ) ^ { - 1 } I \right) } } \end{array}
+$$
+
+In order to make the results comparable across all ensembling techniques, we used the same probabilistic model for all methods, choosing fixed weight decay parameters for each architecture. We used the softmax-based likelihood for all models. We also use the fully-factorized zero-mean Gaussian prior distribution with variances $\sigma ^ { 2 } = ( N \lambda ) ^ { - 1 }$ where the number of objects $N$ and the weight decay scale $\lambda$ are dictated by the particular datasets and neural architectures as defined in the following paragraph.
+
+Conventional networks To train a single network on CIFAR-10/100, we used SGD with batch size of 128, momentum 0.9 and model-specific parameters, i.e. the initial learning rate $( l r _ { i n i t } )$ , the weight decay coefficient (wd), and the number of optimization epochs (epoch). Specific hyperparameters are shown in Table 1. The models were trained with a unified learning rate scheduler that is shown in equation 10. All models have been trained using data augmentation that consists of horizontal flips and a random crop of 32 pixels with a padding of 4 pixels5. The standard data normalization has also been applied. Weight decays, initial learning rates, and the learning rate scheduler were taken from (Garipov et al., 2018) paper. Compared with hyperparameters of (Garipov et al., 2018), we increased the number of optimization epochs since we found that all models were underfitted. While the original WideResNet28x10 network includes a number of dropout layers with $p = 0 . 3$ and is trained for 200 epochs, we find that the WideResNet28x10 underfits in this setting and requires a longer training. Therefore, we used $p = 0$ which reduces training time while bearing no significant effect on final model performance in our experiments.
+
+Table 1: Hyperparameters of models trained on CIFARs for single-model evaluation
+
+| Model | Deep ensemble equivalent score |
| Method | 1 sample 5 samples 10 samples 50 samples 100 samples |
| VGG16 CIFAR-10 | Deep Ensembles | 1.0 5.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 1.0 2.6 | 4.7 | 21.9 | 33.8 |
| Cyclic SGLD | 1.0 1.7 | 2.6 | 7.4 | 10.5 |
| SWA-Gaussian | 1.0 2.2 | 2.9 | 4.4 | 4.6 |
| 1.3 2.0 | | | |
| Fast Geometric Ens. | | 2.4 | 4.7 | 5.8 |
| Dropout | 1.0 1.0 | 1.0 | 1.1 | 1.1 |
| Variational Inf. (FFG) | 1.0 1.8 | 2.0 | 2.5 | 2.6 1.6 |
| KFAC-Laplace Single model | 1.4 1.6 1.1 | 1.6 1.1 | 1.6 1.1 | 1.1 |
| ResNet110 CIFAR-10 | | 1.1 5.0 | | | 94.6 |
| Deep Ensembles | 1.0 1.0 1.0 | 10.0 2.0 | 48.0 | 32.6 |
| Snapshot Ensembles | 1.0 1.0 | 1.6 | 14.6 | 6.0 |
| Cyclic SGLD | 1.3 | | 4.6 | 2.0 |
| SWA-Gaussian | 1.0 | 1.5 | 1.9 | |
| Fast Geometric Ens. | 1.0 1.4 1.0 | 1.7 | 2.4 | 2.9 |
| Variational Inf. (FFG) | 1.0 | 1.2 | 1.4 | 1.5 |
| KFAC-Laplace | 1.0 1.0 | 1.0 | 1.0 | 1.1 |
| Single model | 1.0 1.0 | 1.0 | 1.0 | 1.0 |
| Deep Ensembles | 1.0 5.0 | 10.0 | 43.7 | 100.0 |
| ResNet164 CIFAR-10 | | 1.0 1.1 | 2.3 | 11.3 | 17.2 |
| Snapshot Ensembles Cyclic SGLD | 1.0 1.0 | 1.8 | 4.8 | 6.7 |
| SWA-Gaussian | 1.0 1.8 | 1.9 | 2.5 | 2.7 |
| Fast Geometric Ens. | 1.0 1.6 | 1.8 | 2.5 | 2.9 |
| Variational Inf. (FFG) | 1.0 1.0 | 1.0 | 1.3 | 1.3 |
| 1.0 | 1.0 1.0 | 1.2 | 1.2 |
| KFAC-Laplace | 1.1 | 1.1 1.1 | 1.1 | 1.1 |
| Single model | 1.0 | 10.0 | 48.0 | |
| Deep Ensembles | 5.0 1.0 | 2.5 | 30.6 | 86.0 100.0 |
| Snapshot Ensembles | 1.3 1.0 1.0 | 1.6 | 4.0 | 4.1 |
| Cyclic SGLD SWA-Gaussian | 1.0 | 1.8 | 2.5 | 3.1 3.3 |
| WideResNet Fast Geometric Ens. CIFAR-10 | 1.0 | 1.2 | 1.7 | 2.7 3.2 |
| Dropout | 1.0 | 1.2 | 1.3 1.4 | 1.4 |
| Variational Inf. (FFG) | 1.0 | 1.0 | 1.0 1.3 | 1.4 |
| KFAC-Laplace | 1.0 | 1.0 | 1.0 1.2 | 1.2 |
| Single model | 1.0 | 1.0 | 1.0 | 1.0 |
| VGG16 CIFAR-100 | | | | | 1.0 |
| Deep Ensembles | 1.0 5.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 1.0 2.9 | 5.4 | 18.8 | 26.8 |
| Cyclic SGLD | 1.0 1.8 1.0 2.3 | 3.3 3.4 | 15.9 | 28.2 |
| SWA-Gaussian | 1.2 1.9 | 2.3 | 5.2 | 5.9 5.8 |
| Fast Geometric Ens. | 1.0 1.0 | 1.0 | 4.5 1.0 | 1.1 |
| Dropout | | 1.8 | 2.1 | 2.2 |
| Variational Inf. (FFG) | 1.0 1.6 | | | 1.3 |
| KFAC-Laplace Single model | 1.0 1.0 | 1.2 1.0 | 1.3 1.0 | 1.0 |
| 1.0 1.0 | 1.0 | | |
| ResNet110 CIFAR-100 | | 5.0 | 10.0 | 50.0 | 99.0 |
| Deep Ensembles Snapshot Ensembles | 1.0 1.3 | 1.9 | 4.7 | 6.2 |
| 1.0 | 2.0 | 7.0 | 12.2 |
| Cyclic SGLD | 1.3 | 2.2 | | 3.6 |
| SWA-Gaussian | 1.0 1.7 1.5 2.6 | 3.0 | 3.3 4.2 | 4.7 |
| Fast Geometric Ens. Variational Inf. (FFG) | 1.0 1.2 | 1.3 | 1.5 | 1.5 |
| KFAC-Laplace | 1.0 1.0 | 1.1 | 1.3 | 1.4 |
| Single model | 1.0 1.0 | 1.0 | 1.0 | 1.0 |
| Deep Ensembles | 1.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 5.0 1.0 | | | |
| ResNet164 CIFAR-100 | | 1.3 | 1.9 | 4.8 | 6.3 |
| Cyclic SGLD SWA-Gaussian | 1.0 1.3 | 2.0 | 6.8 | 13.3 |
| Fast Geometric Ens. | 1.0 2.1 1.6 2.6 | 2.6 3.0 | 3.6 4.2 | 3.8 4.8 |
| Variational Inf. (FFG) | 1.0 1.2 | 1.4 | 1.6 | 1.6 |
| KFAC-Laplace | 1.0 1.1 | 1.3 | 1.5 | 1.5 |
| 1.0 | | | |
| Single model | 1.0 | 1.0 | 1.0 | 1.0 |
| WideResNet Fast Geometric Ens. CIFAR-100Dropout Variational Inf. (FFG) KFAC-Laplace | Deep Ensembles | 5.0 | 10.0 | 49.0 | 95.9 |
| Snapshot Ensembles | 1.0 1.0 2.5 | 4.2 | 16.2 | 20.0 |
| Cyclic SGLD SWA-Gaussian | 1.0 1.9 1.0 3.2 | 3.2 4.2 | 13.6 5.7 | 18.8 6.0 |
|
+
+Table 6: Classification error and negative calibrated log-likelihood for different numbers of samples on ImageNet.
+
+| Model | Deep ensemble equivalent score |
| Method | 1 sample 5 samples 10 samples 50 samples 100 samples |
| VGG16 CIFAR-10 | Deep Ensembles | 1.0 5.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 1.0 2.6 | 4.7 | 21.9 | 33.8 |
| Cyclic SGLD | 1.0 1.7 | 2.6 | 7.4 | 10.5 |
| SWA-Gaussian | 1.0 2.2 | 2.9 | 4.4 | 4.6 |
| 1.3 2.0 | | | |
| Fast Geometric Ens. | | 2.4 | 4.7 | 5.8 |
| Dropout | 1.0 1.0 | 1.0 | 1.1 | 1.1 |
| Variational Inf. (FFG) | 1.0 1.8 | 2.0 | 2.5 | 2.6 1.6 |
| KFAC-Laplace Single model | 1.4 1.6 1.1 | 1.6 1.1 | 1.6 1.1 | 1.1 |
| ResNet110 CIFAR-10 | | 1.1 5.0 | | | 94.6 |
| Deep Ensembles | 1.0 1.0 1.0 | 10.0 2.0 | 48.0 | 32.6 |
| Snapshot Ensembles | 1.0 1.0 | 1.6 | 14.6 | 6.0 |
| Cyclic SGLD | 1.3 | | 4.6 | 2.0 |
| SWA-Gaussian | 1.0 | 1.5 | 1.9 | |
| Fast Geometric Ens. | 1.0 1.4 1.0 | 1.7 | 2.4 | 2.9 |
| Variational Inf. (FFG) | 1.0 | 1.2 | 1.4 | 1.5 |
| KFAC-Laplace | 1.0 1.0 | 1.0 | 1.0 | 1.1 |
| Single model | 1.0 1.0 | 1.0 | 1.0 | 1.0 |
| Deep Ensembles | 1.0 5.0 | 10.0 | 43.7 | 100.0 |
| ResNet164 CIFAR-10 | | 1.0 1.1 | 2.3 | 11.3 | 17.2 |
| Snapshot Ensembles Cyclic SGLD | 1.0 1.0 | 1.8 | 4.8 | 6.7 |
| SWA-Gaussian | 1.0 1.8 | 1.9 | 2.5 | 2.7 |
| Fast Geometric Ens. | 1.0 1.6 | 1.8 | 2.5 | 2.9 |
| Variational Inf. (FFG) | 1.0 1.0 | 1.0 | 1.3 | 1.3 |
| 1.0 | 1.0 1.0 | 1.2 | 1.2 |
| KFAC-Laplace | 1.1 | 1.1 1.1 | 1.1 | 1.1 |
| Single model | 1.0 | 10.0 | 48.0 | |
| Deep Ensembles | 5.0 1.0 | 2.5 | 30.6 | 86.0 100.0 |
| Snapshot Ensembles | 1.3 1.0 1.0 | 1.6 | 4.0 | 4.1 |
| Cyclic SGLD SWA-Gaussian | 1.0 | 1.8 | 2.5 | 3.1 3.3 |
| WideResNet Fast Geometric Ens. CIFAR-10 | 1.0 | 1.2 | 1.7 | 2.7 3.2 |
| Dropout | 1.0 | 1.2 | 1.3 1.4 | 1.4 |
| Variational Inf. (FFG) | 1.0 | 1.0 | 1.0 1.3 | 1.4 |
| KFAC-Laplace | 1.0 | 1.0 | 1.0 1.2 | 1.2 |
| Single model | 1.0 | 1.0 | 1.0 | 1.0 |
| VGG16 CIFAR-100 | | | | | 1.0 |
| Deep Ensembles | 1.0 5.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 1.0 2.9 | 5.4 | 18.8 | 26.8 |
| Cyclic SGLD | 1.0 1.8 1.0 2.3 | 3.3 3.4 | 15.9 | 28.2 |
| SWA-Gaussian | 1.2 1.9 | 2.3 | 5.2 | 5.9 5.8 |
| Fast Geometric Ens. | 1.0 1.0 | 1.0 | 4.5 1.0 | 1.1 |
| Dropout | | 1.8 | 2.1 | 2.2 |
| Variational Inf. (FFG) | 1.0 1.6 | | | 1.3 |
| KFAC-Laplace Single model | 1.0 1.0 | 1.2 1.0 | 1.3 1.0 | 1.0 |
| 1.0 1.0 | 1.0 | | |
| ResNet110 CIFAR-100 | | 5.0 | 10.0 | 50.0 | 99.0 |
| Deep Ensembles Snapshot Ensembles | 1.0 1.3 | 1.9 | 4.7 | 6.2 |
| 1.0 | 2.0 | 7.0 | 12.2 |
| Cyclic SGLD | 1.3 | 2.2 | | 3.6 |
| SWA-Gaussian | 1.0 1.7 1.5 2.6 | 3.0 | 3.3 4.2 | 4.7 |
| Fast Geometric Ens. Variational Inf. (FFG) | 1.0 1.2 | 1.3 | 1.5 | 1.5 |
| KFAC-Laplace | 1.0 1.0 | 1.1 | 1.3 | 1.4 |
| Single model | 1.0 1.0 | 1.0 | 1.0 | 1.0 |
| Deep Ensembles | 1.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 5.0 1.0 | | | |
| ResNet164 CIFAR-100 | | 1.3 | 1.9 | 4.8 | 6.3 |
| Cyclic SGLD SWA-Gaussian | 1.0 1.3 | 2.0 | 6.8 | 13.3 |
| Fast Geometric Ens. | 1.0 2.1 1.6 2.6 | 2.6 3.0 | 3.6 4.2 | 3.8 4.8 |
| Variational Inf. (FFG) | 1.0 1.2 | 1.4 | 1.6 | 1.6 |
| KFAC-Laplace | 1.0 1.1 | 1.3 | 1.5 | 1.5 |
| 1.0 | | | |
| Single model | 1.0 | 1.0 | 1.0 | 1.0 |
| WideResNet Fast Geometric Ens. CIFAR-100Dropout Variational Inf. (FFG) KFAC-Laplace | Deep Ensembles | 5.0 | 10.0 | 49.0 | 95.9 |
| Snapshot Ensembles | 1.0 1.0 2.5 | 4.2 | 16.2 | 20.0 |
| Cyclic SGLD SWA-Gaussian | 1.0 1.9 1.0 3.2 | 3.2 4.2 | 13.6 5.7 | 18.8 6.0 |
|
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+ },
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+ "type": "table",
+ "img_path": "images/dd08cc8919404c8521c907c23aa73ed43d249d5e9805df9552f3337025dcc5a3.jpg",
+ "table_caption": [
+ "Table 6: Classification error and negative calibrated log-likelihood for different numbers of samples on ImageNet. "
+ ],
+ "table_footnote": [],
+ "table_body": "| Model | Deep ensemble equivalent score |
| Method | 1 sample 5 samples 10 samples 50 samples 100 samples |
| VGG16 CIFAR-10 | Deep Ensembles | 1.0 5.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 1.0 2.6 | 4.7 | 21.9 | 33.8 |
| Cyclic SGLD | 1.0 1.7 | 2.6 | 7.4 | 10.5 |
| SWA-Gaussian | 1.0 2.2 | 2.9 | 4.4 | 4.6 |
| 1.3 2.0 | | | |
| Fast Geometric Ens. | | 2.4 | 4.7 | 5.8 |
| Dropout | 1.0 1.0 | 1.0 | 1.1 | 1.1 |
| Variational Inf. (FFG) | 1.0 1.8 | 2.0 | 2.5 | 2.6 1.6 |
| KFAC-Laplace Single model | 1.4 1.6 1.1 | 1.6 1.1 | 1.6 1.1 | 1.1 |
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| Deep Ensembles | 1.0 1.0 1.0 | 10.0 2.0 | 48.0 | 32.6 |
| Snapshot Ensembles | 1.0 1.0 | 1.6 | 14.6 | 6.0 |
| Cyclic SGLD | 1.3 | | 4.6 | 2.0 |
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| Fast Geometric Ens. | 1.0 1.4 1.0 | 1.7 | 2.4 | 2.9 |
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| Fast Geometric Ens. | 1.0 1.6 | 1.8 | 2.5 | 2.9 |
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| Dropout | 1.0 | 1.2 | 1.3 1.4 | 1.4 |
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| VGG16 CIFAR-100 | | | | | 1.0 |
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| SWA-Gaussian | 1.2 1.9 | 2.3 | 5.2 | 5.9 5.8 |
| Fast Geometric Ens. | 1.0 1.0 | 1.0 | 4.5 1.0 | 1.1 |
| Dropout | | 1.8 | 2.1 | 2.2 |
| Variational Inf. (FFG) | 1.0 1.6 | | | 1.3 |
| KFAC-Laplace Single model | 1.0 1.0 | 1.2 1.0 | 1.3 1.0 | 1.0 |
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| Deep Ensembles Snapshot Ensembles | 1.0 1.3 | 1.9 | 4.7 | 6.2 |
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| Cyclic SGLD | 1.3 | 2.2 | | 3.6 |
| SWA-Gaussian | 1.0 1.7 1.5 2.6 | 3.0 | 3.3 4.2 | 4.7 |
| Fast Geometric Ens. Variational Inf. (FFG) | 1.0 1.2 | 1.3 | 1.5 | 1.5 |
| KFAC-Laplace | 1.0 1.0 | 1.1 | 1.3 | 1.4 |
| Single model | 1.0 1.0 | 1.0 | 1.0 | 1.0 |
| Deep Ensembles | 1.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 5.0 1.0 | | | |
| ResNet164 CIFAR-100 | | 1.3 | 1.9 | 4.8 | 6.3 |
| Cyclic SGLD SWA-Gaussian | 1.0 1.3 | 2.0 | 6.8 | 13.3 |
| Fast Geometric Ens. | 1.0 2.1 1.6 2.6 | 2.6 3.0 | 3.6 4.2 | 3.8 4.8 |
| Variational Inf. (FFG) | 1.0 1.2 | 1.4 | 1.6 | 1.6 |
| KFAC-Laplace | 1.0 1.1 | 1.3 | 1.5 | 1.5 |
| 1.0 | | | |
| Single model | 1.0 | 1.0 | 1.0 | 1.0 |
| WideResNet Fast Geometric Ens. CIFAR-100Dropout Variational Inf. (FFG) KFAC-Laplace | Deep Ensembles | 5.0 | 10.0 | 49.0 | 95.9 |
| Snapshot Ensembles | 1.0 1.0 2.5 | 4.2 | 16.2 | 20.0 |
| Cyclic SGLD SWA-Gaussian | 1.0 1.9 1.0 3.2 | 3.2 4.2 | 13.6 5.7 | 18.8 6.0 |
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| Method | 1 sample 5 samples 10 samples 50 samples 100 samples |
| VGG16 CIFAR-10 | Deep Ensembles | 1.0 5.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 1.0 2.6 | 4.7 | 21.9 | 33.8 |
| Cyclic SGLD | 1.0 1.7 | 2.6 | 7.4 | 10.5 |
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| ResNet110 CIFAR-10 | | 1.1 5.0 | | | 94.6 |
| Deep Ensembles | 1.0 1.0 1.0 | 10.0 2.0 | 48.0 | 32.6 |
| Snapshot Ensembles | 1.0 1.0 | 1.6 | 14.6 | 6.0 |
| Cyclic SGLD | 1.3 | | 4.6 | 2.0 |
| SWA-Gaussian | 1.0 | 1.5 | 1.9 | |
| Fast Geometric Ens. | 1.0 1.4 1.0 | 1.7 | 2.4 | 2.9 |
| Variational Inf. (FFG) | 1.0 | 1.2 | 1.4 | 1.5 |
| KFAC-Laplace | 1.0 1.0 | 1.0 | 1.0 | 1.1 |
| Single model | 1.0 1.0 | 1.0 | 1.0 | 1.0 |
| Deep Ensembles | 1.0 5.0 | 10.0 | 43.7 | 100.0 |
| ResNet164 CIFAR-10 | | 1.0 1.1 | 2.3 | 11.3 | 17.2 |
| Snapshot Ensembles Cyclic SGLD | 1.0 1.0 | 1.8 | 4.8 | 6.7 |
| SWA-Gaussian | 1.0 1.8 | 1.9 | 2.5 | 2.7 |
| Fast Geometric Ens. | 1.0 1.6 | 1.8 | 2.5 | 2.9 |
| Variational Inf. (FFG) | 1.0 1.0 | 1.0 | 1.3 | 1.3 |
| 1.0 | 1.0 1.0 | 1.2 | 1.2 |
| KFAC-Laplace | 1.1 | 1.1 1.1 | 1.1 | 1.1 |
| Single model | 1.0 | 10.0 | 48.0 | |
| Deep Ensembles | 5.0 1.0 | 2.5 | 30.6 | 86.0 100.0 |
| Snapshot Ensembles | 1.3 1.0 1.0 | 1.6 | 4.0 | 4.1 |
| Cyclic SGLD SWA-Gaussian | 1.0 | 1.8 | 2.5 | 3.1 3.3 |
| WideResNet Fast Geometric Ens. CIFAR-10 | 1.0 | 1.2 | 1.7 | 2.7 3.2 |
| Dropout | 1.0 | 1.2 | 1.3 1.4 | 1.4 |
| Variational Inf. (FFG) | 1.0 | 1.0 | 1.0 1.3 | 1.4 |
| KFAC-Laplace | 1.0 | 1.0 | 1.0 1.2 | 1.2 |
| Single model | 1.0 | 1.0 | 1.0 | 1.0 |
| VGG16 CIFAR-100 | | | | | 1.0 |
| Deep Ensembles | 1.0 5.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 1.0 2.9 | 5.4 | 18.8 | 26.8 |
| Cyclic SGLD | 1.0 1.8 1.0 2.3 | 3.3 3.4 | 15.9 | 28.2 |
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| Fast Geometric Ens. | 1.0 1.0 | 1.0 | 4.5 1.0 | 1.1 |
| Dropout | | 1.8 | 2.1 | 2.2 |
| Variational Inf. (FFG) | 1.0 1.6 | | | 1.3 |
| KFAC-Laplace Single model | 1.0 1.0 | 1.2 1.0 | 1.3 1.0 | 1.0 |
| 1.0 1.0 | 1.0 | | |
| ResNet110 CIFAR-100 | | 5.0 | 10.0 | 50.0 | 99.0 |
| Deep Ensembles Snapshot Ensembles | 1.0 1.3 | 1.9 | 4.7 | 6.2 |
| 1.0 | 2.0 | 7.0 | 12.2 |
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| SWA-Gaussian | 1.0 1.7 1.5 2.6 | 3.0 | 3.3 4.2 | 4.7 |
| Fast Geometric Ens. Variational Inf. (FFG) | 1.0 1.2 | 1.3 | 1.5 | 1.5 |
| KFAC-Laplace | 1.0 1.0 | 1.1 | 1.3 | 1.4 |
| Single model | 1.0 1.0 | 1.0 | 1.0 | 1.0 |
| Deep Ensembles | 1.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 5.0 1.0 | | | |
| ResNet164 CIFAR-100 | | 1.3 | 1.9 | 4.8 | 6.3 |
| Cyclic SGLD SWA-Gaussian | 1.0 1.3 | 2.0 | 6.8 | 13.3 |
| Fast Geometric Ens. | 1.0 2.1 1.6 2.6 | 2.6 3.0 | 3.6 4.2 | 3.8 4.8 |
| Variational Inf. (FFG) | 1.0 1.2 | 1.4 | 1.6 | 1.6 |
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| Cyclic SGLD SWA-Gaussian | 1.0 1.9 1.0 3.2 | 3.2 4.2 | 13.6 5.7 | 18.8 6.0 |
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| Method | 1 sample 5 samples 10 samples 50 samples 100 samples |
| VGG16 CIFAR-10 | Deep Ensembles | 1.0 5.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 1.0 2.6 | 4.7 | 21.9 | 33.8 |
| Cyclic SGLD | 1.0 1.7 | 2.6 | 7.4 | 10.5 |
| SWA-Gaussian | 1.0 2.2 | 2.9 | 4.4 | 4.6 |
| 1.3 2.0 | | | |
| Fast Geometric Ens. | | 2.4 | 4.7 | 5.8 |
| Dropout | 1.0 1.0 | 1.0 | 1.1 | 1.1 |
| Variational Inf. (FFG) | 1.0 1.8 | 2.0 | 2.5 | 2.6 1.6 |
| KFAC-Laplace Single model | 1.4 1.6 1.1 | 1.6 1.1 | 1.6 1.1 | 1.1 |
| ResNet110 CIFAR-10 | | 1.1 5.0 | | | 94.6 |
| Deep Ensembles | 1.0 1.0 1.0 | 10.0 2.0 | 48.0 | 32.6 |
| Snapshot Ensembles | 1.0 1.0 | 1.6 | 14.6 | 6.0 |
| Cyclic SGLD | 1.3 | | 4.6 | 2.0 |
| SWA-Gaussian | 1.0 | 1.5 | 1.9 | |
| Fast Geometric Ens. | 1.0 1.4 1.0 | 1.7 | 2.4 | 2.9 |
| Variational Inf. (FFG) | 1.0 | 1.2 | 1.4 | 1.5 |
| KFAC-Laplace | 1.0 1.0 | 1.0 | 1.0 | 1.1 |
| Single model | 1.0 1.0 | 1.0 | 1.0 | 1.0 |
| Deep Ensembles | 1.0 5.0 | 10.0 | 43.7 | 100.0 |
| ResNet164 CIFAR-10 | | 1.0 1.1 | 2.3 | 11.3 | 17.2 |
| Snapshot Ensembles Cyclic SGLD | 1.0 1.0 | 1.8 | 4.8 | 6.7 |
| SWA-Gaussian | 1.0 1.8 | 1.9 | 2.5 | 2.7 |
| Fast Geometric Ens. | 1.0 1.6 | 1.8 | 2.5 | 2.9 |
| Variational Inf. (FFG) | 1.0 1.0 | 1.0 | 1.3 | 1.3 |
| 1.0 | 1.0 1.0 | 1.2 | 1.2 |
| KFAC-Laplace | 1.1 | 1.1 1.1 | 1.1 | 1.1 |
| Single model | 1.0 | 10.0 | 48.0 | |
| Deep Ensembles | 5.0 1.0 | 2.5 | 30.6 | 86.0 100.0 |
| Snapshot Ensembles | 1.3 1.0 1.0 | 1.6 | 4.0 | 4.1 |
| Cyclic SGLD SWA-Gaussian | 1.0 | 1.8 | 2.5 | 3.1 3.3 |
| WideResNet Fast Geometric Ens. CIFAR-10 | 1.0 | 1.2 | 1.7 | 2.7 3.2 |
| Dropout | 1.0 | 1.2 | 1.3 1.4 | 1.4 |
| Variational Inf. (FFG) | 1.0 | 1.0 | 1.0 1.3 | 1.4 |
| KFAC-Laplace | 1.0 | 1.0 | 1.0 1.2 | 1.2 |
| Single model | 1.0 | 1.0 | 1.0 | 1.0 |
| VGG16 CIFAR-100 | | | | | 1.0 |
| Deep Ensembles | 1.0 5.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 1.0 2.9 | 5.4 | 18.8 | 26.8 |
| Cyclic SGLD | 1.0 1.8 1.0 2.3 | 3.3 3.4 | 15.9 | 28.2 |
| SWA-Gaussian | 1.2 1.9 | 2.3 | 5.2 | 5.9 5.8 |
| Fast Geometric Ens. | 1.0 1.0 | 1.0 | 4.5 1.0 | 1.1 |
| Dropout | | 1.8 | 2.1 | 2.2 |
| Variational Inf. (FFG) | 1.0 1.6 | | | 1.3 |
| KFAC-Laplace Single model | 1.0 1.0 | 1.2 1.0 | 1.3 1.0 | 1.0 |
| 1.0 1.0 | 1.0 | | |
| ResNet110 CIFAR-100 | | 5.0 | 10.0 | 50.0 | 99.0 |
| Deep Ensembles Snapshot Ensembles | 1.0 1.3 | 1.9 | 4.7 | 6.2 |
| 1.0 | 2.0 | 7.0 | 12.2 |
| Cyclic SGLD | 1.3 | 2.2 | | 3.6 |
| SWA-Gaussian | 1.0 1.7 1.5 2.6 | 3.0 | 3.3 4.2 | 4.7 |
| Fast Geometric Ens. Variational Inf. (FFG) | 1.0 1.2 | 1.3 | 1.5 | 1.5 |
| KFAC-Laplace | 1.0 1.0 | 1.1 | 1.3 | 1.4 |
| Single model | 1.0 1.0 | 1.0 | 1.0 | 1.0 |
| Deep Ensembles | 1.0 | 10.0 | 50.0 | 100.0 |
| Snapshot Ensembles | 5.0 1.0 | | | |
| ResNet164 CIFAR-100 | | 1.3 | 1.9 | 4.8 | 6.3 |
| Cyclic SGLD SWA-Gaussian | 1.0 1.3 | 2.0 | 6.8 | 13.3 |
| Fast Geometric Ens. | 1.0 2.1 1.6 2.6 | 2.6 3.0 | 3.6 4.2 | 3.8 4.8 |
| Variational Inf. (FFG) | 1.0 1.2 | 1.4 | 1.6 | 1.6 |
| KFAC-Laplace | 1.0 1.1 | 1.3 | 1.5 | 1.5 |
| 1.0 | | | |
| Single model | 1.0 | 1.0 | 1.0 | 1.0 |
| WideResNet Fast Geometric Ens. CIFAR-100Dropout Variational Inf. (FFG) KFAC-Laplace | Deep Ensembles | 5.0 | 10.0 | 49.0 | 95.9 |
| Snapshot Ensembles | 1.0 1.0 2.5 | 4.2 | 16.2 | 20.0 |
| Cyclic SGLD SWA-Gaussian | 1.0 1.9 1.0 3.2 | 3.2 4.2 | 13.6 5.7 | 18.8 6.0 |
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| while training not done do |
| for all workers in parallel do |
| Wait until a message arrives |
| msg ←the highest priority message from the input queue |
| op ← sink node of the message msg |
| if msg is forward type then Compute forward: out_msgs ← op(msg) |
| Enqueue the resulting message(s) into the queue(s) of the workers hosting the child nodes. |
| else if msg is backward type then |
| Compute backward: out_msgs ← op*(msg,fwd_msg) |
| Enqueue the resulting message(s) into the queue(s) of the workers hosting the parent nodes. |
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+
+Tree-structured neural networks are powerful models used for parsing of natural language and images, semantic representation, and sentiment analysis (Socher et al., 2011; Bowman et al., 2016; Socher et al., 2013; Tai et al., 2015). They require evaluation of (potentially multiple) trees with shared parameters but different topology for each instance. Each tree structure is instance-specific and batching requires nontrivial planning (Looks et al., 2017). A simple form of tree neural network performs a bottom up traversal of the instance, starting from an embedding of the leaves. At each level the values from the child nodes are concatenated and sent through a specialized unit (e.g. LSTM). The result is then propagated further up the tree. Backpropagation over the tree structure is known as backpropagation through structure (Goller and Kuchler, 1996).
+
+Graph neural networks (Scarselli et al., 2009; Li et al., 2015; Gilmer et al., 2017) combine both the temporal recurrence and recurrence over the structure. GNNs can be seen as performing aggregation/distribution operations over a general graph structure with shared parameters.
+
+Apart from the models above, there exist many recently proposed models with flexible control flow (e.g. hierarchical memory networks (Chandar et al., 2016), neural programmer interpreters (Reed and De Freitas, 2015), adaptive computation networks (Graves, 2016; Figurnov et al., 2016), and sparsely-gated mixture of experts (Shazeer et al., 2017), to which our framework can be applied.
+
+# 3 ASYNCHRONOUS MODEL-PARALLEL TRAINING
+
+In AMP training, each node of a computation graph (including control flow nodes – see next section) is associated with a worker, which is an abstraction of a compute device. Neural network training and inference is carried out by message passing among workers following algorithm 1. All workers run in parallel without synchronization. Each message contains a payload (float or int tensor), as well as a state that includes the IDs of the source and sink nodes and a label indicating the type of message (forward, backward, or update). The state is used to keep track of algorithm and control flow information. For example, in a variable-length RNN the state also contains the instance identifier, the current position in the sequence, and the total sequence length for the instance. More generally, if the neural model use (possibly nested) loops, then the state for the messages that arrive to and are produced from nodes that logically belong in loop bodies will contain sets of loop counters that together with the instance id uniquely identify the messages throughout the course of the computation.
+
+When a worker receives a message labeled as forward (or backward) then it performs the operation of the node indicated by the sink node ID on the supplied payload. This produces one or more outgoing messages that are then enqueued into the queues of the workers hosting the next sink nodes in the computation graph. The final loss layer initiates backward propagation. If the message type is update, the worker will carry out weight updates on the sink node using gradients accumulated in the appropriate slot in the worker’s local memory. Since both the operation weights and weight gradients can be stored locally on the worker then workers only need to communicate activations and activation gradients, which are typically an order of magnitude smaller than the weights. The update message is typically sent from the sink node to itself as part of the backward process but it can also be sent from a controller node to simulate synchronous pipelined training.
+
+There are two important details that are not fully spelled out in Algorithm 1. First, since the messages arrive asynchronously (and possibly out of order), any operation that has more than one parent nodes need to store the payload into its local cache until all the parents send the corresponding payloads. Thus output message(s) can only be produced when all the payloads become available. The cache needs to be able to distinguish payloads received from different parents and payloads with different instance ID, and different counters (all encoded in the message states). The same is true in the backward pass for a node with multiple child nodes. Second, ${ \tt o p } ^ { * }$ denotes the adjoint operation of op and takes the backward message msg and potentially the forward message fwd msg stored in the cache. For a nonlinear activation node (e.g., ReLU), the node will not change the state of the message in the forward pass. Thus in the backward pass, the adjoint operation will just multiply the partial derivative of the activation function to the payload of the received backward message keeping the state unchanged. By contrast, an operation that only changes the state of the message in the forward pass (e.g., increment the loop counter) will reverse the change in the backward pass leaving the payload unchanged.
+
+In the experiments we vary two hyper parameters to control the effect of asynchrony:
+
+min update interval: determines the minimum number of gradients that a parameterized operation needs to accumulate before it can update its parameters (using the update message). The staleness of a gradient can be measured by the number of updates between the forward and backward computation that produces the gradient. Small min update interval may increase gradient staleness. On the other hand, large min update interval can reduce the variance of the gradient but can result in very infrequent updates and also slow down convergence.
+
+max active keys: controls the maximum number of active instances that are in-flight at any point in time. By setting max active key $\mathbf { \boldsymbol { s } } = 1$ we restrict to single-instance processing, typically equivalent to synchronous training. More in-flight messages generally increase hardware utilization, but may also increase gradient staleness. Section 6 demonstrates the effects of these parameters in a multi-core CPU runtime.
+
+# 3.1 WORKER AFFINITY
+
+In our asynchronous execution model, the optimal assignment of $N$ neural network computation graph nodes to $W$ workers (referred to as affinitization) is in general a non-trivial scheduling problem. We investigated several heuristics for assigning affinities (for example $\mathbf { k }$ -hop coloring (Ichrak et al., 2012) to ensure subsequent heavy operations were assigned different workers). However, we find that the following procedure achieves high performance in practice in our multi-core CPU implementation, and is adopted throughout our experiments for simplicity. We first partition the nodes into $H$ ’heavy’ operations (namely matrix multiplies) and $( N - H )$ ’light’ operations, and then balance the heavy nodes across the workers by affinitizing the $h ^ { \mathrm { t h } }$ with the $( h \mod W ) ^ { \mathrm { t h } }$ worker. Finally, the light operations are affinitized randomly among the rest of the workers. Note that in scenarios where communication is over a physical network affinitization will become more critical for high performance.
+
+
+Figure 3: Distribution based on dynamic information
+
+# 4 A STATIC INTERMEDIATE REPRESENTATION FOR DYNAMIC CONTROL FLOW
+
+Overview Computation graphs are expressed using a static intermediate representation (IR) that can be a compilation target for high-level libraries (e.g. TensorFlow or our own Python and $\mathrm { C } { + + }$ frontends), and can itself admit multiple backends (e.g. the multi-core CPU runtime in this paper, or a network of accelerators). Static means that the IR graph is instance-independent. Nevertheless, it can execute dynamic and instance-dependent control flow decisions, in a forward and backward manner, by storing instance- and iteration-dependent information as the computation evolves. Each IR node comes with a forward and a backward semantics. A model is specified by (i) an IR graph, and (ii) a specialized controller loop that pumps instances and other data (e.g. initial hidden states or labels), and is responsible for throttling asynchrony. In the rest of this section we discuss the most important IR nodes along with their operational semantics, and show how they are used in the example models.
+
+Payload transformations Parameterized payload transform (PPT) nodes can be used to encode, for instance, fully connected layers. They apply a transform in the forward pass, but also record the activation in order to use it to compute gradients in the backward pass. An activation is recorded by keying on the state of the message, allowing thus to process forward and backwards messages completely asynchronously, and – in the extreme case – out of order. A PPT node requires specification of the forward and the backward transformation. It may decide to independently apply accumulated gradients to update its parameters. For transformations that do not involve parameters (e.g. ReLUs) our IR offers a simpler non-parameterized payload transform.
+
+Loops, state, and control flow A condition node (Cond $f$ ) is parameterized by a function $f$ that queries the state (but not the payload) of the incoming message and, based on the response, routes the input to one of the successor nodes. A join node (Phi) propagates the messages it receives from each of its ancestor nodes but records the origin (using the state of the message as the key) so that in the backward pass it can backpropagate them to the correct origin. An invertible state update node $( \mathtt { I s u } \ f \ f ^ { - 1 } )$ is parameterized by two functions $f$ and $f ^ { - 1 }$ that operate on the state of a message, and satisfy $\bar { f } ^ { - 1 } ( f ( \bar { x } ) ) = x$ . Typically these are loop counter update functions.
+
+Figure 2 shows how to encode an RNN. The loop at the heart of the RNN is implemented with Cond, Phi and Isu nodes. The controller pumps sequences in a lookup table (just another PPT layer), and our Ungroup node (to be described in the next section) generates a stream of tensors each corresponding to a single token, tagged with the current time-step (loop counter). For each forward message, the Isu node increments the time-step, and the conditional node tests whether the end of the sequence has been reached. Depending on the answer it either propagates the hidden state back to Phi, or pushes the hidden state to the final linear and loss layers. In backward mode, messages pass through the Isu (which decrements the time-step), and reach the Phi node. The Phi node will (based on information from the forward phase) either back-propagate to the Cond node, or to the controller to terminate. Hence the loop is executed in both the forward and backward direction.
+
+
+Figure 4: IR graphs for (a) Gated Graph Neural Network and (b) RNN-with-replicas. The RNNCell in (a) denotes a recurrent structure (e.g. GRU, LSTM), the details of which we omit. Cond modId in (b) implements a round-robin branching to different replicas by computing (instance ID mod 3).
+
+Aggregation and disaggregation Our IR offers several constructs for aggregation and disagreggation; for example RNN requires us to concatenate (Concat) hidden states and embeddings with matching timesteps and instance ids (states). We offer a construct for broadcasting (Bcast) a message to multiple successors. We offer an ungrouping construct (Ungroup) that ungroups a matrix and emits all resulting messages tagged with an extra user-provided increasing loop counter. This allows us to insert the stream of token embeddings in the middle of the RNN loop in Figure 2. In backward mode Ungroup groups back all the incoming gradients. A simpler variant of Ungroup is Replicate, which replicates a message with an extra loop counter in the state. In backwards mode Replicate sums up all incoming gradients that correspond to the state without the extra loop counter.
+
+Figure 4(a) describes a GNN that combines aggregation on the structure of a graph instance with an outer loop. The controller pumps data that contain the feature embeddings of all nodes of an input graph. In addition it pumps in a map specifiying graph topology. The Distribute node uses that information along with the graph itself to create sub-matrices (here each corresponding to edge types) and pass them through linear (fully connected) layers. The Collect node collects and re-groups the results based on graph topology. These nodes correspond to a form of dynamic partition and merging. Schematically the Distribute behaviour is given in Figure 3. In backward mode, based on the control information received during the forward pass (gray line) re-groups the gradients and sums together those that correspond to the same index. The Collect operator is essentially symmetric.
+
+# 5 DATA PARALLELISM IN MODEL PARALLELISM
+
+Pipelined model parallelism can often be augmented with forms of data parallelism. Consider the RNN in Fig. 2. The only heavy operation (Linear-1) in the body of the loop will act as a bottleneck for computation. One solution is to split the linear layer into smaller tiles and compute them in parallel. This is expressible in our IR but the linear operation needs to be large enough to benefit from tiling in this way. Another approach is to replicate the linear layer in full. This requires only minimal new machinery – we can replicate the linear layer and place the replicas inside Cond and Phi nodes as in Figure 4(b). Different instances or messages from the same instance but with different position in the sequence can be processed in an (pipeline-)parallel fashion by being sent to one of the replicas chosen by a random or deterministic function of the message state. To enable parameters to be shared among the replicas, we use infrequent end-of-epoch replica synchronization (averaging) that incurs negligible communication cost. We also tried more elaborate message-passing protocols for group synchronization, but found that infrequent global synchronization was sufficient for fast convergence.
+
+# 6 EXPERIMENTS
+
+We evaluate AMPNet using the dynamic models introduced in Section 2. For completeness, we additionally consider a multi-layer perceptron (MLP) as an example of a static, batched network
+
+that AMPNet is not specifically designed to tackle. Brief details of the models and data sets in the experiments are presented below, and further details are given in Appendix B.
+
+| Algorithm 1 Asynchronous Model-Parallel training |
| while training not done do |
| for all workers in parallel do |
| Wait until a message arrives |
| msg ←the highest priority message from the input queue |
| op ← sink node of the message msg |
| if msg is forward type then Compute forward: out_msgs ← op(msg) |
| Enqueue the resulting message(s) into the queue(s) of the workers hosting the child nodes. |
| else if msg is backward type then |
| Compute backward: out_msgs ← op*(msg,fwd_msg) |
| Enqueue the resulting message(s) into the queue(s) of the workers hosting the parent nodes. |
| else if msg isupdate type then |
| Perform parameter updates on op |
| end if |
| end for |
| end while |
",
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+ "type": "text",
+ "text": "Tree-structured neural networks are powerful models used for parsing of natural language and images, semantic representation, and sentiment analysis (Socher et al., 2011; Bowman et al., 2016; Socher et al., 2013; Tai et al., 2015). They require evaluation of (potentially multiple) trees with shared parameters but different topology for each instance. Each tree structure is instance-specific and batching requires nontrivial planning (Looks et al., 2017). A simple form of tree neural network performs a bottom up traversal of the instance, starting from an embedding of the leaves. At each level the values from the child nodes are concatenated and sent through a specialized unit (e.g. LSTM). The result is then propagated further up the tree. Backpropagation over the tree structure is known as backpropagation through structure (Goller and Kuchler, 1996). ",
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+ },
+ {
+ "type": "text",
+ "text": "Graph neural networks (Scarselli et al., 2009; Li et al., 2015; Gilmer et al., 2017) combine both the temporal recurrence and recurrence over the structure. GNNs can be seen as performing aggregation/distribution operations over a general graph structure with shared parameters. ",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Apart from the models above, there exist many recently proposed models with flexible control flow (e.g. hierarchical memory networks (Chandar et al., 2016), neural programmer interpreters (Reed and De Freitas, 2015), adaptive computation networks (Graves, 2016; Figurnov et al., 2016), and sparsely-gated mixture of experts (Shazeer et al., 2017), to which our framework can be applied. ",
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+ {
+ "type": "text",
+ "text": "3 ASYNCHRONOUS MODEL-PARALLEL TRAINING ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 2
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+ {
+ "type": "text",
+ "text": "In AMP training, each node of a computation graph (including control flow nodes – see next section) is associated with a worker, which is an abstraction of a compute device. Neural network training and inference is carried out by message passing among workers following algorithm 1. All workers run in parallel without synchronization. Each message contains a payload (float or int tensor), as well as a state that includes the IDs of the source and sink nodes and a label indicating the type of message (forward, backward, or update). The state is used to keep track of algorithm and control flow information. For example, in a variable-length RNN the state also contains the instance identifier, the current position in the sequence, and the total sequence length for the instance. More generally, if the neural model use (possibly nested) loops, then the state for the messages that arrive to and are produced from nodes that logically belong in loop bodies will contain sets of loop counters that together with the instance id uniquely identify the messages throughout the course of the computation. ",
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+ "text": "When a worker receives a message labeled as forward (or backward) then it performs the operation of the node indicated by the sink node ID on the supplied payload. This produces one or more outgoing messages that are then enqueued into the queues of the workers hosting the next sink nodes in the computation graph. The final loss layer initiates backward propagation. If the message type is update, the worker will carry out weight updates on the sink node using gradients accumulated in the appropriate slot in the worker’s local memory. Since both the operation weights and weight gradients can be stored locally on the worker then workers only need to communicate activations and activation gradients, which are typically an order of magnitude smaller than the weights. The update message is typically sent from the sink node to itself as part of the backward process but it can also be sent from a controller node to simulate synchronous pipelined training. ",
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+ {
+ "type": "text",
+ "text": "There are two important details that are not fully spelled out in Algorithm 1. First, since the messages arrive asynchronously (and possibly out of order), any operation that has more than one parent nodes need to store the payload into its local cache until all the parents send the corresponding payloads. Thus output message(s) can only be produced when all the payloads become available. The cache needs to be able to distinguish payloads received from different parents and payloads with different instance ID, and different counters (all encoded in the message states). The same is true in the backward pass for a node with multiple child nodes. Second, ${ \\tt o p } ^ { * }$ denotes the adjoint operation of op and takes the backward message msg and potentially the forward message fwd msg stored in the cache. For a nonlinear activation node (e.g., ReLU), the node will not change the state of the message in the forward pass. Thus in the backward pass, the adjoint operation will just multiply the partial derivative of the activation function to the payload of the received backward message keeping the state unchanged. By contrast, an operation that only changes the state of the message in the forward pass (e.g., increment the loop counter) will reverse the change in the backward pass leaving the payload unchanged. ",
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+ "type": "text",
+ "text": "In the experiments we vary two hyper parameters to control the effect of asynchrony: ",
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+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "min update interval: determines the minimum number of gradients that a parameterized operation needs to accumulate before it can update its parameters (using the update message). The staleness of a gradient can be measured by the number of updates between the forward and backward computation that produces the gradient. Small min update interval may increase gradient staleness. On the other hand, large min update interval can reduce the variance of the gradient but can result in very infrequent updates and also slow down convergence. ",
+ "bbox": [
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+ "type": "text",
+ "text": "max active keys: controls the maximum number of active instances that are in-flight at any point in time. By setting max active key $\\mathbf { \\boldsymbol { s } } = 1$ we restrict to single-instance processing, typically equivalent to synchronous training. More in-flight messages generally increase hardware utilization, but may also increase gradient staleness. Section 6 demonstrates the effects of these parameters in a multi-core CPU runtime. ",
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+ },
+ {
+ "type": "text",
+ "text": "3.1 WORKER AFFINITY ",
+ "text_level": 1,
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+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "In our asynchronous execution model, the optimal assignment of $N$ neural network computation graph nodes to $W$ workers (referred to as affinitization) is in general a non-trivial scheduling problem. We investigated several heuristics for assigning affinities (for example $\\mathbf { k }$ -hop coloring (Ichrak et al., 2012) to ensure subsequent heavy operations were assigned different workers). However, we find that the following procedure achieves high performance in practice in our multi-core CPU implementation, and is adopted throughout our experiments for simplicity. We first partition the nodes into $H$ ’heavy’ operations (namely matrix multiplies) and $( N - H )$ ’light’ operations, and then balance the heavy nodes across the workers by affinitizing the $h ^ { \\mathrm { t h } }$ with the $( h \\mod W ) ^ { \\mathrm { t h } }$ worker. Finally, the light operations are affinitized randomly among the rest of the workers. Note that in scenarios where communication is over a physical network affinitization will become more critical for high performance. ",
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+ },
+ {
+ "type": "image",
+ "img_path": "images/05546d60c49a3bb63bc9a2abaa8d81d264c7b6f479acb7c5ca1109ff70b82680.jpg",
+ "image_caption": [
+ "Figure 3: Distribution based on dynamic information "
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ ],
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+ {
+ "type": "text",
+ "text": "4 A STATIC INTERMEDIATE REPRESENTATION FOR DYNAMIC CONTROL FLOW ",
+ "text_level": 1,
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "Overview Computation graphs are expressed using a static intermediate representation (IR) that can be a compilation target for high-level libraries (e.g. TensorFlow or our own Python and $\\mathrm { C } { + + }$ frontends), and can itself admit multiple backends (e.g. the multi-core CPU runtime in this paper, or a network of accelerators). Static means that the IR graph is instance-independent. Nevertheless, it can execute dynamic and instance-dependent control flow decisions, in a forward and backward manner, by storing instance- and iteration-dependent information as the computation evolves. Each IR node comes with a forward and a backward semantics. A model is specified by (i) an IR graph, and (ii) a specialized controller loop that pumps instances and other data (e.g. initial hidden states or labels), and is responsible for throttling asynchrony. In the rest of this section we discuss the most important IR nodes along with their operational semantics, and show how they are used in the example models. ",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Payload transformations Parameterized payload transform (PPT) nodes can be used to encode, for instance, fully connected layers. They apply a transform in the forward pass, but also record the activation in order to use it to compute gradients in the backward pass. An activation is recorded by keying on the state of the message, allowing thus to process forward and backwards messages completely asynchronously, and – in the extreme case – out of order. A PPT node requires specification of the forward and the backward transformation. It may decide to independently apply accumulated gradients to update its parameters. For transformations that do not involve parameters (e.g. ReLUs) our IR offers a simpler non-parameterized payload transform. ",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Loops, state, and control flow A condition node (Cond $f$ ) is parameterized by a function $f$ that queries the state (but not the payload) of the incoming message and, based on the response, routes the input to one of the successor nodes. A join node (Phi) propagates the messages it receives from each of its ancestor nodes but records the origin (using the state of the message as the key) so that in the backward pass it can backpropagate them to the correct origin. An invertible state update node $( \\mathtt { I s u } \\ f \\ f ^ { - 1 } )$ is parameterized by two functions $f$ and $f ^ { - 1 }$ that operate on the state of a message, and satisfy $\\bar { f } ^ { - 1 } ( f ( \\bar { x } ) ) = x$ . Typically these are loop counter update functions. ",
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+ },
+ {
+ "type": "text",
+ "text": "Figure 2 shows how to encode an RNN. The loop at the heart of the RNN is implemented with Cond, Phi and Isu nodes. The controller pumps sequences in a lookup table (just another PPT layer), and our Ungroup node (to be described in the next section) generates a stream of tensors each corresponding to a single token, tagged with the current time-step (loop counter). For each forward message, the Isu node increments the time-step, and the conditional node tests whether the end of the sequence has been reached. Depending on the answer it either propagates the hidden state back to Phi, or pushes the hidden state to the final linear and loss layers. In backward mode, messages pass through the Isu (which decrements the time-step), and reach the Phi node. The Phi node will (based on information from the forward phase) either back-propagate to the Cond node, or to the controller to terminate. Hence the loop is executed in both the forward and backward direction. ",
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+ "img_path": "images/09f7aa446951696f32af43edccbcf5dc7b775fea5217c39a8ac79143df2a7033.jpg",
+ "image_caption": [
+ "Figure 4: IR graphs for (a) Gated Graph Neural Network and (b) RNN-with-replicas. The RNNCell in (a) denotes a recurrent structure (e.g. GRU, LSTM), the details of which we omit. Cond modId in (b) implements a round-robin branching to different replicas by computing (instance ID mod 3). "
+ ],
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+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "Aggregation and disaggregation Our IR offers several constructs for aggregation and disagreggation; for example RNN requires us to concatenate (Concat) hidden states and embeddings with matching timesteps and instance ids (states). We offer a construct for broadcasting (Bcast) a message to multiple successors. We offer an ungrouping construct (Ungroup) that ungroups a matrix and emits all resulting messages tagged with an extra user-provided increasing loop counter. This allows us to insert the stream of token embeddings in the middle of the RNN loop in Figure 2. In backward mode Ungroup groups back all the incoming gradients. A simpler variant of Ungroup is Replicate, which replicates a message with an extra loop counter in the state. In backwards mode Replicate sums up all incoming gradients that correspond to the state without the extra loop counter. ",
+ "bbox": [
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+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "Figure 4(a) describes a GNN that combines aggregation on the structure of a graph instance with an outer loop. The controller pumps data that contain the feature embeddings of all nodes of an input graph. In addition it pumps in a map specifiying graph topology. The Distribute node uses that information along with the graph itself to create sub-matrices (here each corresponding to edge types) and pass them through linear (fully connected) layers. The Collect node collects and re-groups the results based on graph topology. These nodes correspond to a form of dynamic partition and merging. Schematically the Distribute behaviour is given in Figure 3. In backward mode, based on the control information received during the forward pass (gray line) re-groups the gradients and sums together those that correspond to the same index. The Collect operator is essentially symmetric. ",
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+ },
+ {
+ "type": "text",
+ "text": "5 DATA PARALLELISM IN MODEL PARALLELISM ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "Pipelined model parallelism can often be augmented with forms of data parallelism. Consider the RNN in Fig. 2. The only heavy operation (Linear-1) in the body of the loop will act as a bottleneck for computation. One solution is to split the linear layer into smaller tiles and compute them in parallel. This is expressible in our IR but the linear operation needs to be large enough to benefit from tiling in this way. Another approach is to replicate the linear layer in full. This requires only minimal new machinery – we can replicate the linear layer and place the replicas inside Cond and Phi nodes as in Figure 4(b). Different instances or messages from the same instance but with different position in the sequence can be processed in an (pipeline-)parallel fashion by being sent to one of the replicas chosen by a random or deterministic function of the message state. To enable parameters to be shared among the replicas, we use infrequent end-of-epoch replica synchronization (averaging) that incurs negligible communication cost. We also tried more elaborate message-passing protocols for group synchronization, but found that infrequent global synchronization was sufficient for fast convergence. ",
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+ },
+ {
+ "type": "text",
+ "text": "6 EXPERIMENTS ",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "We evaluate AMPNet using the dynamic models introduced in Section 2. For completeness, we additionally consider a multi-layer perceptron (MLP) as an example of a static, batched network ",
+ "bbox": [
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+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "table",
+ "img_path": "images/61b96a9172c549ef7fac0f812fcfd89bc724c88ddab6413e94a05f94193bdbdb.jpg",
+ "table_caption": [
+ "that AMPNet is not specifically designed to tackle. Brief details of the models and data sets in the experiments are presented below, and further details are given in Appendix B. "
+ ],
+ "table_footnote": [],
+ "table_body": "| Algorithm 1 Asynchronous Model-Parallel training |
| while training not done do |
| for all workers in parallel do |
| Wait until a message arrives |
| msg ←the highest priority message from the input queue |
| op ← sink node of the message msg |
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+ "html": "| Constraint on δ|Bound on Eq. 22 | Constraint on δ|Bound on Eq. 22 | Sampling from X ~ μ(0) |
| |δl ≤ε | E[f(exp(-eoll-X))]X~μ1(0） | Xi ~ Lap(0,1) iid |
| 1l2≤ e | E(o[f(exp(IX-eoll-X2))]X~μ2（0） | X=RuR~Γ(d,1)u ~U(aB2) |
| ≤e | E[f(exp(X-e1|-X)]X~μ(0） | X=RuR~ Γ(d+1,1)u EU(B) |
| |/ lnuc ≤ e | [f(exp(||U[[s]]vT-e[[eo]|lnuc- |slli)]s~μ1(0)U,v~u(O) | X=U[s]]VTs ~ μ1,U,V ~U(O) |
| ≤e | E [f(exp(|IU[[s]]vT-∈[e]|l, - |sll)]s~μ∞(x）U,v~u(O) | X=U[[s]IVTs ~ μ∞,U,V ~U(O) |
+
+Efficient sampling The only other requirement for obtaining a certificate computationally is to be able to sample from $\mu ( x )$ to estimate $\theta _ { a } , \theta _ { b }$ . Since $\mu ( x )$ is log-concave, there are general purpose polynomial time algorithms for sampling from this measure. However, for most norms, more efficient methods exist, as outlined below.
+
+The random variable $X \sim \mu ( x )$ can be obtained as $X = x + Z$ with $Z \sim \mu ( 0 )$ . Thus, to sample from $\mu ( x )$ for any $x$ it is enough to be able to sample from $\mu ( 0 )$ . For $\left\| \cdot \right\| _ { 1 }$ , this reduces to sampling from a Laplace distribution which can be done easily. For $\left\| \cdot \right\| _ { \infty }$ , (Steinke & Ullman, 2016) give the following efficient sampling procedure: first sample $r$ from a Gamma distribution with shape $d + 1$ and mean $d + 1$ , i.e. $\bar { r } \sim \bar { \Gamma } ( \bar { d } + 1 , 1 )$ , and then sample each $Z _ { i }$ , $i \in [ d ]$ , uniformly from $[ - r , r ]$ . Theorem 9 gives a short proof of correctness for this procedure. Theorem 10 also has a similar result for the case of $\lVert \cdot \rVert _ { 2 }$ and Table 5 lists the sampling procedures for several norms.
+
+Theorem 9. The random variable $Z \in \mathbb { R } ^ { d }$ obtained by first sampling $R \sim \Gamma ( d + 1 , 1 )$ and then sampling each $Z _ { i }$ , $i \in [ d ]$ , uniformly from $[ - R , R ]$ has density $\propto e ^ { - \| z \| _ { \infty } }$ .
+
+Proof. We first compute the normalization constant for a density of the form $\propto e ^ { - \| z \| _ { \infty } }$ as follows:
+
+$$
+\int _ { \mathbb { R } ^ { d } } e ^ { - \| z \| _ { \infty } } d z = \int _ { 0 } ^ { \infty } \left( \int _ { \mathbb { R } ^ { d } } 1 [ \| z \| _ { \infty } = t ] d z \right) e ^ { - t } d t = \int _ { 0 } ^ { \infty } 2 ^ { d } d t ^ { d - 1 } e ^ { - t } d t = d ! 2 ^ { d } \ .
+$$
+
+Next we show the density of $Z$ satisfies $p _ { Z } ( z ) = e ^ { - \| z \| _ { \infty } } / ( d ! 2 ^ { d } )$ by noting that conditioned on $R = r$ we have $p _ { Z | R = r } ( \bar { z } ) = 1 [ \| z \| _ { \infty } \leq r ] / ( 2 r ) ^ { d }$ because of the uniform sampling used in each
+
+coordinate, and integrating over $R$ sampled from a Gamma distribution with shape $d + 1$ and mean $d + 1$ yields
+
+$$
+_ { _ { D G } } ( z ) = \int _ { 0 } ^ { \infty } p _ { Z | R = r } ( z ) p _ { R } ( r ) d r = \int _ { 0 } ^ { \infty } \frac { 1 [ \| z \| _ { \infty } \leq r ] } { ( 2 r ) ^ { d } } \frac { r ^ { d } e ^ { - r } } { d ! } d r = \frac { 1 } { d ! 2 ^ { d } } \int _ { \| z \| _ { \infty } } ^ { \infty } e ^ { - r } d r = \frac { e ^ { - \| z \| _ { \infty } } } { d ! 2 ^ { d } } .
+$$
+
+Theorem 10. The random variable $Z \in \mathbb { R } ^ { d }$ obtained by first sampling $Z ^ { \prime } \sim \mathcal { N } ( 0 , I )$ and $R \sim$ $\Gamma _ { p } ( d , 1 )$ and then taking $\begin{array} { r } { Z = R \frac { Z ^ { \prime } } { \Vert Z ^ { \prime } \Vert _ { 2 } } } \end{array}$ has density $\propto e ^ { - \| z \| _ { 2 } ^ { p } }$ . Here $\Gamma _ { p } ( d , a )$ denotes the generalized Gamma distribution of order $p > 0$ with shape $d$ and scale $a$ .
+
+Proof. First note that $\begin{array} { r } { W = \frac { Z ^ { \prime } } { \left. Z ^ { \prime } \right. _ { 2 } } \sim \mathcal { U } ( B _ { 2 } ) } \end{array}$ ; i.e. it is uniform on the $\ell _ { 2 }$ ball of radius 1. Therefore, $R W$ is uniform on the $\ell _ { 2 }$ ball of radius $R$ and the conditional density of $Z$ given $R$ is given by $\begin{array} { r } { p _ { Z | R = r } ( z ) = \frac { 1 [ \| z \| _ { 2 } = r ] \Gamma ( d / 2 ) } { 2 \pi ^ { d / 2 } r ^ { d - 1 } } } \end{array}$ . Since $R$ has density $p _ { R } ( r ) \propto r ^ { d - 1 } e ^ { - r ^ { p } }$ , we get
+
+$$
+\begin{array} { l } { { \displaystyle p _ { Z } ( z ) = \int _ { 0 } ^ { \infty } p _ { Z | R = r } ( z ) p _ { R } ( r ) d r } } \\ { { \displaystyle \propto \int _ { 0 } ^ { \infty } \frac { 1 [ \| z \| _ { 2 } = r ] \Gamma ( d / 2 ) } { 2 \pi ^ { d / 2 } r ^ { d - 1 } } r ^ { d - 1 } e ^ { - r ^ { p } } d r } } \\ { { \displaystyle \propto e ^ { - \| z \| _ { 2 } ^ { p } } } . } \end{array}
+$$
+
+# A.9 ALGORITHM FOR FULL-INFORMATION CERTIFICATION
+
+In this section we describe the subroutines used in Algorithm 1. First we describe a generic procedure to provide high-probability confidence intervals for estimating expectations, then we give a detailed description of the subroutines.
+
+# A.9.1 HIGH-CONFIDENCE ESTIMATES OF EXPECTED VALUES
+
+Let Let $Z _ { 1 } , \dots , Z _ { N }$ be indepel mean be distributed random variables and the empirical variance be $R$ $m$ $\begin{array} { r } { \bar { Z } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \bar { Z _ { i } } } \end{array}$ $\begin{array} { r } { \bar { \sigma } ^ { 2 } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } ( Z _ { i } - \bar { Z } ) ^ { 2 } } \end{array}$ Applying Bernstein’s inequality to the sum and the sum of the squares of these random variables, we get the empirical Bernstein bound (Audibert et al., 2009), which states that with probability at least $1 - \zeta$ ,
+
+$$
+| \bar { Z } - m | \leq \sqrt { \frac { 2 \bar { \sigma } ^ { 2 } \log ( 3 / \zeta ) } { N } } + \frac { 3 R \log ( 3 / \zeta ) } { t } .
+$$
+
+The main benefit of the above inequality is that as long as the variance of the sample $Z _ { 1 } , \dots , Z _ { N }$ is small, the convergence rate becomes essentially $O ( 1 / N )$ instead of the standard $O ( 1 / \sqrt { N } )$ . Also, since Eq. 23 only contains empirical quantities apart from the range $R$ , it can be used to obtain computable bounds for the expectation $\mu$ : with probability at least $1 - \zeta$ ,
+
+$$
+\bar { Z } - \sqrt { \frac { 2 \bar { \sigma } ^ { 2 } \log ( 3 / \zeta ) } { N } } - \frac { 3 R \log ( 3 / \zeta ) } { N } \leq m \leq \bar { Z } + \sqrt { \frac { 2 \bar { \sigma } ^ { 2 } \log ( 3 / \zeta ) } { N } } + \frac { 3 R \log ( 3 / \zeta ) } { N } .
+$$
+
+# A.9.2 SUBROUTINES FOR ALGORITHM 1
+
+The bound in Eq. 24 can be applied to approximate the expectation in Eq. 6 with high probability for given values of $\lambda$ and $\kappa$ . More specifically, if the function $f _ { \lambda } ^ { * } ( \kappa - \phi ( \cdot ) )$ is bounded with range $R$ , then taking $N$ samples $X _ { 1 } , \ldots , X _ { N }$ independently from $\rho$ , and defining $Z _ { i } = f _ { \lambda } ^ { * } ( \kappa - \phi ( X _ { i } ) )$ , and $\bar { Z }$ and $\bar { \sigma } ^ { 2 }$ as above, Eq. 24 implies that with probability at least $1 - \zeta$ ,
+
+$$
+\operatorname { E } _ { X \sim \rho } ^ { } [ f _ { \lambda } ^ { * } ( \kappa - \phi ( X _ { i } ) ) ] \leq \bar { Z } + \sqrt { \frac { 2 \bar { \sigma } ^ { 2 } \log ( 3 / \zeta ) } { N } } + \frac { 3 R \log ( 3 / \zeta ) } { N } .
+$$
+
+Plugging in this bound to Eq. 6 gives a high-probability lower bound for the function to be maximized for any given $\lambda$ and $\kappa$ .
+
+Details of the above procedures are given in Algorithm 3. We use an off-the-shelf convex optimization solver (Diamond & Boyd, 2016) in the ESTIMETEOPT subroutine.
+
+# Algorithm 3 Subroutines for Algorithm 1
+
+function ESTIMAT $\mathrm { E O P T } ( \rho , \phi , N , \{ f _ { i } \} , \{ \epsilon _ { i } \} )$
+
+Sample $X ^ { 1 } , \ldots , X ^ { N } \sim \rho$ and obtain $\kappa ^ { * }$ and $\lambda ^ { * }$ by solving Eq. 6 with $\operatorname { E } _ { X \sim \rho } [ f _ { \lambda } ^ { * } ( \kappa - \phi ( X ) ) ]$ replaced by $\begin{array} { r } { \frac { 1 } { N } \sum _ { i = 1 } ^ { N } f _ { \lambda } ^ { * } \big ( \kappa - \phi \big ( X ^ { i } \big ) \big ) } \end{array}$ and the additional constraints $f _ { \lambda ^ { * } } ^ { * } ( \kappa - a ) < \infty , f _ { \lambda ^ { * } } ^ { * } ( \kappa - b ) <$ $\infty$ .
+
+return $\kappa ^ { * } , \lambda ^ { * }$
+
+function UPPERCON $\mathrm { F I D E N C E B O U N D } ( \rho , \phi , \tilde { N } , \{ f _ { i } \} _ { i = 1 } ^ { M } , \{ \epsilon _ { i } \} _ { i = 1 } ^ { M } , a , b , \lambda , \kappa , \zeta )$
+
+Sample $X ^ { 1 } , \dotsc , X ^ { \tilde { N } } \dotsc \rho$ and compute $Z ^ { i } \gets f _ { \lambda } ^ { * } \big ( \kappa - \phi \big ( X ^ { i } \big ) \big )$ for $i = 1 , \ldots , \tilde { N }$
+
+$$
+E _ { u b } \bar { Z } + \sqrt { \frac { 2 \bar { \sigma } ^ { 2 } \log ( 3 / \zeta ) } { \tilde { N } } } + \frac { 3 R \log ( 3 / \zeta ) } { \tilde { N } } .
+$$
+
+return $E _ { u b }$ end function
+
+# A.10 $\ell _ { 0 }$ SMOOTHING MEASURE
+
+We can also handle discrete perturbations in our framework. A natural case to consider is $\ell _ { 0 }$ perturbations. In this case, we assume that $\mathcal { X } = A ^ { d }$ where
+
+$$
+A = \{ 1 , \ldots , K \}
+$$
+
+is a discrete set. Then, we can choose
+
+$$
+\mu ( x ) [ z ] = \prod _ { i = 1 } ^ { d } p ^ { 1 [ z _ { i } = x _ { i } ] } { \left( { \frac { q } { K - 1 } } \right) } ^ { 1 [ z _ { i } \neq x _ { i } ] }
+$$
+
+where and $\frac { q } { K - 1 }$ $p + q = 1$ denotes a uniform probability of switching it to a different value. In this case, it can be , $p \geq q \geq 0$ , and $p$ denotes the probability that the measure retains the value of $x$ shown that for every $\alpha > 0$ that
+
+$$
+R _ { \alpha } ( \mu ( x ^ { \prime } ) \| \mu ( x ) ) = \| x - x ^ { \prime } \| _ { 0 } \left( \frac { \log \left( p \left( \frac { q } { ( K - 1 ) p } \right) ^ { ( \alpha ) } + \frac { q } { K - 1 } \left( \frac { ( K - 1 ) p } { q } \right) ^ { ( \alpha ) } + \left( \frac { K - 2 } { K - 1 } \right) q \right) } { \alpha - 1 } \right)
+$$
+
+so that we can derive a certificate with respect to $\ell _ { 0 }$ perturbations using any set of Renyi divergences ´ (or combinations of theses).
+
+This can be extended to structured discrete perturbations by introducing coupling terms between the perturbations:
+
+$$
+\mu ( x ) [ z ] \propto \prod _ { i = 1 } ^ { d } p ^ { 1 [ z _ { i } = x _ { i } ] } \bigg ( \frac { q } { K - 1 } \bigg ) ^ { 1 [ z _ { i } \neq x _ { i } ] } \exp \left( \sum _ { i = 1 } ^ { d - 1 } \eta 1 [ z _ { i } = x _ { i } ] 1 [ z _ { i + 1 } = x _ { i + 1 } ] \right) \ .
+$$
+
+This would correlate perturbations between adjacent features (which for example may be useful to model correlated perturbations for time series data). Since this can be viewed as a Markov Chain, Renyi divergences between ´ $\mu ( x ) , \mu ( x ^ { \prime } )$ are still easy to compute.
+
+A.11 COMPARISON WITH LECUYER ET AL. (2019) ON $\ell _ { 1 }$ PERTURBATIONS
+
+Here we compare our certificates to Lecuyer et al. (2019) on the MNIST, CIFAR-10 and ImageNet datasets. The smoothing distribution is as described in A.8; a zero mean Laplacian distribution with smoothing value defined by the scale of the distribution. We first describe the hyperparameters used in training and certification for each of the datasets. For all datasets, images were normalized into a [0,1] range.
+
+MNIST hyperparameters: We trained a standard three layer CNN ReLU classifier for 50,000 steps with a batch size of 128 and a learning rate of 0.001. The smoothing value during training was set to 1.0. For certification we use $N \overset { \cdot } { = } 1 K , \tilde { N } = 1 0 M , \zeta = . 9 9$ , and sweep over a range of smoothing values between 0.5 and 1.5 and report the best certificate found. Certified accuracy is reported on 1,000 MNIST test set images.
+
+CIFAR-10 hyperparameters: We trained a Wide ResNet classifier for 50,000 training steps with a batch size of 32 and a learning rate of 0.001. The smoothing value during training was set to 0.2. For certification we use $N = 1 K , \tilde { N } = 1 M , \zeta = . 9 9$ , and sweep over a range of smoothing values between 0.1 and 0.5 and report the best certificate found. Certified accuracy is reported on 1,000 CIFAR-10 test set images.
+
+ImageNet hyperparameters: We trained a ResNet-152 classifier for 1 million training steps with a batch size of 16 and an initial learning rate of 0.1 that is decayed by a factor of ten every 25,000 steps. The smoothing value during training was set to 0.1. For certification we use $N = 1 K , \tilde { N } =$ $1 0 0 K , \zeta = . 9 9$ , and sweep over a range of smoothing values between 0.05 and 0.25 and report the best certificate found. Certified accuracy is reported on 500 ImageNet validation set images.
+
+
+Figure 3: Certified accuracy under $\ell _ { 1 }$ perturbations on CIFAR-10 and ImageNet.
+
+Results for MNIST can be seen in Table 6, CIFAR-10 and ImageNet results are shown in Figure 3.
+We significantly outperform Lecuyer et al. (2019) on all three datasets.
\ No newline at end of file
diff --git a/parse/train/SJlKrkSFPH/SJlKrkSFPH_content_list.json b/parse/train/SJlKrkSFPH/SJlKrkSFPH_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..133bc986ec1a3c6ffe630f0a7a74f65ed93522bb
--- /dev/null
+++ b/parse/train/SJlKrkSFPH/SJlKrkSFPH_content_list.json
@@ -0,0 +1,4743 @@
+[
+ {
+ "type": "text",
+ "text": "A FRAMEWORK FOR ROBUSTNESS CERTIFICATION OF SMOOTHED CLASSIFIERS USING F-DIVERGENCES ",
+ "text_level": 1,
+ "bbox": [
+ 174,
+ 99,
+ 823,
+ 146
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Krishnamurthy (Dj) Dvijotham† Jamie Hayes†‡∗ Borja Balle† J. Zico Kolterac \nChongli Qin† Andras Gy ´ orgy ¨ † Kai Xiaob†∗ Sven Gowal† Pushmeet Kohli† \n†DeepMind, London, UK ‡University College London, UK \na Carnegie Mellon University, USA b Massachusetts Institute of Technology, USA \nc Bosch Center for AI, USA \n{dvij, bballe, chongliqin, agyorgy, sgowal, pushmeet}@google.com \n{zkolter}@cs.cmu.edu {j.hayes}@ucl.ac.uk {kaix}@mit.edu ",
+ "bbox": [
+ 181,
+ 169,
+ 802,
+ 272
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "ABSTRACT ",
+ "text_level": 1,
+ "bbox": [
+ 454,
+ 308,
+ 544,
+ 321
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Formal verification techniques that compute provable guarantees on properties of machine learning models, like robustness to norm-bounded adversarial perturbations, have yielded impressive results. Although most techniques developed so far require knowledge of the architecture of the machine learning model and remain hard to scale to complex prediction pipelines, the method of randomized smoothing has been shown to overcome many of these obstacles. By requiring only black-box access to the underlying model, randomized smoothing scales to large architectures and is agnostic to the internals of the network. However, past work on randomized smoothing has focused on restricted classes of smoothing measures or perturbations (like Gaussian or discrete) and has only been able to prove robustness with respect to simple norm bounds. In this paper we introduce a general framework for proving robustness properties of smoothed machine learning models in the black-box setting. Specifically, we extend randomized smoothing procedures to handle arbitrary smoothing measures and prove robustness of the smoothed classifier by using $f$ -divergences. Our methodology improves upon the state of the art in terms of computation time or certified robustness on several image classification tasks and an audio classification task, with respect to several classes of adversarial perturbations. ",
+ "bbox": [
+ 233,
+ 339,
+ 766,
+ 590
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "1 INTRODUCTION ",
+ "text_level": 1,
+ "bbox": [
+ 176,
+ 621,
+ 336,
+ 636
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Predictors obtained from machine learning algorithms have been shown to be vulnerable to making errors when the inputs are perturbed by carefully chosen small but imperceptible amounts (Szegedy et al., 2014; Biggio et al., 2013). This has motivated significant amount of research in improving adversarial robustness of a machine learning model (see, e.g. Goodfellow et al., 2015; Madry et al., 2018). While significant advances have been made, it has been shown that models that were estimated to be robust have later been broken by stronger attacks (Athalye et al., 2018; Uesato et al., 2018). This has led to the need for methods that offer provable guarantees that the predictor cannot be forced to misclassify an example by any attack algorithm restricted to produce perturbations within a certain set (for example, within an $\\ell _ { p }$ norm ball). While progress has been made leading to methods that are able to compute provable guarantees for several image and text classification tasks (Wong & Kolter, 2018; Wong et al., 2018; Raghunathan et al., 2018; Dvijotham et al., 2018; Katz et al., 2017; Huang et al., 2019; Jia et al., 2019), these methods require extensive knowledge of the architecture of the predictor and are not easy to extend to new models or architectures, requiring specialized algorithms for each new class of models. Furthermore, the computational complexity of these methods grows significantly with input dimension and model size. ",
+ "bbox": [
+ 174,
+ 655,
+ 825,
+ 861
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "To deal with these obstacles, recent work has proposed the randomized smoothing strategy for verifying the robustness of classifiers. Specifically, Lecuyer et al. (2019) and Cohen et al. (2019) ",
+ "bbox": [
+ 176,
+ 869,
+ 823,
+ 897
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "have shown that robustness properties can be more easily verified for the smoothed version of a base classifier $h$ producing labels in some set $\\mathcal { V }$ : ",
+ "bbox": [
+ 173,
+ 103,
+ 823,
+ 132
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "equation",
+ "img_path": "images/ffae80575ac3e04b65ca0265c693acd9bbebfd42241c9d584e622d28c70312fe.jpg",
+ "text": "$$\nh _ { s } ( x ) = \\underset { y \\in \\mathcal { V } } { \\arg \\operatorname* { m a x } } \\ \\underset { X \\sim \\mu ( x ) } { \\mathbb { P } } \\big [ h ( X ) = y \\big ] \\ ,\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 369,
+ 138,
+ 627,
+ 165
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "where the labels returned by the smoothed classifier $h _ { s }$ are obtained by taking a “majority vote” over the predictions of the original classifier $h$ on random inputs drawn from a probability distribution $\\mu ( x )$ , called the smoothing measure. Lecuyer et al. (2019) showed that verifying the robustness of this smoothed classifier is significantly simpler than verifying the original classifier $h$ and only requires estimating the distribution of outputs of the classifier under random perturbations of the input, but does not require access to the internals of the classifier $h$ . We refer to this as black-box verification. ",
+ "bbox": [
+ 173,
+ 172,
+ 825,
+ 270
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "In this work, we develop a general framework for black-box verification that recovers prior work as special cases, and improves upon previous results in various ways. ",
+ "bbox": [
+ 173,
+ 276,
+ 823,
+ 306
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "Contributions Our contributions are summarized as follows: ",
+ "bbox": [
+ 174,
+ 320,
+ 586,
+ 335
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "1. We formulate the general problem of black-box verification via a generalized randomized smoothing procedure, which extends existing approaches to allow for arbitrary smoothing measures. Specifically, we show that robustness certificates for smoothed classifiers can be obtained by solving a small convex optimization problem when allowed adversarial perturbations can be characterized via divergence-based bounds on the smoothing measure. ",
+ "bbox": [
+ 186,
+ 347,
+ 825,
+ 417
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "2. We prove that our certificates generalize previous results obtained in related work (Lecuyer et al., 2019; Cohen et al., 2019; Li et al., 2019), and vastly extend the class of perturbations and smoothing measures that can be used while still allowing certifiable guarantees. ",
+ "bbox": [
+ 184,
+ 422,
+ 821,
+ 464
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "3. We introduce the notion of full-information and information-limited settings, and show that the information-limited setting that has been the main focus of prior work leads to weaker certificates for smoothed probabilistic classifiers, and can be improved by using additional information (the distribution of label scores under randomized smoothing). ",
+ "bbox": [
+ 186,
+ 469,
+ 826,
+ 525
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "4. We evaluate our framework experimentally on image and classification tasks, obtaining robustness certificates that improve upon other black-box methods either in terms of certificate tightness or computation time on robustness to $\\ell _ { 0 } , \\ell _ { 1 }$ or $\\ell _ { 2 }$ perturbations on MNIST, CIFAR-10 and ImageNet. $\\ell _ { 2 }$ perturbations result from worst-case realizations of white noise that is common in many image, speech and video processing. $\\ell _ { 0 }$ perturbations can model missing data (missing pixels in an image, or samples in a time-domain audio signal) while $\\ell _ { 1 }$ perturbations can be used to model convex combinations of discrete perturbations in text classification (Jia et al., 2019). We also obtain the first, to the best of our knowledge, certifiably robust model for an audio classification task, Librispeech (Panayotov et al., 2015), with variable-length inputs. ",
+ "bbox": [
+ 186,
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+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "2 BLACK-BOX VERIFICATION FOR SMOOTHED CLASSIFIERS ",
+ "text_level": 1,
+ "bbox": [
+ 174,
+ 676,
+ 679,
+ 691
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "Consider a binary classifier $h : \\mathcal { X } \\{ \\pm 1 \\}$ given to us as a black box, so we can only access the inputs and outputs of $h$ but not its internals. We are interested in investigating the robustness of the smoothed classifier $h _ { s }$ (defined in Eq. 1) against adversarial perturbations of size at most $\\epsilon$ with respect to a given norm $\\left\\| \\cdot \\right\\|$ . To determine whether a norm-bounded adversarial attack on a fixed input $x \\in \\mathcal { X }$ with $h _ { s } ( x ) = + 1$ could be successful, we can solve the optimization problem ",
+ "bbox": [
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+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "equation",
+ "img_path": "images/488b1e3ac23c10ba3be97a0376ce5ef0c389de876510b99f1269c2115dfd78f1.jpg",
+ "text": "$$\n\\operatorname* { m i n } _ { \\| \\boldsymbol { x ^ { \\prime } } - \\boldsymbol { x } \\| \\le \\epsilon { \\boldsymbol X ^ { \\prime } } \\sim \\mu ( \\boldsymbol { x ^ { \\prime } } ) } [ h ( \\boldsymbol { X ^ { \\prime } } ) = + 1 ] ,\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 385,
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+ 809
+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "and check whether the minimum value can be smaller than $\\frac { 1 } { 2 }$ . This is a non-convex optimization problem for which we may not even be able to compute gradients since we only have black-box access to $h$ . While techniques have been developed to address this problem, obtaining provable guarantees on whether these algorithms actually find the worst-case adversarial perturbation is difficult since we do not know anything about the nature of $h$ . ",
+ "bbox": [
+ 174,
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+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "Motivated by this difficulty, we take a different approach: Rather than studying the adversarial attack in the input space $\\mathcal { X }$ , we study it in the space of probability measures over inputs, denoted by $\\mathcal { P } ( \\mathcal { X } )$ . ",
+ "bbox": [
+ 173,
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+ ],
+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "Formally, this amounts to rewriting Eq. 2 as ",
+ "bbox": [
+ 174,
+ 103,
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+ 118
+ ],
+ "page_idx": 2
+ },
+ {
+ "type": "equation",
+ "img_path": "images/406031201bef6a18ca0d598138c006751f745df1e6b0807723299cd2dc109d75.jpg",
+ "text": "$$\n\\operatorname* { m i n } _ { \\nu \\in \\{ \\mu ( x ^ { \\prime } ) \\colon \\| x ^ { \\prime } - x \\| \\leq \\epsilon \\} } \\operatorname* { l p } _ { X ^ { \\prime } \\sim \\nu } [ h ( X ^ { \\prime } ) = + 1 ] .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "This is an infinite dimensional optimization problem over the space of probability measures $\\nu \\in { \\mathcal { P } } ( { \\mathcal { X } } )$ subject to the constraint $\\nu \\in \\mathcal { D } = \\{ \\mu ( x ^ { \\prime } ) : \\| x ^ { \\prime } - x \\| \\leq \\epsilon \\}$ . While this set is still intractable to deal with, we can consider relaxations of this set defined by divergence constraints between $\\nu$ and $\\rho = \\mu ( x )$ , i.e., $\\mathcal { D } \\subseteq \\{ \\nu : D ( \\nu \\| \\rho ) \\leq \\epsilon _ { D } \\}$ where $D$ denotes some divergence between probability distributions. We will show in Section 3 that for several commonly used divergences (in fact, for any $f$ -divergence; cf. Ali & Silvey, 1966), the relaxed problem can be solved efficiently. ",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "2.1 A GENERAL FRAMEWORK FOR ROBUSTNESS CERTIFICATION ",
+ "text_level": 1,
+ "bbox": [
+ 174,
+ 256,
+ 630,
+ 270
+ ],
+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "To formulate the general verification problem, consider a specification $\\phi : \\mathcal { X } \\to \\mathcal { Z } \\subseteq \\mathbb { R }$ : a generic function over the input space (that typically is a function of the classifier output) that we want to verify has certain properties. Unless otherwise specified, we will assume that $\\mathcal { \\hat { X } } \\subseteq \\mathbb { R } ^ { d }$ (we work in a $d$ dimensional input space). Our framework also involves a reference measure $\\rho$ (in the above example we would take $\\rho = \\mu ( x ) )$ and a collection of perturbed distributions $\\mathcal { D }$ (in the above example we would take $\\mathcal { D } = \\mathcal { D } _ { x , \\epsilon } = \\{ \\mu ( x ^ { \\prime } ) : \\| x ^ { \\prime } - x \\| \\le \\bar { \\epsilon } \\}$ ). ",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Verifying that a given specification $\\phi$ is robustly certified is equivalent to checking whether the optimal value of the optimization problem ",
+ "bbox": [
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+ ],
+ "page_idx": 2
+ },
+ {
+ "type": "equation",
+ "img_path": "images/35fc9d3f11482f78fa99ee5d084323f2aec2d43ac4feba6a89099cfae2a7e1a3.jpg",
+ "text": "$$\n{ \\mathrm { O P T } } ( \\phi , \\rho , { \\mathcal { D } } ) : = \\operatorname* { m i n } _ { \\nu \\in { \\mathcal { D } } } \\operatorname { E } _ { X \\sim \\nu } [ \\phi ( X ) ] \\ ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "is non-negative. Solving problems of this form is the key workhorse of our general framework for black-box certification of adversarial robustness for smoothed classifiers. ",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Using these ingredients we introduce two closely related certification problems: information-limited robust certification and full-information robust certification. In the former case, we assume that we are given only given access to $\\mathbb { P } _ { X \\sim \\rho } [ \\phi ( X ) = + 1 ] , \\mathbb { P } _ { X \\sim \\nu } [ \\phi ( X ) = + 1 ]$ . In the latter case, we are given full-access to specification $\\phi$ . The definitions are below. ",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Definition 2.1 (Information-limited robust certification). Given reference distribution $\\rho \\in { \\mathcal { P } } ( { \\mathcal { X } } )$ , probabilities $\\theta _ { a } , \\theta _ { b }$ that satisfy $\\theta _ { a } , \\theta _ { b } \\geq 0 , \\theta _ { a } + \\theta _ { b } \\leq 1$ and collection of perturbed distributions $\\mathcal { D } \\subset \\mathcal { P } ( \\mathcal { X } )$ containing $\\rho$ , define the class of specifications $S$ as ",
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+ "type": "equation",
+ "img_path": "images/fa2b5ecc9101a201a2d31b04eca2bee40ec61351ed76c8f5010a8bdb06885eef.jpg",
+ "text": "$$\nS = \\left\\{ \\phi : \\mathcal { X } \\to \\{ - 1 , 0 , + 1 \\} \\mathrm { ~ s . t . ~ } \\operatorname { \\mathbb { P } } _ { X \\sim \\rho } [ \\phi ( X ) = + 1 ] \\geq \\theta _ { a } , \\operatorname { \\mathbb { P } } _ { X \\sim \\rho } [ \\phi ( X ) = - 1 ] \\leq \\theta _ { b } \\right\\}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "We say that $S$ is information-limited robustly certified at $\\rho$ with respect to $\\mathcal { D }$ if the following condition holds: $\\operatorname { E } _ { X \\sim \\nu } [ \\phi ( X ) ] \\geq 0$ for all $\\nu \\in \\mathcal { D } , \\phi \\in S$ . ",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Note since we don’t have access to $\\phi$ , we need to prove that ${ \\mathrm { E } } _ { X \\sim \\nu } [ \\phi ( X ) ] \\geq 0 \\quad \\forall \\nu \\in { \\mathcal { D } }$ is satisfied for all specifications in set $S$ . Although the information-limited case may seem challenging because we need to provide guarantees that hold simultaneously over a whole class of specifications, it turns out that, for perturbation sets $\\mathcal { D }$ specified by an $f$ -divergence bound, this certification task can be solved efficiently using convex optimization. ",
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Definition 2.2 (Full-information robust certification). Given a reference distribution $\\rho \\in { \\mathcal { P } } ( { \\mathcal { X } } )$ , a specification $\\phi : \\mathcal { X } \\to \\mathcal { Z } \\subseteq \\mathbb { R }$ and a collection of perturbed distributions $\\mathcal { D } \\subset \\mathcal { P } ( \\mathcal { X } )$ containing $\\rho$ , we say that $\\phi$ is full-information robustly certified at $\\rho$ with respect to $\\mathcal { D }$ if the following condition holds: $\\operatorname { E } _ { X \\sim \\nu } [ \\phi ( X ) ] \\geq 0$ for all $\\nu \\in \\mathcal { D }$ . ",
+ "bbox": [
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+ },
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+ "type": "text",
+ "text": "Most often we are dealing with the case where we have full access to the specification $\\phi$ , thus we should be able to certify using full-information robust certification. However, prior works, Cohen et al. (2019) and Lecuyer et al. (2019), have only provided solutions to certify with respect to the information-limited case where we cannot use all of the information about $\\phi$ . The framework we develop is a more general method that can be used in both information-limited and full-information scenarios. We will demonstrate that our framework recovers certificates provided by Cohen et al. (2019), Li et al. (2019) and dominates Lecuyer et al. (2019) in the information-limited setting (see section 5). Further, it can utilize full-information about the specification $\\phi$ to provide tighter certificates for smoothed probabilistic classifiers (see section 6). ",
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+ "type": "text",
+ "text": "We first note that the definitions above are sufficient to capture the standard usage of randomized smoothing as it has been used in past work (e.g. Lecuyer et al., 2019; Cohen et al., 2019) to verify the robustness of smoothed multi-class classifiers. Specifically, consider smoothing a classifier $h : \\mathcal { X } \\to \\mathcal { Y }$ with a finite set of labels $\\mathcal { V }$ using a smoothing measure $\\mu : \\mathcal X \\mapsto \\mathcal P ( \\mathcal X )$ . The resulting randomly smoothed classifier $h _ { s }$ is defined in Eq. 1. Our goal is to certify that the prediction $h _ { s } ( x )$ is robust to perturbations of size at most $\\epsilon$ measured by distance function1 $d : \\mathcal { X } \\times \\mathcal { X } \\mapsto \\mathbb { R } _ { + }$ , i.e., ",
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+ "type": "equation",
+ "img_path": "images/6d041ce45084b5b6fb732d7b7aad962753203e718f12289be9047943f9a87cc2.jpg",
+ "text": "$$\nh _ { s } ( x ^ { \\prime } ) = h _ { s } ( x ) \\quad \\forall x ^ { \\prime } \\ \\mathrm { s u c h ~ t h a t } \\ d ( x , x ^ { \\prime } ) \\leq \\epsilon .\n$$",
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+ "text": "To pose this question within our framework, we choose the reference distribution $\\rho = \\mu ( x )$ , the set of perturbed distributions $\\mathcal { D } _ { x , \\epsilon } = \\{ \\mu ( x ^ { \\prime } ) : d ( x , x ^ { \\prime } ) \\leq \\epsilon \\}$ , and the following specifications. Let $c = h _ { s } ( x )$ . For every $c ^ { \\prime } \\in \\mathcal { V } \\setminus \\{ c \\}$ , we define the specification $\\phi _ { c , c ^ { \\prime } } : \\mathcal { X } \\mapsto \\{ - \\bar { 1 } , \\bar { 0 } , + 1 \\}$ as follows: ",
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+ "text": "$$\n\\phi _ { c , c ^ { \\prime } } ( x ) = \\left\\{ \\begin{array} { l l } { { + 1 } } & { { \\mathrm { ~ i f ~ } h ( x ) = c ~ , } } \\\\ { { - 1 } } & { { \\mathrm { ~ i f ~ } h ( x ) = c ^ { \\prime } ~ , } } \\\\ { { 0 } } & { { \\mathrm { ~ o t h e r w i s e ~ . } } } \\end{array} \\right.\n$$",
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+ "text": "Then, Eq. 5 holds if and only if every $\\phi _ { c , c ^ { \\prime } }$ , $c ^ { \\prime } \\neq c$ , is robustly certified at $\\mu ( x )$ with respect to $\\mathcal { D } _ { x , \\epsilon }$ (see Appendix A.1). ",
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+ "type": "text",
+ "text": "2.2 CONSTRAINT SETS FROM F-DIVERGENCES ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Dealing with the set $\\mathcal { D } _ { x , \\epsilon }$ directly is difficult due to its possibly non-convex geometry. In this section, we discuss specific relaxations of this set, i.e., choices for sets $\\mathcal { D }$ such that $\\mathcal { D } _ { x , \\epsilon } \\subseteq \\mathcal { D }$ that are easier to optimize over. In particular, we focus on a general family of constraint sets defined in terms of $f$ -divergences. These divergences satisfy a number of useful properties and include many well-known instances (e.g. relative entropy, total variation); see Appendix A.2 for details. ",
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+ "type": "text",
+ "text": "Definition 2.3. $f$ -divergence constraint set). Given $\\rho , \\nu \\in \\mathcal { P } ( \\mathcal { X } )$ , their $f$ -divergence is defined as ",
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+ "text": "$$\nD _ { f } ( \\nu \\| \\rho ) = \\underset { X \\sim \\rho } { \\mathrm { E } } \\Bigg [ f \\Bigg ( \\frac { \\nu ( X ) } { \\rho ( X ) } \\Bigg ) \\Bigg ] ~ ,\n$$",
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+ "type": "text",
+ "text": "where $f : \\mathbb { R } _ { + } \\mapsto \\mathbb { R }$ is a convex function with $f ( 1 ) = 0$ . Given a reference distribution $\\rho$ , an $f$ -divergence $D _ { f }$ and a bound $\\epsilon _ { f } \\geq 0$ , we define the $f$ -divergence constraint set to be: ",
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+ "text": "$$\n\\mathcal { D } _ { f } = \\{ \\nu \\in \\mathcal { P } ( \\mathcal { X } ) : D _ { f } ( \\nu \\| \\rho ) \\leq \\epsilon _ { f } \\} \\ .\n$$",
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+ "text": "Technically, this definition depends on the Radon-Nikodym derivative of $\\nu$ with respect to $\\rho$ , but we ignore measure-theoretic issues in this paper for simplicity of exposition. For continuous distributions, $\\nu$ and $\\rho$ should be treated as densities, and for discrete distributions as probability mass functions. ",
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+ "text": "Relaxations using $f$ -divergence This construction immediately allows us to obtain relaxations of $\\mathcal { D } _ { x , \\epsilon }$ . For example, by choosing $f ( u ) = u \\log ( u )$ , we have the KL-divergence. Using KL-divergence yields the following relaxation between norm-based and divergence-based constraint sets for Gaussian smoothing measures, i.e. $\\mu ( x ) = \\mathcal { N } ( x , \\sigma ^ { 2 } I )$ : ",
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+ "img_path": "images/497076acbf7ca0680384e81321a4d54f8679a7044a4fdc43971d505af2085111.jpg",
+ "text": "$$\n\\begin{array} { r } { { \\mathcal { D } } _ { x , \\epsilon } = \\{ \\mu ( x ^ { \\prime } ) : \\| x - x ^ { \\prime } \\| _ { 2 } \\leq \\epsilon \\} \\subseteq \\left\\{ \\nu : \\mathrm { K L } ( \\nu \\| \\mu ( x ) ) \\leq \\epsilon ^ { 2 } / ( 2 \\sigma ^ { 2 } ) \\right\\} \\ . } \\end{array}\n$$",
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+ "text": "Tighter relaxations can be constructed by combining multiple divergence-based constraints. In particular, suppose $\\mathcal { F }$ is a collection of convex functions each defining an $f$ -divergence, and assume each $f \\in { \\mathcal { F } }$ has a bound $\\epsilon _ { f }$ associated with it. Then we can define the constraint set containing perturbed distributions where all the bounds hold simultaneously (Fig. 1): ",
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+ "text": "$$\n{ \\mathcal { D } } _ { { \\mathcal { F } } } : = \\bigcap _ { f \\in { \\mathcal { F } } } { \\mathcal { D } } _ { f } = \\left\\{ \\nu : \\forall f \\in { \\mathcal { F } } \\ D _ { f } ( \\nu \\| \\rho ) \\leq \\epsilon _ { f } \\right\\} \\ .\n$$",
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+ "type": "image",
+ "img_path": "images/780bf0bf4f2c683a963451dd5c5948364e7681d2494ad1b2600b2a70aae3d94a.jpg",
+ "image_caption": [
+ "Figure 1: Intersecting $f$ -divergence constraints to obtain better relaxations g. \\`0). $\\mathcal { D } _ { \\mathcal { F } }$ (depicted by the orange region) of $\\mathcal { D } _ { x , \\epsilon }$ . "
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+ "type": "text",
+ "text": "In this paper, we work with the following divergences: ",
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+ "type": "text",
+ "text": "(1) Renyi: ´ $R _ { \\alpha } ( \\nu | | \\rho ) = \\log ( 1 + D _ { f } ( \\nu | | \\rho ) ) / ( \\alpha - 1 )$ where $f ( x ) = x ^ { \\alpha } - 1$ (for $\\alpha \\geq 1 \\gamma$ ), and $\\tilde { R _ { \\alpha } } ( \\rho \\| \\nu ) = \\log ( 1 - D _ { f } ( \\nu \\| \\rho ) ) / ( \\alpha - 1 )$ with $f ( x ) = 1 - x ^ { \\alpha }$ (for $0 \\leq$ $\\alpha \\leq 1 _ { \\cdot }$ ). The limit $\\alpha \\to \\infty$ yields the infinite order Renyi divergence ´ $R _ { \\infty } ( \\nu \\| \\rho ) = \\operatorname* { s u p } _ { x } ( \\nu ( x ) / \\rho ( x ) )$ . ",
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+ "type": "text",
+ "text": "(2) $\\mathrm { K L } ( \\nu \\| \\rho ) = D _ { f } ( \\nu \\| \\rho )$ with $f ( x ) = x \\log ( x )$ ",
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+ "type": "text",
+ "text": "(3) Hockey-Stick: $D _ { \\mathrm { H S } , \\beta } ( \\nu \\| \\rho ) = D _ { f } ( \\nu \\| \\rho )$ with $f ( x ) = \\mathrm { m a x } ( x - \\beta , 0 ) - \\mathrm { m a x } ( 1 - \\beta , 0 ) .$ ",
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+ "type": "text",
+ "text": "It turns out that the Renyi and KL divergences are computationally attractive for a broad class of ´ smoothing measures, while the Hockey-Stick divergences are theoretically attractive as they lead to optimal certificates in the information-limited setting. However, Hockey-Stick divergences are harder to estimate in general, so we only use them for Gaussian smoothing measures. ",
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+ "type": "text",
+ "text": "2.3 COMPUTING F-DIVERGENCE BOUNDS ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "In general, our framework can be used with any family of smoothing measures and any family of $f$ divergences such that an upper bound on $\\mathrm { m a x } _ { \\nu \\in \\mathcal { D } _ { x , \\epsilon } } D _ { f } ( \\nu \\| \\rho )$ can be estimated efficiently. We describe how $f$ -divergence bounds can be obtained for several classes of smoothing measures: ",
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+ "type": "text",
+ "text": "Product measures Product measures are of the form $\\mu ( \\boldsymbol { x } ) = \\otimes _ { i = 1 } ^ { d } \\mu _ { i } ( x _ { i } )$ where $\\begin{array} { r } { \\mathcal { X } = \\prod _ { i = 1 } ^ { d } \\mathcal { X } _ { i } } \\end{array}$ and $\\mu _ { i }$ is a smoothing measure on $\\mathcal { X } _ { i }$ . We note that the discrete smoothing measure used in (Lee et al., 2019), the Gaussian measure used in (Cohen et al., 2019) and the Laplacian measure used in (Li et al., 2019) are all of this form. For such measures, one can construct bounds on Renyi- ´ divergences subject to any $\\ell _ { p }$ norm constraint using a Lagrangian relaxation of the optimization problem $\\begin{array} { r } { \\operatorname { 1 a x } _ { x ^ { \\prime } : \\| x - x ^ { \\prime } \\| _ { p } \\leq \\epsilon } R _ { \\alpha } ( \\mu ( x ^ { \\prime } ) \\| \\mu ( x ) ) } \\end{array}$ (see Appendix A.3 for details). ",
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+ "text": "Norm-based smoothing measures Appendix A.9.1 also shows how we can obtain bounds on the infinite-order Renyi divergence ´ $R _ { \\infty }$ , as well as on several classes of $f$ -divergences, for norm-based smoothing measures of the form $\\mu ( x ) [ X ] \\propto \\exp ( { - \\| X - x \\| } )$ . ",
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+ "type": "text",
+ "text": "3 OUR CERTIFICATION PROCEDURES ",
+ "text_level": 1,
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+ "text": "We now show how to reduce the problems of full-information and information-limited robust blackbox certification to simple convex optimization problems for general constraint sets $\\mathcal { D }$ defined in terms of $f$ -divergences. This allows us, by extension, to solve the problem for related divergences like Renyi divergences. The following two theorems provide the main foundation for the verification ´ procedures in the paper. ",
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+ "text": "Theorem 1 (Verifying full-information robust certification). Let $\\mathcal { D } _ { \\mathcal { F } }$ be the constraint set defined by $\\mathcal { F } = \\{ f _ { 1 } , \\ldots , f _ { M } \\}$ and $\\epsilon _ { f _ { i } } = \\epsilon _ { i }$ . Define $\\begin{array} { r } { f _ { \\lambda } ( u ) = \\sum _ { i = 1 } ^ { M } \\lambda _ { i } f _ { i } ( u ) } \\end{array}$ and denote its convex conjugate2 by $f _ { \\lambda } ^ { * }$ . The specification $\\phi$ is robustly certified at $\\rho$ with respect to $\\mathcal { D } _ { \\mathcal { F } }$ (cf. Definition 2.2) if and only $i f$ the optimal value of the following convex optimization problem is non-negative: ",
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+ "img_path": "images/d09b0f98805b182956667fae3f6d7cf5ed5e474dedec02afe488e65d2c27a26f.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { \\lambda _ { 1 } , \\dots , \\lambda _ { M } \\ge 0 , \\kappa } \\kappa - \\sum _ { i = 1 } ^ { M } \\lambda _ { i } \\epsilon _ { i } - \\underset { X \\sim \\rho } { \\mathrm { E } } [ f _ { \\lambda } ^ { * } ( \\kappa - \\phi ( X ) ) ] .\n$$",
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+ "text": "The proof of Theorem 1, given in Appendix A.4, uses standard duality results to show that the dual of the verification optimization problem has the desired form. We note that the special case where $M = 1$ reduces to Proposition 1 of Duchi & Namkoong (2018), although the result is used in a completely different context in that work. ",
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+ "text": "To build a practical certification algorithm from Theorem 1, we must do two things: 1) compute the optimal values of $\\lambda$ and $\\kappa$ ; and 2) estimate the expectation in Eq. 6. Since the estimation of the expectation cannot be done in closed form (due to the black-box nature of $\\phi$ ), we must rely on sampling. In step 1 of Algorithm 1, we use $N$ samples taken independently from $\\rho$ to estimate the expectation and solve the “sampled” optimization problem using an off-the-shelf solver (Diamond ",
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+ "text": "Algorithm 1 Full information certification (see appendix A.9 for details of subroutines) ",
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+ "type": "text",
+ "text": "Inputs: Query access to specification $\\phi : \\mathcal { X } [ a , b ]$ , sampling access to reference distribution $\\rho$ , divergences $f _ { i }$ and bounds $\\epsilon _ { i }$ , sample sizes $N$ , $\\tilde { N }$ , confidence level $\\zeta$ . \n1: $\\kappa ^ { * }$ $\\lambda ^ { * } \\gets \\mathrm { E S T I M A T E O P T } ( \\rho , \\phi , N , \\{ f _ { i } \\} _ { i = 1 } ^ { M } , \\{ \\epsilon _ { i } \\} _ { i = 1 } ^ { M } ) .$ . \n2: $E _ { u b } \\gets$ UPPERCONFIDENCEBOUND(ρ, φ, N, ˜ {fi}Mi=1, $\\{ \\epsilon _ { i } \\} _ { i = 1 } ^ { M } , a , b , \\lambda ^ { * } , \\kappa ^ { * } , \\zeta ) .$ \n3: If $\\begin{array} { r } { \\kappa ^ { * } - \\sum _ { i = 1 } ^ { M } \\lambda _ { i } ^ { * } \\epsilon _ { i } - E _ { u b } \\geq 0 } \\end{array}$ return CERTIFIED else return NOT CERTIFIED. ",
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+ "type": "text",
+ "text": "& Boyd, 2016). This gives us $\\kappa ^ { * } , \\lambda ^ { * }$ , the estimated optimal values of $\\kappa$ and $\\lambda$ , respectively. Then we take these values and compute a high-confidence lower bound on the objective function of Eq. 6, which is then used to verify robustness. In particular, in step 2, we compute a high-confidence upper bound $E _ { u b }$ on the expectation term in the objective such that $E _ { u b } \\ge \\mathrm { E } _ { X \\sim \\rho } [ f _ { \\lambda } ^ { * } ( \\kappa ^ { * } - \\phi ( X ) ) ]$ with probability at least $\\zeta$ ; this computation involves taking $\\tilde { N }$ independent samples from $\\rho$ and finding a confidence interval around the resulting empirical estimate of the expectation (for details, see Eq. 25 in Appendix A.9.1). Plugging in this estimate back into Eq. 6 gives the desired high-confidence lower bound in step 3. Details of both subroutines ESTIMATEOPT and UPPERCONFIDENCEBOUND used in Algorithm 1 are given in Algorithm 3 in Appendix A.9.2. ",
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+ "type": "text",
+ "text": "Our next theorem concerns the specialization of this verification procedure to the information-limited setting. ",
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+ "text": "Theorem 2 (Verifying information-limited robust certification). Let $\\mathcal { D } _ { \\mathcal { F } }$ be as in Theorem 1, and $S$ and $\\theta _ { a }$ , $\\theta _ { b }$ be as in Definition 2.1. The class of specifications $S$ is information-limited robustly certified at $\\rho$ with respect to $\\mathcal { D } _ { \\mathcal { F } }$ (cf. Definition 2.1) if and only if the optimal value of the following convex optimization problem is non-negative: ",
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+ "type": "equation",
+ "img_path": "images/392c4613b2cdeb1d1f9c9bca7fa2a97540eb13589960aba2f5a78dfa03703916.jpg",
+ "text": "$$\n\\begin{array} { r l } { \\underset { \\zeta _ { a } , \\zeta _ { b } , \\zeta _ { c } \\geq 0 } { \\operatorname* { m i n } } } & { \\zeta _ { a } - \\zeta _ { b } } \\\\ { \\mathrm { S u b j e c t ~ t o } } & { \\zeta _ { a } + \\zeta _ { b } + \\zeta _ { c } = 1 } \\end{array} , \\quad D _ { f _ { i } } ( \\zeta \\| \\theta ) \\leq \\epsilon _ { i } \\quad i = 1 , \\dots , M \\ ,\n$$",
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+ "type": "text",
+ "text": "where $\\theta = ( \\theta _ { a } , \\theta _ { b } , 1 - \\theta _ { a } - \\theta _ { b } )$ and $\\zeta = ( \\zeta _ { a } , \\zeta _ { b } , \\zeta _ { c } )$ are interpreted as probability distributions. ",
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+ "text": "The proof of Theorem 2 is presented in Appendix A.5. It is based on the fact that in the informationlimited setting, it is possible to directly compute the expectation in Eq. 6, and in fact this expectation only depends on $\\phi$ via the probabilities $\\theta _ { a }$ and $\\theta _ { b }$ . ",
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+ "text": "Theorem 2 naturally leads to a certification algorithm, presented in Algorithm 2. It simply uses the same procedure as Cohen et al. (2019, Section 3.2) to compute a high-confidence lower bound $\\underline { { \\theta } } _ { a }$ on the probability of the correct class under randomized smoothing and then solves the convex optimization problem Eq. 7. Again, we can use an off-the-shelf solver CVXPY (Diamond & Boyd, 2016) in step 2 for the general $M > 1$ case, but closed-form solutions are also available for $M = 1$ ; these are given in Table 4 in Appendix A.6. ",
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+ "type": "text",
+ "text": "Algorithm 2 Information-limited certification ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Inputs: Query access to classifier $h$ , correct label $y$ , sampling access to reference distribution $\\rho$ divergences $f _ { i }$ and bounds $\\epsilon _ { i }$ , sample sizes $N , { \\tilde { N } }$ , confidence level $\\zeta$ . ",
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+ "text": "1: Sample $X ^ { 1 } , \\ldots , X ^ { N } \\sim \\rho$ and test3 whether $y = \\arg \\operatorname* { m a x } _ { y ^ { \\prime } \\in \\mathcal { V } } \\mathbb { P } _ { X \\sim \\rho } [ h ( X ) = y ^ { \\prime } ] .$ . \n2: Sample $X ^ { 1 } , \\dotsc , X ^ { \\tilde { N } } \\dotsc \\rho$ and compute4 a bound $\\mathbb { P } _ { X \\sim \\rho } [ h ( X ) = y ^ { \\prime } ] \\geq \\underline { { \\theta } } _ { a }$ with confidence $\\zeta$ . \n3: Obtain $o ^ { * }$ by solving Eq. 7 with $\\theta _ { a } \\gets \\underline { { \\theta _ { a } } }$ and $\\theta _ { b } \\gets 1 - \\theta _ { a }$ . \n4: If $o ^ { * } \\geq 0$ return CERTIFIED else return NOT CERTIFIED. ",
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+ {
+ "type": "text",
+ "text": "4 THEORETICAL ANALYSIS OF CERTIFICATION METHODS ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "We now present theoretical results characterizing our certification methods and show the following: ",
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+ "text": "1. For smoothed probabilistic classifiers, the full-information certificate dominates the informationlimited one. \n2. In the information-limited setting, if we define the $f$ -divergence relaxation $\\mathcal { D } _ { \\mathcal { F } }$ using HockeyStick divergences with specific parameters, then the computed certificate is provably tight. ",
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+ "type": "text",
+ "text": "4.1 ADVANTAGE OF FULL-INFORMATION CERTIFICATION ",
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+ "text": "Consider a soft binary classifier $H : \\mathcal { X } [ 0 , 1 ]$ that outputs the probability of label $+ 1$ and consider a point $x \\in \\mathcal { X }$ with $\\dot { H } ( x ) > 1 / 2$ . We define the specification $\\begin{array} { r } { \\phi ( \\bar { x } ) = H ( x ) - \\frac { 1 } { 2 } } \\end{array}$ . Then, the smoothed classifier $H _ { s } ( x ) = \\operatorname { E } _ { X \\sim \\mu ( x ) } [ H ( X ) ]$ predicts label $+ 1$ for all $x ^ { \\prime }$ with $\\left\\| x ^ { \\prime } - x \\right\\| ^ { - } \\leq \\epsilon$ if and only if $\\phi$ is full-information robustly certified at $\\mu ( x )$ with respect to $\\mathcal { D } _ { x , \\epsilon } = \\{ \\mu ( x ^ { \\prime } ) : \\| x ^ { \\prime } - x \\| \\le \\epsilon \\}$ . Note that the optimization in Theorem 1 depends on the full distribution of $\\phi ( X ) \\in [ - 1 / 2 , 1 / 2 ]$ , $X \\sim \\mu ( x )$ . On the other hand, to certify this robustness in the information-limited setting is equivalent to taking the specification $\\phi ( x ) = 1 [ \\dot { H } ( x ) > 1 / 2 ]$ (the indicator function of the event $\\mathbf { \\bar { \\boldsymbol { H } } } ( x ) \\mathbf { \\bar { \\boldsymbol { { \\Sigma } } } } > 1 / 2$ ), in which case the only information available is $\\begin{array} { r } { \\dot { \\theta _ { a } } = H _ { s } ( x ) = \\operatorname { E } _ { X \\sim \\mu ( x ) } [ H ( X ) ] } \\end{array}$ . ",
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+ "type": "text",
+ "text": "To compare the two approaches, consider the objective of Eq. 6 with a single $f$ -divergence constraint $D _ { f } ( \\nu \\| \\bar { \\rho } ) \\leq \\epsilon$ . Then, we have ",
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+ "type": "equation",
+ "img_path": "images/b5bd4199b09b54ebba6a7b65ec6c4c9d5a46818a3714bff2523de2703bc1fbe9.jpg",
+ "text": "$$\n\\begin{array} { l } { { \\displaystyle \\kappa - \\lambda \\epsilon - \\underset { X \\sim \\mu ( x ) } { \\mathrm { E } } [ f _ { \\lambda } ^ { * } ( \\kappa - \\phi ( X ) ) ] = \\kappa - \\lambda \\epsilon - \\underset { X \\sim \\mu ( x ) } { \\mathrm { E } } \\biggl [ f _ { \\lambda } ^ { * } \\biggl ( \\kappa - H ( X ) + \\frac 1 2 \\biggr ) \\biggr ] } } \\\\ { { \\displaystyle = \\kappa - \\lambda \\epsilon - \\underset { X \\sim \\mu ( x ) } { \\mathrm { E } } \\biggl [ f _ { \\lambda } ^ { * } \\biggl ( \\biggl ( \\kappa + \\frac 1 2 - 1 \\biggr ) H ( X ) + \\biggl ( \\kappa + \\frac 1 2 \\biggr ) ( 1 - H ( X ) ) \\biggr ) \\biggr ] } } \\\\ { { \\displaystyle \\geq \\kappa - \\lambda \\epsilon - \\underset { X \\sim \\mu ( X ) } { \\mathrm { E } } \\biggl [ f _ { \\lambda } ^ { * } \\biggl ( \\kappa - \\frac 1 2 \\biggr ) H ( X ) + f _ { \\lambda } ^ { * } \\biggl ( \\kappa + \\frac 1 2 \\biggr ) ( 1 - H ( X ) ) \\biggr ] } } \\\\ { { \\displaystyle = \\kappa - \\lambda \\epsilon - \\theta _ { a } f _ { \\lambda } ^ { * } \\biggl ( \\kappa - \\frac 1 2 \\biggr ) - ( 1 - \\theta _ { a } ) f _ { \\lambda } ^ { * } \\biggl ( \\kappa + \\frac 1 2 \\biggr ) \\enspace , } } \\end{array}\n$$",
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+ "type": "text",
+ "text": "where the third line follows from Jensen’s inequality. The proof of Theorem 2 shows that maximizing the final expression above with respect to $\\kappa , \\lambda$ is equivalent to the dual of the information-limited certification problem Eq. 7. Thus, the information-limited setting computes a weaker certificate than the full-information setting for soft classifiers: ",
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+ "type": "text",
+ "text": "Corollary 3. The optimization problem of Eq. 6 with the specification $\\phi$ defined above has an optimal value that is greater than or equal to that of the optimization problem defined in Eq. 7. ",
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+ "type": "text",
+ "text": "4.2 TIGHT RELAXATIONS FOR INFORMATION-LIMITED ROBUST CERTIFICATION ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "Ideally, we would like to certify robustness of specifications with respect to sets of the form $\\mathcal { D } _ { x , \\epsilon } =$ $\\{ \\mu ( x ^ { \\prime } ) : d ( x , x ^ { \\prime } ) \\leq \\epsilon \\}$ . The following result shows that the gap between the ideal $\\mathcal { D } _ { x , \\epsilon }$ and the tractable constraint sets $\\mathcal { D } _ { \\mathcal { F } }$ can be closed in the context of information-limited robust certification provided that we can measure hockey-stick divergences of every non-negative order $\\beta \\geq 0$ . The proof is given in Appendix A.7. ",
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+ "text": "Theorem 4. Let $\\theta _ { a } , \\theta _ { b } , \\rho , \\mathcal { D } , S$ be as in Definition 2.1. Define $\\begin{array} { r } { \\epsilon _ { \\beta } = \\operatorname* { m a x } _ { \\nu \\in \\mathcal { D } } D _ { \\mathrm { H S } , \\beta } ( \\nu \\| \\rho ) } \\end{array}$ for all $\\beta \\geq 0$ and let $\\beta _ { a } ^ { * } , \\beta _ { b } ^ { * }$ be chosen as follows: ",
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+ "text": "$$\n\\beta _ { a } ^ { * } , \\beta _ { b } ^ { * } = \\operatorname * { a r g m a x } _ { \\beta _ { a } \\ge \\beta _ { b } \\ge 0 } ~ 1 - \\beta _ { a } ( 1 - \\theta _ { a } ) - \\beta _ { b } \\theta _ { b } - \\left( \\epsilon _ { \\beta _ { a } } + [ 1 - \\beta _ { a } ] _ { + } \\right) - \\left( \\epsilon _ { \\beta _ { b } } + [ 1 - \\beta _ { b } ] _ { + } \\right) ~ .\n$$",
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+ "type": "text",
+ "text": "Define the constraint set ",
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+ "img_path": "images/05c76d3cca17d0e7601088a458bb148e7c8408896dc247c73aa909fd7020141b.jpg",
+ "text": "$$\n\\begin{array} { r } { \\mathcal { D } _ { \\mathrm { H S } } = \\left\\{ \\nu \\in \\mathcal { P } ( \\mathcal { X } ) : D _ { \\mathrm { H S } ; \\beta _ { \\alpha } ^ { * } } \\left( \\nu \\mid \\mid \\rho \\right) \\leq \\epsilon _ { \\beta _ { \\alpha } ^ { * } } \\right\\} \\cap \\left\\{ \\nu \\in \\mathcal { P } ( \\mathcal { X } ) : D _ { \\mathrm { H S } ; \\beta _ { b } ^ { * } } \\left( \\nu \\mid \\mid \\rho \\right) \\leq \\epsilon _ { \\beta _ { b } ^ { * } } \\right\\} \\ . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Then, $S$ is information-limited robustly certified at $\\rho$ with respect to $\\mathcal { D }$ if and only if $S$ is informationlimited robustly certified at $\\rho$ with respect to ${ \\mathcal { D } } _ { \\mathrm { H S } }$ . Thus, the optimal information-limited certificate in this case can be obtained by applying theorem 2 to ${ \\mathcal { D } } _ { \\mathrm { H S } }$ . ",
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+ {
+ "type": "text",
+ "text": "5 CONNECTIONS WITH PRIOR WORK ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "Table 1 summarizes the differences between our work and prior work in terms of the set of smoothing measures admitted, the offline computation cost of the certification procedure (which needs to be performed once for every possible perturbation size and choice of smoothing measure), the perturbations considered, whether they can use information beyond $\\theta _ { a } , \\theta _ { b }$ to improve the certificates and whether they compute optimal certificates for a given smoothing measure in the informationlimited setting. ",
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+ "type": "table",
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+ "table_caption": [
+ "Table 1: Comparison between black-box verification methods. \\* Technically, Lee et al. (2019) can handle smoothing measures such that the likelihood ratios take a finite set of values, but most practical instances of this correspond to finite-support distributions. "
+ ],
+ "table_footnote": [],
+ "table_body": "| Constraint on δ|Bound on Eq. 22 | Constraint on δ|Bound on Eq. 22 | Sampling from X ~ μ(0) |
| |δl ≤ε | E[f(exp(-eoll-X))]X~μ1(0） | Xi ~ Lap(0,1) iid |
| 1l2≤ e | E(o[f(exp(IX-eoll-X2))]X~μ2（0） | X=RuR~Γ(d,1)u ~U(aB2) |
| ≤e | E[f(exp(X-e1|-X)]X~μ(0） | X=RuR~ Γ(d+1,1)u EU(B) |
| |/ lnuc ≤ e | [f(exp(||U[[s]]vT-e[[eo]|lnuc- |slli)]s~μ1(0)U,v~u(O) | X=U[s]]VTs ~ μ1,U,V ~U(O) |
| ≤e | E [f(exp(|IU[[s]]vT-∈[e]|l, - |sll)]s~μ∞(x）U,v~u(O) | X=U[[s]IVTs ~ μ∞,U,V ~U(O) |
",
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+ "page_idx": 22
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+ {
+ "type": "text",
+ "text": "Efficient sampling The only other requirement for obtaining a certificate computationally is to be able to sample from $\\mu ( x )$ to estimate $\\theta _ { a } , \\theta _ { b }$ . Since $\\mu ( x )$ is log-concave, there are general purpose polynomial time algorithms for sampling from this measure. However, for most norms, more efficient methods exist, as outlined below. ",
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+ "type": "text",
+ "text": "The random variable $X \\sim \\mu ( x )$ can be obtained as $X = x + Z$ with $Z \\sim \\mu ( 0 )$ . Thus, to sample from $\\mu ( x )$ for any $x$ it is enough to be able to sample from $\\mu ( 0 )$ . For $\\left\\| \\cdot \\right\\| _ { 1 }$ , this reduces to sampling from a Laplace distribution which can be done easily. For $\\left\\| \\cdot \\right\\| _ { \\infty }$ , (Steinke & Ullman, 2016) give the following efficient sampling procedure: first sample $r$ from a Gamma distribution with shape $d + 1$ and mean $d + 1$ , i.e. $\\bar { r } \\sim \\bar { \\Gamma } ( \\bar { d } + 1 , 1 )$ , and then sample each $Z _ { i }$ , $i \\in [ d ]$ , uniformly from $[ - r , r ]$ . Theorem 9 gives a short proof of correctness for this procedure. Theorem 10 also has a similar result for the case of $\\lVert \\cdot \\rVert _ { 2 }$ and Table 5 lists the sampling procedures for several norms. ",
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+ "type": "text",
+ "text": "Theorem 9. The random variable $Z \\in \\mathbb { R } ^ { d }$ obtained by first sampling $R \\sim \\Gamma ( d + 1 , 1 )$ and then sampling each $Z _ { i }$ , $i \\in [ d ]$ , uniformly from $[ - R , R ]$ has density $\\propto e ^ { - \\| z \\| _ { \\infty } }$ . ",
+ "bbox": [
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+ "page_idx": 22
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+ {
+ "type": "text",
+ "text": "Proof. We first compute the normalization constant for a density of the form $\\propto e ^ { - \\| z \\| _ { \\infty } }$ as follows: ",
+ "bbox": [
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+ "img_path": "images/b760f4fd1dd9d5eb592c9d225b8a48b89541f8684a236f84bb1a025ea84d30c9.jpg",
+ "text": "$$\n\\int _ { \\mathbb { R } ^ { d } } e ^ { - \\| z \\| _ { \\infty } } d z = \\int _ { 0 } ^ { \\infty } \\left( \\int _ { \\mathbb { R } ^ { d } } 1 [ \\| z \\| _ { \\infty } = t ] d z \\right) e ^ { - t } d t = \\int _ { 0 } ^ { \\infty } 2 ^ { d } d t ^ { d - 1 } e ^ { - t } d t = d ! 2 ^ { d } \\ .\n$$",
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+ "type": "text",
+ "text": "Next we show the density of $Z$ satisfies $p _ { Z } ( z ) = e ^ { - \\| z \\| _ { \\infty } } / ( d ! 2 ^ { d } )$ by noting that conditioned on $R = r$ we have $p _ { Z | R = r } ( \\bar { z } ) = 1 [ \\| z \\| _ { \\infty } \\leq r ] / ( 2 r ) ^ { d }$ because of the uniform sampling used in each ",
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+ "type": "text",
+ "text": "coordinate, and integrating over $R$ sampled from a Gamma distribution with shape $d + 1$ and mean $d + 1$ yields ",
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+ "type": "equation",
+ "img_path": "images/0ac67627adfa178cf788354344afd6452f56064aa85ab1a8f8ad1232f6020a93.jpg",
+ "text": "$$\n_ { _ { D G } } ( z ) = \\int _ { 0 } ^ { \\infty } p _ { Z | R = r } ( z ) p _ { R } ( r ) d r = \\int _ { 0 } ^ { \\infty } \\frac { 1 [ \\| z \\| _ { \\infty } \\leq r ] } { ( 2 r ) ^ { d } } \\frac { r ^ { d } e ^ { - r } } { d ! } d r = \\frac { 1 } { d ! 2 ^ { d } } \\int _ { \\| z \\| _ { \\infty } } ^ { \\infty } e ^ { - r } d r = \\frac { e ^ { - \\| z \\| _ { \\infty } } } { d ! 2 ^ { d } } .\n$$",
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+ "type": "text",
+ "text": "Theorem 10. The random variable $Z \\in \\mathbb { R } ^ { d }$ obtained by first sampling $Z ^ { \\prime } \\sim \\mathcal { N } ( 0 , I )$ and $R \\sim$ $\\Gamma _ { p } ( d , 1 )$ and then taking $\\begin{array} { r } { Z = R \\frac { Z ^ { \\prime } } { \\Vert Z ^ { \\prime } \\Vert _ { 2 } } } \\end{array}$ has density $\\propto e ^ { - \\| z \\| _ { 2 } ^ { p } }$ . Here $\\Gamma _ { p } ( d , a )$ denotes the generalized Gamma distribution of order $p > 0$ with shape $d$ and scale $a$ . ",
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+ "type": "text",
+ "text": "Proof. First note that $\\begin{array} { r } { W = \\frac { Z ^ { \\prime } } { \\left. Z ^ { \\prime } \\right. _ { 2 } } \\sim \\mathcal { U } ( B _ { 2 } ) } \\end{array}$ ; i.e. it is uniform on the $\\ell _ { 2 }$ ball of radius 1. Therefore, $R W$ is uniform on the $\\ell _ { 2 }$ ball of radius $R$ and the conditional density of $Z$ given $R$ is given by $\\begin{array} { r } { p _ { Z | R = r } ( z ) = \\frac { 1 [ \\| z \\| _ { 2 } = r ] \\Gamma ( d / 2 ) } { 2 \\pi ^ { d / 2 } r ^ { d - 1 } } } \\end{array}$ . Since $R$ has density $p _ { R } ( r ) \\propto r ^ { d - 1 } e ^ { - r ^ { p } }$ , we get ",
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+ "img_path": "images/3c852d1069c9943e6818d72302d180128c04ff20a2be5610c75d134675538dc6.jpg",
+ "text": "$$\n\\begin{array} { l } { { \\displaystyle p _ { Z } ( z ) = \\int _ { 0 } ^ { \\infty } p _ { Z | R = r } ( z ) p _ { R } ( r ) d r } } \\\\ { { \\displaystyle \\propto \\int _ { 0 } ^ { \\infty } \\frac { 1 [ \\| z \\| _ { 2 } = r ] \\Gamma ( d / 2 ) } { 2 \\pi ^ { d / 2 } r ^ { d - 1 } } r ^ { d - 1 } e ^ { - r ^ { p } } d r } } \\\\ { { \\displaystyle \\propto e ^ { - \\| z \\| _ { 2 } ^ { p } } } . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "A.9 ALGORITHM FOR FULL-INFORMATION CERTIFICATION ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 23
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+ "type": "text",
+ "text": "In this section we describe the subroutines used in Algorithm 1. First we describe a generic procedure to provide high-probability confidence intervals for estimating expectations, then we give a detailed description of the subroutines. ",
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+ "type": "text",
+ "text": "A.9.1 HIGH-CONFIDENCE ESTIMATES OF EXPECTED VALUES ",
+ "text_level": 1,
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+ "text": "Let Let $Z _ { 1 } , \\dots , Z _ { N }$ be indepel mean be distributed random variables and the empirical variance be $R$ $m$ $\\begin{array} { r } { \\bar { Z } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\bar { Z _ { i } } } \\end{array}$ $\\begin{array} { r } { \\bar { \\sigma } ^ { 2 } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } ( Z _ { i } - \\bar { Z } ) ^ { 2 } } \\end{array}$ Applying Bernstein’s inequality to the sum and the sum of the squares of these random variables, we get the empirical Bernstein bound (Audibert et al., 2009), which states that with probability at least $1 - \\zeta$ , ",
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+ "text": "$$\n| \\bar { Z } - m | \\leq \\sqrt { \\frac { 2 \\bar { \\sigma } ^ { 2 } \\log ( 3 / \\zeta ) } { N } } + \\frac { 3 R \\log ( 3 / \\zeta ) } { t } .\n$$",
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+ "type": "text",
+ "text": "The main benefit of the above inequality is that as long as the variance of the sample $Z _ { 1 } , \\dots , Z _ { N }$ is small, the convergence rate becomes essentially $O ( 1 / N )$ instead of the standard $O ( 1 / \\sqrt { N } )$ . Also, since Eq. 23 only contains empirical quantities apart from the range $R$ , it can be used to obtain computable bounds for the expectation $\\mu$ : with probability at least $1 - \\zeta$ , ",
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+ "text": "$$\n\\bar { Z } - \\sqrt { \\frac { 2 \\bar { \\sigma } ^ { 2 } \\log ( 3 / \\zeta ) } { N } } - \\frac { 3 R \\log ( 3 / \\zeta ) } { N } \\leq m \\leq \\bar { Z } + \\sqrt { \\frac { 2 \\bar { \\sigma } ^ { 2 } \\log ( 3 / \\zeta ) } { N } } + \\frac { 3 R \\log ( 3 / \\zeta ) } { N } .\n$$",
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+ "type": "text",
+ "text": "A.9.2 SUBROUTINES FOR ALGORITHM 1 ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "The bound in Eq. 24 can be applied to approximate the expectation in Eq. 6 with high probability for given values of $\\lambda$ and $\\kappa$ . More specifically, if the function $f _ { \\lambda } ^ { * } ( \\kappa - \\phi ( \\cdot ) )$ is bounded with range $R$ , then taking $N$ samples $X _ { 1 } , \\ldots , X _ { N }$ independently from $\\rho$ , and defining $Z _ { i } = f _ { \\lambda } ^ { * } ( \\kappa - \\phi ( X _ { i } ) )$ , and $\\bar { Z }$ and $\\bar { \\sigma } ^ { 2 }$ as above, Eq. 24 implies that with probability at least $1 - \\zeta$ , ",
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+ "img_path": "images/1aee9b25cb65041a4d9c7abaece9b937ebe0f666843838ed656046d1bffa9793.jpg",
+ "text": "$$\n\\operatorname { E } _ { X \\sim \\rho } ^ { } [ f _ { \\lambda } ^ { * } ( \\kappa - \\phi ( X _ { i } ) ) ] \\leq \\bar { Z } + \\sqrt { \\frac { 2 \\bar { \\sigma } ^ { 2 } \\log ( 3 / \\zeta ) } { N } } + \\frac { 3 R \\log ( 3 / \\zeta ) } { N } .\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "Plugging in this bound to Eq. 6 gives a high-probability lower bound for the function to be maximized for any given $\\lambda$ and $\\kappa$ . ",
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+ "type": "text",
+ "text": "Details of the above procedures are given in Algorithm 3. We use an off-the-shelf convex optimization solver (Diamond & Boyd, 2016) in the ESTIMETEOPT subroutine. ",
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+ "type": "text",
+ "text": "Algorithm 3 Subroutines for Algorithm 1 ",
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+ "type": "text",
+ "text": "function ESTIMAT $\\mathrm { E O P T } ( \\rho , \\phi , N , \\{ f _ { i } \\} , \\{ \\epsilon _ { i } \\} )$ ",
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+ "text": "Sample $X ^ { 1 } , \\ldots , X ^ { N } \\sim \\rho$ and obtain $\\kappa ^ { * }$ and $\\lambda ^ { * }$ by solving Eq. 6 with $\\operatorname { E } _ { X \\sim \\rho } [ f _ { \\lambda } ^ { * } ( \\kappa - \\phi ( X ) ) ]$ replaced by $\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } f _ { \\lambda } ^ { * } \\big ( \\kappa - \\phi \\big ( X ^ { i } \\big ) \\big ) } \\end{array}$ and the additional constraints $f _ { \\lambda ^ { * } } ^ { * } ( \\kappa - a ) < \\infty , f _ { \\lambda ^ { * } } ^ { * } ( \\kappa - b ) <$ $\\infty$ . ",
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+ "text": "function UPPERCON $\\mathrm { F I D E N C E B O U N D } ( \\rho , \\phi , \\tilde { N } , \\{ f _ { i } \\} _ { i = 1 } ^ { M } , \\{ \\epsilon _ { i } \\} _ { i = 1 } ^ { M } , a , b , \\lambda , \\kappa , \\zeta )$ ",
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+ "text": "Sample $X ^ { 1 } , \\dotsc , X ^ { \\tilde { N } } \\dotsc \\rho$ and compute $Z ^ { i } \\gets f _ { \\lambda } ^ { * } \\big ( \\kappa - \\phi \\big ( X ^ { i } \\big ) \\big )$ for $i = 1 , \\ldots , \\tilde { N }$ ",
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+ "type": "equation",
+ "img_path": "images/7d72494dae529baa82ecf6ea3a088348f3a05f1141002622c8fdb97ada2ace9b.jpg",
+ "text": "$$\nE _ { u b } \\bar { Z } + \\sqrt { \\frac { 2 \\bar { \\sigma } ^ { 2 } \\log ( 3 / \\zeta ) } { \\tilde { N } } } + \\frac { 3 R \\log ( 3 / \\zeta ) } { \\tilde { N } } .\n$$",
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+ "text": "A.10 $\\ell _ { 0 }$ SMOOTHING MEASURE ",
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+ "text": "We can also handle discrete perturbations in our framework. A natural case to consider is $\\ell _ { 0 }$ perturbations. In this case, we assume that $\\mathcal { X } = A ^ { d }$ where ",
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+ "text": "$$\nA = \\{ 1 , \\ldots , K \\}\n$$",
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+ "type": "text",
+ "text": "is a discrete set. Then, we can choose ",
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+ "text": "$$\n\\mu ( x ) [ z ] = \\prod _ { i = 1 } ^ { d } p ^ { 1 [ z _ { i } = x _ { i } ] } { \\left( { \\frac { q } { K - 1 } } \\right) } ^ { 1 [ z _ { i } \\neq x _ { i } ] }\n$$",
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+ "type": "text",
+ "text": "where and $\\frac { q } { K - 1 }$ $p + q = 1$ denotes a uniform probability of switching it to a different value. In this case, it can be , $p \\geq q \\geq 0$ , and $p$ denotes the probability that the measure retains the value of $x$ shown that for every $\\alpha > 0$ that ",
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+ "img_path": "images/6cfaf7e14b50761ec8aa248e1c83a6b460e78bc0acba5504bb9f04f2bd97d7a3.jpg",
+ "text": "$$\nR _ { \\alpha } ( \\mu ( x ^ { \\prime } ) \\| \\mu ( x ) ) = \\| x - x ^ { \\prime } \\| _ { 0 } \\left( \\frac { \\log \\left( p \\left( \\frac { q } { ( K - 1 ) p } \\right) ^ { ( \\alpha ) } + \\frac { q } { K - 1 } \\left( \\frac { ( K - 1 ) p } { q } \\right) ^ { ( \\alpha ) } + \\left( \\frac { K - 2 } { K - 1 } \\right) q \\right) } { \\alpha - 1 } \\right)\n$$",
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+ "type": "text",
+ "text": "so that we can derive a certificate with respect to $\\ell _ { 0 }$ perturbations using any set of Renyi divergences ´ (or combinations of theses). ",
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+ "text": "This can be extended to structured discrete perturbations by introducing coupling terms between the perturbations: ",
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+ "text": "$$\n\\mu ( x ) [ z ] \\propto \\prod _ { i = 1 } ^ { d } p ^ { 1 [ z _ { i } = x _ { i } ] } \\bigg ( \\frac { q } { K - 1 } \\bigg ) ^ { 1 [ z _ { i } \\neq x _ { i } ] } \\exp \\left( \\sum _ { i = 1 } ^ { d - 1 } \\eta 1 [ z _ { i } = x _ { i } ] 1 [ z _ { i + 1 } = x _ { i + 1 } ] \\right) \\ .\n$$",
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+ "type": "text",
+ "text": "This would correlate perturbations between adjacent features (which for example may be useful to model correlated perturbations for time series data). Since this can be viewed as a Markov Chain, Renyi divergences between ´ $\\mu ( x ) , \\mu ( x ^ { \\prime } )$ are still easy to compute. ",
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+ "type": "text",
+ "text": "A.11 COMPARISON WITH LECUYER ET AL. (2019) ON $\\ell _ { 1 }$ PERTURBATIONS ",
+ "bbox": [
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+ "page_idx": 25
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+ "type": "text",
+ "text": "Here we compare our certificates to Lecuyer et al. (2019) on the MNIST, CIFAR-10 and ImageNet datasets. The smoothing distribution is as described in A.8; a zero mean Laplacian distribution with smoothing value defined by the scale of the distribution. We first describe the hyperparameters used in training and certification for each of the datasets. For all datasets, images were normalized into a [0,1] range. ",
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+ "type": "text",
+ "text": "MNIST hyperparameters: We trained a standard three layer CNN ReLU classifier for 50,000 steps with a batch size of 128 and a learning rate of 0.001. The smoothing value during training was set to 1.0. For certification we use $N \\overset { \\cdot } { = } 1 K , \\tilde { N } = 1 0 M , \\zeta = . 9 9$ , and sweep over a range of smoothing values between 0.5 and 1.5 and report the best certificate found. Certified accuracy is reported on 1,000 MNIST test set images. ",
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+ "type": "text",
+ "text": "CIFAR-10 hyperparameters: We trained a Wide ResNet classifier for 50,000 training steps with a batch size of 32 and a learning rate of 0.001. The smoothing value during training was set to 0.2. For certification we use $N = 1 K , \\tilde { N } = 1 M , \\zeta = . 9 9$ , and sweep over a range of smoothing values between 0.1 and 0.5 and report the best certificate found. Certified accuracy is reported on 1,000 CIFAR-10 test set images. ",
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+ "type": "text",
+ "text": "ImageNet hyperparameters: We trained a ResNet-152 classifier for 1 million training steps with a batch size of 16 and an initial learning rate of 0.1 that is decayed by a factor of ten every 25,000 steps. The smoothing value during training was set to 0.1. For certification we use $N = 1 K , \\tilde { N } =$ $1 0 0 K , \\zeta = . 9 9$ , and sweep over a range of smoothing values between 0.05 and 0.25 and report the best certificate found. Certified accuracy is reported on 500 ImageNet validation set images. ",
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+ "type": "image",
+ "img_path": "images/91632107e779409bf7a1723095f72ac1b1035eae560bf1b3ef171f5a1d760330.jpg",
+ "image_caption": [
+ "Figure 3: Certified accuracy under $\\ell _ { 1 }$ perturbations on CIFAR-10 and ImageNet. "
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+ "text": "Results for MNIST can be seen in Table 6, CIFAR-10 and ImageNet results are shown in Figure 3. \nWe significantly outperform Lecuyer et al. (2019) on all three datasets. ",
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diff --git a/parse/train/SJlKrkSFPH/SJlKrkSFPH_middle.json b/parse/train/SJlKrkSFPH/SJlKrkSFPH_middle.json
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index 0000000000000000000000000000000000000000..3b74677d7b10941a1b398cff58436b5d2715f635
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+ "html": "| Setting | Method | Test loss | Test accuracy |
| CIFAR-100, ResNet-18 | 10-fold CVGDPlug-in | 5.023 ± 0.0834.463 ± 0.0384.964 ± 0.057 | 1.59 ± 0.15 (top-5: 6.47 ± 0.52)3.68 ± 0.52 (top-5: 15.22 ± 1.24)1.68 ± 0.24 (top-5: 7.05 ± 0.71) |
| CIFAR-100, ResNet-34 | 10-fold CVGDPlug-in | 4.062 ± 0.0914.592 ± 0.1794.134± 0.185 | 9.62 ± 1.08 (top-5: 32.06 ± 1.47)10.41 ± 1.40 (top-5: 36.92 ± 1.20)10.11 ± 1.60 (top-5: 34.19 ± 2.10) |
| CIFAR-10, VGG-13 | 10-fold CVGDPlug-in | 2.126 ± 0.0632.519 ± 0.0622.195 ± 0.142 | 34.88 ± 1.6636.98 ± 0.7735.40 ± 3.00 |
| MNIST, AlexNet | 10-fold CVGDPlug-in | 0.656 ± 0.0340.654 ± 0.0310.639 ± 0.029 | 97.28 ± 0.2097.32 ± 0.2797.31 ± 0.15 |
+
+Table 6: The loss and accuracy on the test set comparing 10-fold cross validation and gradient disparity as early stopping criterion when the available dataset is noisy. In all the experiments, $50 \%$ of the available data has random labels. The corresponding curves during training are presented in Figure 9. The above results are obtained by stopping the optimization when the metric (either validation loss or gradient disparity) has been increased for five epochs from the beginning of training. The last row in each setting, which we call plug-in, refers to when we plug-in the epoch suggested by 10-fold CV and then report the test loss and accuracy at that epoch for a network trained on the entire set. In all these settings, using GD would still result in a higher test accuracy and therefore the advantage of GD over 10-fold CV is a better characterization of overfitting.
+
+test AUC score for this task. With further tuning, and combining the predictions found on two other MRI planes of each patient (axial and coronal), our final prediction results could even be improved.
+
+| Setting | Method | Test loss | Test accuracy |
| CIFAR-100, ResNet-18 | 10-fold CVGDPlug-in | 5.023 ± 0.0834.463 ± 0.0384.964 ± 0.057 | 1.59 ± 0.15 (top-5: 6.47 ± 0.52)3.68 ± 0.52 (top-5: 15.22 ± 1.24)1.68 ± 0.24 (top-5: 7.05 ± 0.71) |
| CIFAR-100, ResNet-34 | 10-fold CVGDPlug-in | 4.062 ± 0.0914.592 ± 0.1794.134± 0.185 | 9.62 ± 1.08 (top-5: 32.06 ± 1.47)10.41 ± 1.40 (top-5: 36.92 ± 1.20)10.11 ± 1.60 (top-5: 34.19 ± 2.10) |
| CIFAR-10, VGG-13 | 10-fold CVGDPlug-in | 2.126 ± 0.0632.519 ± 0.0622.195 ± 0.142 | 34.88 ± 1.6636.98 ± 0.7735.40 ± 3.00 |
| MNIST, AlexNet | 10-fold CVGDPlug-in | 0.656 ± 0.0340.654 ± 0.0310.639 ± 0.029 | 97.28 ± 0.2097.32 ± 0.2797.31 ± 0.15 |
",
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+ "type": "text",
+ "text": "Table 6: The loss and accuracy on the test set comparing 10-fold cross validation and gradient disparity as early stopping criterion when the available dataset is noisy. In all the experiments, $50 \\%$ of the available data has random labels. The corresponding curves during training are presented in Figure 9. The above results are obtained by stopping the optimization when the metric (either validation loss or gradient disparity) has been increased for five epochs from the beginning of training. The last row in each setting, which we call plug-in, refers to when we plug-in the epoch suggested by 10-fold CV and then report the test loss and accuracy at that epoch for a network trained on the entire set. In all these settings, using GD would still result in a higher test accuracy and therefore the advantage of GD over 10-fold CV is a better characterization of overfitting. ",
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+ "table_caption": [
+ "test AUC score for this task. With further tuning, and combining the predictions found on two other MRI planes of each patient (axial and coronal), our final prediction results could even be improved. "
+ ],
+ "table_footnote": [],
+ "table_body": " | | method | BAT | AVT |
| Random | Adaptive | Random | Adaptive |
| (n,K) | 8 | rej probs | E[stop] | rej probs | E[stop] | rej probs | E[stop] | rej probs | E[stop] |
| S1 | (200,5) | 0.000.100.150.200.250.30 | 5.2(1.1)27.5(2.2)45.5(2.5)62.5(2.4)80.2(2.0)88.2(1.6) | 1763(8) | 6.2(1.2) | 1762(8) | 0.2(0.2) | 1800(0) | 51.2(2.5) | 1586(11) |
| 1644(15) | 26.0(2.2) | 1647(14) | 6.8(1.3) | 1771(6) | 89.0(1.6) | 1344（9) |
| 1511(17)1391(18)1263(17)1176(14) | 44.2(2.5) | 1527(17) | 19.0(2.0) | 1718(10) | 98.0(0.7) | 1250(6) |
| 64.2(2.4) | 1383(18) | 42.8(2.5) | 1581(15) | 99.5(0.4) | 1196(6) |
| 78.8(2.0) | 1266(17) | 72.8(2.2) | 1394(17) | 99.8(0.2) | 1145(5)1091(5) |
| 88.8(1.6) | 1179(14) | 89.2(1.5) | 1216(14) | 100.0(0.0) |
| (20,50) | 0.000.100.150.200.250.30 | 5.8(1.2)27.5(2.2)45.5(2.5)67.0(2.4)83.8(1.8)92.0(1.4) | 1933(9)1771(18)1617(22)1446(22)1287(19)1182(16) | 5.0(1.1) | 1936(9) | 0.2(0.2) | 1978(1) | 51.5(2.5) | 1621(17) |
| 27.5(2.2) | 1771(18) | 8.0(1.4) | 1929(9) | 89.2(1.5) | 1276(13) |
| 45.8(2.5) | 1630(21) | 25.2(2.2) | 1826(15) | 97.8(0.7) | 1166(7) |
| 65.5(2.4)82.5(1.9)91.5(1.4) | 1459(22) | 53.8(2.5) | 1617(20) | 99.0(0.5) | 1105(6) |
| 1288(19) | 79.0(2.0) | 1379(19) | 99.8(0.2) | 1061(4)1030(2) |
| 1193(16) | 94.0(1.2) | 1187(15) | 100.0(0.0) |
| S2 | (200,5) | 0.000.100.150.200.250.30 | 5.2(1.1)18.2(1.9)29.0(2.3)40.5(2.5)50.5(2.5)62.5(2.4) | 1763(8)1692(12) | 6.2(1.2)16.8(1.9) | 1762(8)1699(12) | 0.2(0.2) | 1800(0) | 51.2(2.5)82.2(1.9) | 1586(11)1406(10) |
| 3.0(0.9) | 1788(3) |
| 1633(15)1559(17)1489(18)1407(18) | 25.2(2.2) | 1642(15) | 8.5(1.4) | 1762（7) | 91.8(1.4) | 1323（9) |
| 42.0(2.5)49.8(2.5)62.5(2.4) | 1548(17)1492(18)1413(18) | 17.2(1.9) | 1724(10) | 97.8(0.7) | 1257(7) |
| 33.0(2.4)53.5(2.5) | 1641(14) | 99.0(0.5)99.5(0.4) | 1218(6)1181(6) |
| 62.5(2.4) | 1522(16) |
| (20,50) | 0.000.100.150.200.250.30 | 5.8(1.2)19.0(2.0)28.5(2.3)39.0(2.4)50.7(2.5)65.2(2.4) | 1933(9)1839(16) | 5.0(1.1) | 1936(9) | 0.2(0.2) | 1978(1) | 51.5(2.5) | 1621(17)1360(15) |
| 19.0(2.0) | 1837(16) | 3.5(0.9) | 1961(5) | 81.0(2.0) |
| 1763(19)1680(21) | 28.0(2.2) | 1771(18) | 11.5(1.6) | 1911(10) | 90.8(1.4) | 1256(12)1171(8)1124(6)1092(5) |
| 1680(21) | 41.8(2.5) | 1685(20) | 24.0(2.1) | 1830(15) | 97.5(0.8) |
| 1592(22)1479(22) | 52.5(2.5) | 1568(22) | 44.2(2.5) | 1688(19)1539(20) | 99.0(0.5)99.2(0.4) |
| 63.7(2.4) | 1481(22) | 63.5(2.4) |
+
+
+Figure 5: Alpha spending functions when $\theta = 0 . 5 , \gamma = 1 . 0$ .
\ No newline at end of file
diff --git a/parse/train/yxafu6ZtUux/yxafu6ZtUux_content_list.json b/parse/train/yxafu6ZtUux/yxafu6ZtUux_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..776f24d4639cfe4878865c214a8f404a5b4bd7d7
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+[
+ {
+ "type": "text",
+ "text": "AN ONLINE SEQUENTIAL TEST FOR QUALITATIVETREATMENT EFFECTS",
+ "text_level": 1,
+ "bbox": [
+ 176,
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+ ],
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+ },
+ {
+ "type": "text",
+ "text": "Anonymous authors Paper under double-blind review ",
+ "bbox": [
+ 183,
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+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "ABSTRACT ",
+ "text_level": 1,
+ "bbox": [
+ 454,
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+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Tech companies (e.g., Google or Facebook) often use randomized online experiments and/or A/B testing primarily based on the average treatment effects to compare their new product with an old one. However, it is also critically important to detect qualitative treatment effects such that the new one may significantly outperform the existing one only under some specific circumstances. The aim of this paper is to develop a powerful testing procedure to efficiently detect such qualitative treatment effects. We propose a scalable online updating algorithm to implement our test procedure. It has three novelties including adaptive randomization, sequential monitoring, and online updating with guaranteed type-I error control. We also thoroughly examine the theoretical properties of our testing procedure including the limiting distribution of test statistics and the justification of an efficient bootstrap method. Extensive empirical studies are conducted to examine the finite sample performance of our test procedure. ",
+ "bbox": [
+ 233,
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+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "1 INTRODUCTION ",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 0
+ },
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+ "type": "text",
+ "text": "Tech companies use randomized online experiments, or A/B testing to compare their new product with a well-established one. Most works in the literature focus on the average treatment effects (ATE) between the new and existing products (see Kharitonov et al., 2015; Johari et al., 2015; 2017; Yang et al., 2017; Ju et al., 2019, and the references therein). In addition to ATE, sometimes we are interested in locating the subgroup (if exists) that the new product performs significantly better than the existing one, as early as possible. Consider a ride-hailing company (e.g., Uber). Suppose some passengers are in the recession state (at a high risk of stopping using the companys app) and the company comes up with certain strategy to intervene the recession process. We would like to if there are some subgroups that are sensitive to the strategy and pin-point these subgroups if exists. It motivates us to consider the null hypothesis that the treatment effect is nonpositive for all passenger. ",
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+ "text": "Such a null hypothesis is closely related to the notion of qualitative treatment effects in medical studies (QTE, Gail & Simon, 1985; Roth & Simon, 2018; Shi et al., 2020a), and conditional moment inequalities in economics (see for example, Andrews & Shi, 2013; 2014; Chernozhukov et al., 2013; Armstrong & Chan, 2016; Chang et al., 2015; Hsu, 2017). However, these tests are computed offline and might not be suitable to implement in online settings. Moreover, it is assumed in those papers that observations are independent. In online experiment, one may wish to adaptively allocate the treatment based on the observed data stream in order to maximize the cumulative reward or to detect the alternative more efficiently. The independence assumption is thus violated. In addition, an online experiment is desired to be terminated as early as possible in order to save time and budget. Sequential testing for qualitative treatment effects has been less explored. ",
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+ "text": "In the literature, there is a line of research on estimation and inference of the heterogeneous treatment effects (HTE) (Athey & Imbens, 2016; Taddy et al., 2016; Wager & Athey, 2018; Yu et al., 2020). In particular, Yu et al. (2020) proposed an online test for HTE. We remark that HTE and QTE are related yet fundamentally different hypotheses. There are cases where HTE exists whereas QTE does not. See Figure 1 for an illustration. Consequently, applying their test will fail in our setting. ",
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+ "text": "The contributions of this paper are summarized as follows. First, we propose a new testing procedure for treatment comparison based on the notion of QTE. When the null hypothesis is not rejected, the new product is no better than the control for any realization of covariates, and thus it is not useful at all. Otherwise, the company could implement different products according to the auxiliary covariates observed, to maximize the average reward obtained. We remark that there are plenty cases where the treatment effects are always nonpositive (see Section 5 of Chang et al., 2015; Shi et al., 2020a). A by-product of our test is that it yields a decision rule to implement personalization when the null is rejected (see Section 3.1 for details). Although we primarily focus on QTE in this paper, our procedure can be easily extended to testing ATE as well (see Appendix D for details). ",
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+ "Figure 1: Plots demonstrating QTE. $X$ denotes the observed covariates, $A$ denotes the received treatment and $Y$ denotes the associated reward. In the ride-hailing example, $X$ is a feature vector describing the characteristics of a passenger, $A$ is a binary strategy indicator and $Y$ is the passenger’s number of rides in the following two weeks. In the left panel, the treatment effect does not depend on $X$ . Neither HTE nor QTE exists in this case. In the middle panel, HTE exists. However, the treatment effect is always negative. As such, QTE does not exist. In the right penal, both QTE and HTE exist. "
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+ "text": "Second, we propose a scalable online updating algorithm to implement our test. To allow for sequential monitoring, our procedure leverages idea from the $\\alpha$ spending function approach (Lan & DeMets, 1983) originally designed for sequential analysis in a clinical trial (see Jennison & Turnbull, 1999, for an overview). Classical sequential tests focus on ATE. The test statistic at each interim stage is asymptotically normal and the stopping boundary can be recursively updated via numerical integration. However, the limiting distribution of the proposed test statistic does not have a tractable analytical form, making the numerical integration method difficult to apply. To resolve this issue, we propose a scalable bootstrap-assisted procedure to determine the stopping boundary. ",
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+ "text": "Third, we adopt a theoretical framework that allows the maximum number of interim analyses $K$ to diverge as the number of observations increases, since tech companies might analyze the results every few minutes (or hours) to determine whether to stop the experiment or continue collecting more data. It is ultimately different from classical sequential analysis where $K$ is fixed. Moreover, the derivation of the asymptotic property of the proposed test is further complicated due to the adaptive randomization procedure, which makes observations dependent of each other. Despite these technical challenges, we establish a nonasymptotic upper bound on the type-I error rate by explicitly characterizing the conditions needed on randomization procedure, $K$ and the number of samples observed at the initial decision point to ensure the validity of our test. ",
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+ "type": "text",
+ "text": "2 BACKGROUND AND PROBLEM FORMULATION",
+ "text_level": 1,
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+ "text": "We propose a potential outcome framework (Rubin, 2005) to formulate our problem. Suppose that we have two products including the control and the treatment. The observed data at time point $t$ consists of a sequence of triples $\\{ ( X _ { i } , A _ { i } , Y _ { i } ) \\} _ { i = 1 } ^ { N ( t ) }$ , where $N ( \\cdot )$ is a counting process that is independent of the data stream $\\{ ( X _ { i } , A _ { i } , Y _ { i } ) \\} _ { i = 1 } ^ { + \\infty }$ , $A _ { i }$ is a binary random variable indicating the product executed for the $i$ -th experiment, $X _ { i } \\in \\mathbb { R } ^ { p }$ denotes the associated covariates, and $Y _ { i }$ stands for the associated reward (the larger the better by convention). We allow $A _ { i }$ to depend on $X _ { i }$ and past observations $\\{ ( X _ { j } , A _ { j } , Y _ { j } ) \\} _ { j < i }$ so that the randomization procedure can be adaptively changed. In corresponding product is executed for the are independently and identically distribut addition, define $Y _ { i } ^ { * } ( 0 )$ and $Y _ { i } ^ { * } ( 1 )$ to be the potential outcome that would have been observed if the $i$ -th experimd copies of $\\{ ( X _ { i } , Y _ { i } ^ { * } ( 0 ) , Y _ { i } ^ { * } ( 1 ) ) \\} _ { i = 1 } ^ { + \\infty }$ $( X , Y ^ { * } \\bar { ( 0 ) } , Y ^ { * } ( 1 ) )$ $\\mathbb { X }$ $X$ and $\\bar { Q } _ { 0 } ( x , a ) \\stackrel { \\cdot } { = } \\mathrm { E } \\{ Y ^ { * } ( a ) | X \\stackrel { \\cdot } { = } x \\}$ for $a = 0 , 1$ , we focus on testing the following hypotheses: ",
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+ "text": "$$\nH _ { 0 } : Q _ { 0 } ( x , 1 ) \\leq Q _ { 0 } ( x , 0 ) , \\forall x \\in \\mathbb { X } { \\mathrm { ~ ~ v e r s u s ~ } } H _ { 1 } : Q _ { 0 } ( x , 1 ) > Q _ { 0 } ( x , 0 ) , \\exists x \\in \\mathbb { X } .\n$$",
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+ "text": "Notice that when there are no covariates, i.e., $\\mathbb { X } = \\emptyset$ , the hypotheses are reduced to $H _ { 0 } : \\tau _ { 0 } \\leq 0$ versus $H _ { 1 } : \\tau _ { 0 } > 0$ , where $\\tau _ { 0 }$ corresponds to ATE, i.e, $\\tau _ { 0 } \\stackrel { } { = } \\mathrm { E } \\{ Y ^ { \\ast } ( 1 ) - Y ^ { \\ast } ( 0 ) \\}$ . In general, we require $\\mathbb { X }$ to be a compact set. We consider a large linear approximation space $\\mathcal { Q }$ for the conditional mean function $Q _ { 0 }$ . Specifically, let $\\mathcal { Q } = \\{ Q \\in \\langle x , a ; \\beta _ { 0 } , \\hat { \\beta _ { 1 } } \\rangle = \\varphi ^ { \\top } ( x ) \\hat { \\beta _ { a } } : \\beta _ { 0 } , \\beta _ { 1 } \\in \\mathbb { R } ^ { q } \\}$ be the approximation space, where $\\varphi ( x )$ is a $q$ -dimensional vector composed of basis functions on $\\mathbb { X }$ . The dimension $q$ is allowed to diverge with the number of observations in order to alleviate the effects of model misspecification. The use of linear approximation space simplifies the computation of our testing procedure. When $Q _ { 0 }$ is well approximated, it suffices to test ",
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+ "text": "$$\nH _ { 0 } : \\varphi ^ { \\top } ( x ) ( \\beta _ { 1 } ^ { * } - \\beta _ { 0 } ^ { * } ) \\leq 0 , \\forall x \\in \\mathbb { X } \\mathrm { ~ \\mathrm { ~ v e r s u s ~ } ~ } H _ { 1 } : \\varphi ^ { \\top } ( x ) ( \\beta _ { 1 } ^ { * } - \\beta _ { 0 } ^ { * } ) > 0 , \\exists x \\in \\mathbb { X } .\n$$",
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+ "text": "For clarity, here we assume $Q _ { 0 } ( x , a ) = Q ( x , a ; \\beta _ { 0 } ^ { * } , \\beta _ { 1 } ^ { * } )$ for some $\\beta _ { 0 } ^ { * }$ and $\\beta _ { 1 } ^ { * }$ . In Appendix B, we allow the approximation error $\\begin{array} { r } { \\operatorname* { i n f } _ { \\beta _ { 0 } , \\beta _ { 1 } \\in \\mathbb { R } ^ { p } } \\operatorname* { s u p } _ { x \\in \\mathbb { X } , a \\in \\{ 0 , 1 \\} } | Q _ { 0 } ( x , a ) - Q ( x , a ; \\beta _ { 0 } , \\beta _ { 1 } ) | } \\end{array}$ to be nonzero. ",
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+ "text": "Let ${ \\mathcal { F } } _ { j }$ denote the sub-dataset $\\{ ( X _ { i } , A _ { i } , Y _ { i } ) \\} _ { 1 \\leq i \\leq j }$ for $j \\geq 1$ and $\\mathcal { F } _ { 0 } = \\emptyset$ . Throughout this paper, we assume that the following two assumptions hold. ",
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+ "text": "(A1) $Y _ { i } = A _ { i } Y _ { i } ^ { * } ( 1 ) + ( 1 - A _ { i } ) Y _ { i } ^ { * } ( 0 )$ for $\\forall i \\geq 1$ . (A2) $A _ { i }$ is independent of $Y _ { i } ^ { * } ( 0 ) , Y _ { i } ^ { * } ( 1 )$ , $\\{ ( X _ { k } , Y _ { k } ^ { * } ( 0 ) , Y _ { k } ^ { * } ( 1 ) ) \\} _ { k > i }$ given $X _ { i }$ and $\\mathcal { F } _ { i - 1 }$ , for any $i$ ",
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+ "text": "Assumption (A1) is referred to be the stable unit treatment value assumption (Rubin, 1974) and Assumption (A2) is the sequential randomization assumption (Zhang et al., 2013) and is automatically satisfied in a randomized study where the treatments are independently generated of the observed data. (A2) essentially assumes there is no unmeasured confounders. These assumptions guarantee that both regression coefficients (defined through potential outcomes) are estimable from the observed dataset as shown in the following lemma. ",
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+ "text": "Lemma 1 Let $\\mathbb { I } ( \\cdot )$ denotes the indicator function. Under (A1)-(A2), we have ",
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+ "text": "$$\nE [ \\mathbb { I } ( A _ { i } = a ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } ^ { * } \\} ] = 0 , \\forall a \\in \\{ 0 , 1 \\} , i \\geq 1 .\n$$",
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+ "text": "3 ONLINE SEQUENTIAL TESTING FOR QTE ",
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+ "text": "3.1 TEST STATISTICS AND THEIR LIMITING DISTRIBUTION ",
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+ "text": "We first present our test statistic for testing $H _ { 0 }$ . In view of Lemma 1, we estimate $\\beta _ { a }$ by using the ordinary least squares estimator ",
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+ "text": "$$\n\\widehat { \\beta } _ { a } ( t ) = { \\widehat \\Sigma } _ { a } ^ { - 1 } ( t ) \\left\\{ \\frac { 1 } { N ( t ) } \\sum _ { i = 1 } ^ { N ( t ) } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) Y _ { i } \\right\\}\n$$",
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+ "text": "at each time point $t$ for $a \\in \\{ 0 , 1 \\}$ , where $\\begin{array} { r } { \\widehat { \\Sigma } _ { a } ( t ) = N ^ { - 1 } ( t ) \\sum _ { i = 1 } ^ { N ( t ) } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) } \\end{array}$ . A generalized inverse might be used even if $\\widehat { \\Sigma } _ { a } ( t )$ is not invertible. Consider the following test statistic $\\begin{array} { r } { S ( t ) = \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) \\{ \\widehat \\beta _ { 1 } ( t ) - \\widehat \\beta _ { 0 } ( t ) \\} } \\end{array}$ . Under $H _ { 0 }$ , we expect $S ( t )$ to be small. A large $S ( t )$ can be interpreted as the evidence against $H _ { 0 }$ . As such, we reject $H _ { 0 }$ for large $S ( t )$ . We remark that when $H _ { 0 }$ is rejected, we can apply the decision rule $d ( x ) = \\arg \\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\varphi ^ { \\top } ( x ) \\widehat { \\beta } _ { a } ( t )$ for personalized recommendation. ",
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+ "text": "To determine the rejection region, we next discuss the limiting distribution of $S ( t )$ . Under $H _ { 0 }$ , ",
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+ "text": "$$\n\\begin{array} { r } { S ( t ) \\le \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) \\{ \\widehat { \\beta } _ { 1 } ( t ) - \\beta _ { 1 } ^ { * } - \\widehat { \\beta } _ { 0 } ( t ) + \\beta _ { 0 } ^ { * } \\} + \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) ( \\beta _ { 1 } ^ { * } - \\beta _ { 0 } ^ { * } ) } \\\\ { \\le \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) \\{ \\widehat { \\beta } _ { 1 } ( t ) - \\beta _ { 1 } ^ { * } - \\widehat { \\beta } _ { 0 } ( t ) + \\beta _ { 0 } ^ { * } \\} . } \\end{array}\n$$",
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+ "text": "Both equalities hold when $\\beta _ { 0 } ^ { * } ~ = ~ \\beta _ { 1 } ^ { * }$ . Suppose there exists some function $\\pi ^ { * } ( \\cdot , \\cdot )$ defined on $\\{ 0 , 1 \\} \\times \\mathbb { X }$ that satisfies $\\begin{array} { r } { { \\mathrm { E } } ^ { X } | \\sum _ { i = 1 } ^ { n } n ^ { - 1 } \\pi _ { i - 1 } ( a , X ) - \\pi ^ { * } ( a , X ) | \\ \\stackrel { P } { \\to } \\ 0 , \\forall a \\ \\in \\ \\{ 0 , 1 \\} } \\end{array}$ as $n \\infty$ , where $\\pi _ { n } ( \\cdot , \\cdot ) = \\operatorname* { P r } ( A _ { n } = a | X _ { n } = x , { \\mathcal { F } } _ { n - 1 } )$ , and the expectation $\\operatorname { E } ^ { X }$ is taken with respect to $X$ . This condition implies that the treatment assignment mechanism cannot be arbitrary (see the discussion below Theorem 1 for details). Then we will show ",
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+ "text": "$$\nB ( t ) \\equiv \\sqrt { N ( t ) } \\{ \\widehat { \\beta } _ { 1 } ( t ) - \\beta _ { 1 } ^ { * } - \\widehat { \\beta } _ { 0 } ( t ) + \\beta _ { 0 } ^ { * } \\} \\stackrel { d } { } N ( 0 , \\sum _ { a \\in \\{ 0 , 1 \\} } \\Sigma _ { a } ^ { - 1 } \\Phi _ { a } \\Sigma _ { a } ^ { - 1 } ) , \\mathrm { a s } N ( t ) \\infty ,\n$$",
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+ "text": "where $\\Sigma _ { a } = \\mathrm { E } \\pi ^ { * } ( a , X ) \\varphi ( X ) \\varphi ^ { \\top } ( X )$ , $\\Phi _ { a } = \\mathrm { E } \\pi ^ { * } ( a , X ) \\sigma ^ { 2 } ( a , X ) \\varphi ( X ) \\varphi ^ { \\top } ( X )$ , and $\\sigma ^ { 2 } ( a , x ) =$ $\\operatorname { E } [ \\{ Y ^ { * } ( a ) - \\varphi ^ { \\top } ( X ) \\beta _ { a } \\} ^ { 2 } | X = x ]$ , for any $x \\in \\mathbb { X }$ . According to equation 3, the right-hand-side (RHS) of equation 2 is to converge in distribution to the maximum of some Gaussian random variables. This observation forms the basis of our test. ",
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+ "text": "We next discuss the sequential implementation of our test. Assume that the interim analyses are conducted at time points $t _ { 1 } , t _ { 2 } , \\ldots , t _ { K } \\in [ 0 , \\ldots , T ]$ such that $0 < t _ { 1 } < t _ { 2 } < \\cdots < t _ { K } = T$ . We allow $K$ to grow with the number of observations. In the most extreme case, one may set $t _ { k } = \\operatorname* { i n f } _ { t } \\{ N ( t ) \\geq N ( t _ { k - 1 } ) + 1 \\} , \\forall k \\geq 2$ . That is, we make a decision regarding the null hypothesis upon the arrival of each observation. In addition, we assume that $t _ { 1 }$ is large so that there are enough number of samples $N ( t _ { 1 } )$ to guarantee the validity of the normal approximation for $B ( t _ { 1 } )$ . We remark that in typical tech companies such as Amazon, Facebook, etc., massive data are collected even within a short time interval. Large sample approximation is validated in these applications. ",
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+ "text": "To guarantee our test controls the type-I error, we reject $H _ { 0 }$ and terminate the experiment at $t _ { k }$ if $\\sqrt { N ( t _ { k } ) } S ( t _ { k } ) \\geq z _ { k }$ for some $k = 1 , \\ldots , K$ with some suitably chosen $z _ { 1 } , \\dotsc , z _ { K } > 0$ that satisfy ",
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+ "text": "$$\n\\operatorname* { P r } \\bigg ( \\operatorname* { m a x } _ { k \\in \\{ 1 , \\dots , K \\} } \\{ \\sqrt { N ( t _ { k } ) } S ( t _ { k } ) - z _ { k } \\} > 0 \\bigg ) \\leq \\alpha + o ( 1 )\n$$",
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+ "text": "for a given significance level $\\alpha > 0$ under $H _ { 0 }$ . In view of equation 2, it suffices to find $\\{ z _ { k } \\} _ { k }$ that satisfy ",
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+ "text": "$$\n\\operatorname* { P r } \\left\\{ \\operatorname* { m a x } _ { k \\in \\{ 1 , \\ldots , K \\} } \\left( \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) B ( t _ { k } ) - z _ { k } \\right) > 0 \\right\\} \\leq \\alpha + o ( 1 ) ,\n$$",
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+ "text": "where the stochastic process $B ( \\cdot )$ is defined in equation 3. ",
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+ "text": "To determine $\\{ z _ { k } \\} _ { k }$ , we need to derive the asymptotic distribution of the left-hand-side (LHS) of equation 4. To this end, define a mean-zero Gaussian process $G ( t )$ with covariance function ",
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+ "text": "$$\n\\mathrm { C o v } ( G ( t ) , G ( t ^ { \\prime } ) ) = N ^ { 1 / 2 } ( t ) N ^ { - 1 / 2 } ( t ^ { \\prime } ) \\sum _ { a \\in \\{ 0 , 1 \\} } \\Sigma _ { a } ^ { - 1 } \\Phi _ { a } \\Sigma _ { a } ^ { - 1 } , \\forall 0 < t \\leq t ^ { \\prime } .\n$$",
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+ "text": "In the following, we show that the LHS of equation 4 can be uniformly approximated by $G ( \\cdot )$ , for any $\\{ z _ { k } \\} _ { k = 1 , . . . , K }$ . To establish our theoretical results, we need some regularity conditions on $\\dot { \\varphi } ( \\cdot )$ . To save space, we summarize these assumptions in (A3) and put them in Appendix $\\mathbf { B }$ . ",
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+ "text": "Theorem 1 Assume (A1)-(A3) hold. For $a = 0 , 1$ , assume $\\operatorname* { i n f } _ { x \\in \\mathbb { X } } \\pi ^ { * } ( a , x ) > 0$ and $\\left| Y ^ { * } ( a ) \\right|$ is bounded almost surely. Assume there exists some $0 < \\alpha _ { 0 } \\le 1$ such that for any sequence $\\{ j _ { n } \\} _ { n }$ that satisfies $j _ { n } ^ { \\alpha _ { 0 } } / \\log ^ { \\alpha _ { 0 } } j _ { n } \\gg q ^ { 2 }$ , the following event occurs with probability at least $1 - O \\big ( j _ { n } ^ { - \\alpha _ { 0 } } \\big )$ , ",
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+ "text": "$$\n\\operatorname* { s u p } _ { a \\in \\{ 0 , 1 \\} } E \\left| \\sum _ { i = 1 } ^ { k } \\{ \\pi _ { i - 1 } ( a , x ) - \\pi ^ { * } ( a , x ) \\} \\right| \\leq O ( 1 ) q k ^ { 1 - \\alpha _ { 0 } } \\log ^ { \\alpha _ { 0 } } k , \\forall k \\geq j _ { n } ,\n$$",
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+ "text": "where $O ( 1 )$ denotes some positive constant. Assume $N ^ { \\alpha _ { 0 } } ( t _ { 1 } ) / \\log ^ { \\alpha _ { 0 } } N ( t _ { 1 } ) \\gg q ^ { 2 }$ and $N ( t _ { 1 } ) \\gg$ $\\log N ( T )$ almost surely. Then conditional on the counting process $N ( \\cdot )$ , there exists some constant $c > 0$ such that ",
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+ "text": "$$\n\\begin{array} { r l r } { { \\operatorname* { s u p } _ { 1 , \\ldots , z _ { K } } P r \\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } ( \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) B ( t _ { K } ) - z _ { k } ) > 0 \\} - P r \\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } ( \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { K } ) - z _ { k } ) \\} } } \\\\ & { } & { \\leq c q ^ { 3 / 4 } N ^ { - 1 / 8 } ( t _ { 1 } ) \\log ^ { 1 5 / 8 } \\{ K N ( t _ { 1 } ) \\} + q N ^ { - \\alpha _ { 0 } / 3 } ( t _ { 1 } ) \\log ^ { ( 5 + \\alpha _ { 0 } ) / 3 } \\{ K N ( t _ { 1 } ) \\} } \\end{array}\n$$",
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+ "text": "Theorem 1 implies that the approximation error depends on the number of observations obtained up to the first decision point $\\bar { N ( t _ { 1 } ) }$ , the maximum number of interim analyses $K$ , the total number of basis functions $q$ , and $\\alpha _ { 0 }$ , which characterizes the convergence rate of the treatment assignment mechanism $\\scriptstyle \\sum _ { i = 1 } ^ { n } { n ^ { - 1 } \\pi _ { i - 1 } }$ . Clearly, the error will decay to zero when the followings hold with probability tending to 1, ",
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+ "text": "$$\nq = O ( N ^ { \\alpha _ { * } } ( t _ { 1 } ) ) , \\mathrm { f o r s o m e } 0 \\leq \\alpha ^ { * } < \\operatorname* { m i n } ( 1 / 6 , \\alpha _ { 0 } / 3 ) ,\n$$",
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+ "text": "$$\n\\log ( K ) \\ll \\operatorname * { m i n } \\{ N ^ { 1 / 1 5 - 2 \\alpha ^ { * } / 5 } ( t _ { 1 } ) , N ^ { ( \\alpha _ { 0 } - 3 \\alpha ^ { * } ) / ( 5 + \\alpha _ { 0 } ) } ( t _ { 1 } ) \\} .\n$$",
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+ "text": "In Appendix C, we show that $\\alpha _ { 0 } = 1 / 2$ , when an $\\epsilon$ -greedy strategy is used for randomization to balance the trade-off between exploration and exploitation. In this case, 6 requires $q$ to grow at a slower rate than $N ^ { 1 / 6 } ( t _ { 1 } )$ . This condition is automatically satisfied when $q$ is bounded. Condition 7 is satisfied when $K$ grows polynomially fast with respect to $N ( t _ { 1 } )$ . In addition to $\\epsilon$ -greedy, other adaptive allocation procedures (e.g., upper confidence bound or Thompson sampling) could be applied as well. ",
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+ "text": "As discussed in the introduction, the derivation of Theorem 1 is nontrivial. One way to obtain the magnitude of the approximation error is to apply the strong approximation theorem for multidimensional martingales (see Morrow & Philipp, 1982; Zhang, 2004). However, the rate of approximation typically depends on the dimension and decays fast as the dimension increases. To derive Theorem 1, we view $\\{ \\varphi ^ { \\top } ( x ) B ( t _ { K } ) \\} _ { x \\in \\mathbb { X } , k \\in \\{ 1 , \\dots , \\kappa \\} }$ as a high-dimensional martingale and adopt the Gaussian approximation techniques that have been recently developed by Belloni $\\&$ Oliveira (2018). In view of equation 2, an application of Theorem 1 yields the following result. ",
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+ "text": "Theorem 2 Assume that the conditions of Theorem $^ { l }$ hold, equation $6$ and equation 7 hold with probability tending to 1. Then for any $z _ { 1 } , \\ldots , z _ { k }$ that satisfy ",
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+ "text": "$$\nP r \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - z _ { k } \\right) > 0 \\right\\} = \\alpha + o ( 1 ) ,\n$$",
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+ "type": "text",
+ "text": "as $N ( t _ { 1 } )$ diverges to infinity, we have under $H _ { 0 }$ , ",
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+ "text": "$$\nP r \\left( \\operatorname* { m a x } _ { k \\in \\{ 1 , \\dots , K \\} } \\{ \\sqrt { N ( t _ { k } ) } S ( t _ { k } ) - z _ { k } \\} > 0 \\right) \\leq \\alpha + o ( 1 ) .\n$$",
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+ "text": "The above equality holds when $\\beta _ { 0 } ^ { * } = \\beta _ { 1 } ^ { * }$ . ",
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+ "text": "Theorem 2 suggests that the type-I error rate of the proposed test can be well controlled. It remains to find critical values $\\left\\{ z _ { k } \\right\\} _ { 1 \\leq k \\leq K }$ that satisfy equation 8. In the next section, we propose a bootstrapassisted procedure to determine these critical values. ",
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+ "type": "text",
+ "text": "3.2 BOOTSTRAP STOPPING BOUNDARY ",
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+ "text": "We first outline a method based on the wild bootstrap (Wu, 1986) to approximate the limiting distribution of $\\{ S ( t _ { k } ) \\} _ { k }$ . Then we discuss its limitation and present our proposal, a scalable bootstrap algorithm to determine the stopping boundary. ",
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+ "text": "The idea is to generate bootstrap samples $\\{ \\widehat { \\beta } _ { a } ^ { \\mathrm { M B } } ( t _ { k } ) \\} _ { a , k }$ that have asymptotically the same joint distribution as $\\{ \\widehat { \\beta } _ { a } ( t _ { k } ) - \\beta _ { a } ^ { * } \\} _ { a , k }$ . Then the joint distribution of $\\{ S ( t _ { k } ) \\} _ { k }$ can be well-approximated by the conditional distribution of $\\{ \\widehat { S } ^ { \\mathrm { M B } } ( t _ { k } ) \\} _ { k }$ given the data, where $\\begin{array} { r } { \\widehat { S } ^ { \\mathrm { M B } } ( t ) = \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) \\{ \\widehat { \\beta } _ { 1 } ^ { \\mathrm { M B } } ( t ) - } \\end{array}$ $\\widehat { \\beta } _ { 0 } ^ { \\mathrm { M B } } ( t ) \\}$ for any $t$ . Specifically, let $\\{ \\xi _ { i } \\} _ { i = 1 } ^ { + \\infty }$ be a sequence of i.i.d. standard normal random variables b independent of $\\{ ( X _ { i } , A _ { i } , Y _ { i } ) \\} _ { i = 1 } ^ { + \\infty }$ . For $a \\in \\{ 0 , 1 \\}$ , define ",
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+ "text": "$$\n\\widehat { \\beta } _ { a } ^ { \\mathrm { M B } } ( t ) = \\widehat { \\Sigma } _ { a } ^ { - 1 } ( t ) \\left[ \\frac { 1 } { N ( t ) } \\sum _ { i = 1 } ^ { N ( t ) } \\mathbb { I } ( A _ { i } = a ) \\varphi _ { i } ( X ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\widehat { \\beta } ( t ) \\} \\xi _ { i } \\right] , \\forall a \\in \\{ 0 , 1 \\} .\n$$",
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+ "text": "Both the asymptotic means of $\\sqrt { N ( t ) } \\widehat { \\beta } _ { a } ^ { \\mathrm { M B } } ( t )$ and $\\sqrt { N ( t ) } ( \\widehat { \\beta } _ { a } ( t ) - \\beta _ { a } ^ { * } )$ are zero. In addition, their covariance functions are asymptotically the same. By design, $\\{ \\widehat { \\beta } _ { a } ^ { \\mathrm { M B } } ( t _ { k } ) \\} _ { a , k }$ is multivariate normal. Similar to equation 3, we can show $\\{ \\widehat { \\beta } _ { a } ( t _ { k } ) - \\beta _ { a } ^ { * } \\} _ { a , k }$ is asymptotically multivariate normal. Consequently, the limiting distributions of $\\{ \\widehat { \\beta } _ { a } ^ { \\mathrm { M B } } ( t _ { k } ) \\} _ { a , k }$ and $\\{ \\widehat { \\beta } _ { a } ( t _ { k } ) - \\beta _ { a } ^ { * } \\} _ { a , k }$ are asymptotically equivalent. As such, the bootstrap approximation is valid. ",
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+ "text": "However, calculating $\\widehat { \\beta } _ { a } ^ { \\mathrm { M B } } ( t _ { k } )$ requires $O ( N ( t _ { k } ) )$ operations. The time complexity of the resulting bootstrap algorithm is $O ( B N ( t _ { k } ) )$ up to the $k$ -th interim stage, where $B$ is the total number of bootstrap samples. This can be time consuming when $\\{ N ( t _ { k } ) - \\mathbf { \\bar { \\phi } } N ( t _ { k - 1 } ) \\} _ { k = 1 } ^ { K }$ are large. To facilitate the computation, we observe that in the calculation of ${ \\widehat { \\beta } } _ { a } ^ { \\mathrm { M B } }$ , the random noise is generated upon the arrival of each observation. This is unnecessary as we aim to approximate the distribution of ${ \\widehat \\beta } _ { a } ( \\cdot )$ only at finitely many time points. ",
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+ "text": "We next present our proposal. Let $\\{ e _ { i , a } \\} _ { i = 1 , . . . , K , a = 0 , 1 }$ be a sequence of i.i.d $N ( 0 , I _ { q } )$ random vectors independent of the observed data, where $I _ { q }$ denotes the $q \\times q$ identity matrix. At the $k$ - th interim stage, we compute $\\begin{array} { r } { \\widehat { S } ^ { \\mathtt { M B } * } ( t _ { k } ) = \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\bar { \\varphi } ^ { \\top } ( x ) \\{ \\widehat { \\beta } _ { 1 } ^ { \\mathtt { M B } * } ( t _ { k } ) - \\widehat { \\beta } _ { 0 } ^ { \\mathtt { M B } * } ( t _ { k } ) \\} } \\end{array}$ , where $\\widehat { \\beta } _ { a } ^ { \\mathrm { M B * } } ( t _ { k } )$ equals ",
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+ "text": "$$\n\\frac { 1 } { \\mathrm { V } ( t _ { k } ) } \\sum _ { j = 1 } ^ { k } \\left( \\sum _ { i = N ( t _ { j - 1 } ) + 1 } ^ { N ( t _ { j } ) } \\widehat { \\Sigma } _ { a } ^ { - 1 } ( t _ { j } ) \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ( X _ { i } ) ^ { \\top } \\widehat { \\beta } _ { a } ( t _ { j } ) \\} ^ { 2 } \\widehat { \\Sigma } _ { a } ^ { - 1 } ( t _ { j } ) \\right) ^ { 1 / 2 } e _ { j , a } .\n$$",
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+ "text": "For any $k _ { 1 }$ and $k _ { 2 }$ , the conditional covariance of $\\sqrt { N ( t _ { k _ { 1 } } ) } \\{ \\widehat { \\beta } _ { 1 } ^ { \\mathrm { M B } * } ( t _ { k _ { 1 } } ) \\ - \\ \\widehat { \\beta } _ { 0 } ^ { \\mathrm { M B } * } ( t _ { k _ { 1 } } ) \\}$ and $\\sqrt { N ( t _ { k _ { 2 } } ) } \\{ \\widehat { \\beta } _ { 1 } ^ { \\mathrm { M B } * } ( t _ { k _ { 2 } } ) - \\widehat { \\beta } _ { 0 } ^ { \\mathrm { M B } * } ( t _ { k _ { 2 } } ) \\}$ equals ",
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+ "text": "$$\n\\frac { 1 } { \\sqrt { N ( t _ { k _ { 1 } } ) N ( t _ { k _ { 2 } } ) } } \\sum _ { a = 0 } ^ { 1 } \\sum _ { j = 1 } ^ { k _ { 1 } } \\sum _ { i = N ( t _ { j - 1 } ) + 1 } ^ { N ( t _ { j } ) } \\widehat { \\Sigma } _ { a } ^ { - 1 } ( t _ { j } ) \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\widehat { \\beta } _ { a } ( t _ { j } ) \\} ^ { 2 } \\widehat { \\Sigma } _ { a } ^ { - 1 } ( t _ { k _ { 1 } } ) ,\n$$",
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+ "type": "text",
+ "text": "Under the given conditions in Theorem 1, it is to converge to ",
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+ "text": "$$\n\\frac { \\sqrt { N ( t _ { k _ { 1 } } ) } } { \\sqrt { N ( t _ { k _ { 2 } } ) } } \\sum _ { a = 0 } ^ { 1 } \\Sigma _ { a } ^ { - 1 } \\Phi ( a ) \\Sigma _ { a } ^ { - 1 } = \\mathrm { C o v } ( G ( t _ { k _ { 1 } } ) , G ( t _ { k _ { 2 } } ) ) .\n$$",
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+ "text": "This means $\\{ \\sqrt { N ( t _ { k } ) } ( \\widehat { \\beta } _ { 1 } ^ { \\mathrm { M B } ^ { * } } ( t _ { k } ) - \\widehat { \\beta } _ { 0 } ^ { \\mathrm { M B } ^ { * } } ( t _ { k } ) ) \\} _ { k }$ and $\\{ G ( t _ { k } ) \\} _ { k }$ have the same asymptotic distribution. Consequently, $\\{ \\sqrt { N ( t _ { k } ) } \\widehat { S } ^ { \\mathrm { M B } * } ( t _ { k } ) \\} _ { k = 1 } ^ { K }$ can be used to approximate the joint distribution of $\\{ \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) \\} _ { k = 1 } ^ { \\cdot }$ . ",
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+ "type": "text",
+ "text": "To choose $\\{ z _ { k } \\} _ { k }$ that satisfies equation 8, we adopt the $\\alpha$ -spending approach that allocates the total allowable type I error at each interim stage according to an error-spending function. This guarantees our test controls the type-I error. We begin by specifying an $\\alpha$ spending function $\\alpha ( t )$ that is nonincreasing and satisfies $\\alpha ( 0 ) = 0$ , $\\alpha ( T ) = \\alpha$ . Popular choices of $\\alpha ( \\cdot )$ include ",
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+ "text": "$$\n\\alpha _ { 1 } ( t ) = \\alpha \\log \\left( 1 + ( e - 1 ) \\frac { t } { T } \\right) , \\quad \\quad \\quad \\quad \\alpha _ { 2 } ( t ) = 2 - 2 \\Phi \\left( \\frac { \\Phi ^ { - 1 } ( 1 - \\alpha / 2 ) \\sqrt { T } } { \\sqrt { t } } \\right) ,\n$$",
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+ "text": "$$\n\\alpha _ { 3 } ( t ) = \\alpha \\left( \\frac { t } { T } \\right) ^ { \\theta } , \\mathrm { f o r } \\theta > 0 , \\alpha _ { 4 } ( t ) = \\alpha \\frac { 1 - \\exp ( - \\gamma t / T ) } { 1 - \\exp ( - \\gamma ) } , \\mathrm { f o r } \\gamma \\neq 0 ,\n$$",
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+ "type": "text",
+ "text": "where $\\Phi ( \\cdot )$ denotes the cumulative distribution function of a standard normal variable and $\\Phi ^ { - 1 } ( \\cdot )$ is its quantile function. Based on $\\alpha ( \\cdot )$ , we iteratively calculate $\\widehat { z } _ { k } , k = 1 , \\ldots , K$ as the solution of ",
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+ "text": "$$\n\\mathrm { P r } ^ { * } \\left\\{ \\operatorname* { m a x } _ { j \\in \\{ 1 , \\dots , k - 1 \\} } \\left( \\sqrt { N ( t _ { j } ) } \\widehat { S } ^ { \\mathrm { M B } * } ( t _ { j } ) - \\widehat { z } _ { j } \\right) \\leq 0 , \\sqrt { N ( t _ { k } ) } \\widehat { S } ^ { \\mathrm { M B } * } ( t _ { k } ) > \\widehat { z } _ { k } \\right\\} = \\alpha ( t _ { k } ) - \\alpha ( t _ { k - 1 } ) ,\n$$",
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+ "type": "text",
+ "text": "and reject $H _ { 0 }$ when $\\sqrt { N ( t _ { k } ) } S ( t _ { k } ) > \\widehat { z } _ { k }$ holds for some $k$ . ",
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+ "type": "text",
+ "text": "The validity of the bootstrap test is summarized in Theorems 3 and 4 below. ",
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+ "text": "Theorem 3 Assume the conditions in Theorem $^ { l }$ hold. Assume $q = O ( N ^ { \\alpha ^ { * } } ( t _ { 1 } ) )$ for some $0 ~ <$ $\\alpha ^ { * } < 1 / 3$ , almost surely. Then conditional on the counting process $N ( \\cdot )$ , we have ",
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+ "text": "$$\n\\begin{array} { r } { \\underset { 1 , \\cdots , z _ { K } } { \\operatorname* { s u p } } \\left| P r ^ { * } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\sqrt { N ( t _ { k } ) } \\widehat { S } ^ { M B * } ( t _ { k } ) - z _ { k } \\right) > 0 \\right\\} - P r \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - z _ { k } \\right) \\right\\} \\right| } \\\\ { \\leq c \\left[ q ^ { 1 / 2 } N ^ { - 1 / 6 } ( t _ { 1 } ) \\log ^ { 1 1 / 6 } \\{ K N ( t _ { 1 } ) \\} + q N ^ { - \\alpha _ { 0 } / 3 } ( t _ { 1 } ) \\log ^ { ( 5 + \\alpha _ { 0 } ) / 3 } \\{ K \\} \\right] , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "for some constant $c > 0$ with probability at least $1 - O ( N ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ , where $\\textstyle P r ^ { * } ( \\cdot )$ denotes the probability measure conditional on the data stream $\\{ X _ { i } , A _ { i } , Y _ { i } \\} _ { i = 1 } ^ { + \\infty }$ . ",
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+ "text": "Theorem 4 Assume the conditions in Theorem 3 hold. Then conditional on $N ( \\cdot )$ , the critical values $\\{ \\widehat { z } _ { k } \\} _ { k }$ satisfy ",
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+ "text": "$$\n\\begin{array} { r l } & { \\left| P r \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - \\widehat z _ { k } \\right) > 0 \\right\\} - \\alpha \\right| } \\\\ { \\leq } & { c \\left[ q ^ { 1 / 2 } N ^ { - 1 / 6 } ( t _ { 1 } ) \\log ^ { 1 1 / 6 } \\{ K N ( t _ { 1 } ) \\} + q N ^ { - \\alpha _ { 0 } / 3 } ( t _ { 1 } ) \\log ^ { ( 5 + \\alpha _ { 0 } ) / 3 } \\{ K N ( t _ { 1 } ) \\} \\right] , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "for some constant $c > 0$ . ",
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+ "text": "When the RHS of equation 11 is $o _ { p } ( 1 )$ , it follows from Theorems 2 and 4 that our test is valid. The conditional distribution in equation 10 can be approximated by the empirical distribution of Bootstrap samples. ",
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+ "text": "Finally, we remark that our test can be online updated as batches of observations arrive at the end of each interim stage. A pseudocode summarizing our procedure is given in Algorithm 1 in the appendix. The spatial complexity of the proposed algorithm is $O ( B )$ , where $B$ is the number of bootstrap samples. The time complexity is $\\bar { O ( } B k + \\bar { N } ( t _ { k } ) )$ up to the $k$ -th interim stage. Suppose $N ( t _ { j } ) - N ( t _ { j - 1 } ) = n$ for any $1 \\leq j \\leq K$ , we have $B k + N ( t _ { k } ) = ( B + n ) k \\ll B n k = B N ( t _ { k } )$ for large $n$ and $B$ . Hence, our procedure is much faster compared to the standard wild bootstrap. ",
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+ "type": "text",
+ "text": "4 NUMERICAL STUDIES ",
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+ "text": "4.1 SIMULATION STUDIES ",
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+ "text": "In this section, we conduct Monte Carlo simulations to examine the finite sample properties of the proposed test. We generated the potential outcomes as $Y _ { i } ^ { * } ( a ) = 1 + ( X _ { i 1 } - X _ { i 2 } ) / 2 + a \\tau ( X _ { i } ) + \\varepsilon _ { i }$ , where $\\varepsilon _ { i }$ ’s are i.i.d $N ( 0 , 0 . 5 ^ { 2 } )$ . The covariates $X _ { i } ~ = ~ ( X _ { i 1 } , X _ { i 2 } , X _ { i 3 } ) ^ { \\top }$ were generated as follows. We first generated $X _ { i } ^ { * } ~ = ~ ( X _ { i 1 } ^ { * } , X _ { i 2 } ^ { * } , X _ { i 3 } ^ { * } ) ^ { \\top }$ from a multivariate normal distribution with zero mean and covariance matrix equal to $\\{ 0 . 5 ^ { | i - j | } \\} _ { i , j }$ . Then we set $X _ { i j } ~ = ~ X _ { i j } ^ { * } \\mathbb { I } ( X _ { i j } ^ { * } | ~ \\le$ $2 ) + 2 \\mathrm { s g n } ( X _ { i j } ^ { * } ) \\mathbb { I } ( X _ { i j } ^ { * } | > 2 )$ . We consider two randomization designs. In the first design, the treatment assignment is nondynamic and completely random. Specifically we set $\\textstyle \\pi _ { i } ( a , x ) = 0 . 5$ , for any $a , x$ and $i$ . In the second design, we use an $\\epsilon$ -greedy strategy to generate the treatment with $\\varepsilon = 0 . 3$ . In addition, we set $N ( T _ { 1 } ) = 2 0 0 0$ and $\\bar { N ( T _ { j } ) } - \\bar { N } \\bar { ( T _ { j - 1 } ) } ^ { - } = 2 \\bar { n }$ for $2 \\leq j \\leq K$ and some $n > 0$ . We consider two combinations of $( n , K )$ , corresponding to $( n , K ) = ( 2 0 0 , 5 )$ and (20, 50). ",
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+ "text": "We set the significance level √ $\\alpha = 0 . 0 5$ and choose $B \\ = \\ 1 0 0 0 0$ . We set $\\tau ( X _ { i } ) = \\phi _ { \\delta } \\{ ( X _ { i 1 } +$ $X _ { i 2 } ) / \\sqrt { 2 } \\} X _ { i 3 } ^ { 2 }$ for some function $\\phi _ { \\delta }$ parameterized by some $\\delta \\geq 0$ . We consider two scenarios for $\\phi _ { \\delta }$ . Specifically, we set $\\phi _ { \\delta } ( x ) = \\delta x ^ { 2 } / 3$ in Scenario 1 and $\\phi _ { \\delta } = \\delta \\cos ( \\pi x )$ in Scenario 2. For each setting, we further consider four cases by setting $\\delta = 0 , 0 . 1 , 0 . 1 5 , 0 . 2 , 0 . 2 5$ and 0.3. When $\\delta = 0$ , $H _ { 0 }$ holds. Otherwise, $H _ { 1 }$ holds. For all settings, we construct the basis function $\\varphi ( \\cdot )$ using additive cubic splines. For each univariate spline, we set the number of internal knots to be 4. These knots are equally spaced between $[ - 2 , 2 ]$ . ",
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+ "text": "We denote our test by BAT, short for bootstrap-assisted test. We run our experiments on a single computer instance with 40 Intel(R) Xeon(R) 2.20GHz CPUs. It takes 1-2 seconds on average to compute each test. In Table 1 (see Appendix G), we report the rejection probabilities and average stopping times (defined as the average number of samples consumed when the experiment is terminated) of the proposed test aggregated over 400 simulations when $\\alpha _ { 1 } ( \\cdot )$ is chosen as the spending function. In Figure 2, we plot the rejection probabilities of our tests and the average stopping times of the experiments. It can be seen that the type-I error rates are close to the nominal level in all cases. The power of our test increases as $\\delta$ increases, demonstrating its consistency. In addition, when $\\delta > 0$ , our experiments are stopped early in all cases. ",
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+ "type": "text",
+ "text": "To further evaluate our method, we compare it with a test based on the law of iterated logarithm (denoted by LIL). LIL determines the decision boundary based on an always valid finite error bound (see Appendix $\\mathrm { F }$ for details about the competing method). It can be seen from Figure 2 that our method has much larger power than the law of iterated logarithm approach. ",
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+ "type": "text",
+ "text": "4.2 REAL DATA ANALYSIS ",
+ "text_level": 1,
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+ "text": "In this section, we apply the proposed method to a Yahoo! Today Module user click log dataset1, which contains 45,811,883 user visits to the Today Module, during the first ten days in May 2009. For the $i$ th visit, the dataset contains an ID of the new article recommended to the user, a binary response variable $Y _ { i }$ indicating whether the user clicked the article or not, and a five dimensional feature vector summarizing information of the user. Due to privacy concerns, feature definitions and article names were not included in the data. Each feature vector sums up to 1. Therefore, we took the first three and the fifth elements to form the covariates $X _ { i }$ . For illustration, we only consider a subset of data that contains visits on May 1st where the recommended article ID is either 109510 or 109520. These two articles were being recommended most on that day. This gives us a total of 405888 visits. On the reduced dataset, define $A _ { i } = 1$ if the recommended article is 109510 and $A _ { i } = 0$ otherwise. ",
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+ "Figure 2: Rejection probabilities and average stopping times of the proposed test when $\\alpha _ { 1 } ( \\cdot )$ is chosen as the spending function. From left to right: Scenario 1 with random design, Scenario 1 with $\\epsilon$ -greedy design, Scenario 2 with random design and Scenario 2 with $\\epsilon$ -greedy design. "
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+ "type": "text",
+ "text": "We first conduct A/A experiments (which compare these two articles against themselves) to examine the validity of our test. The A/A experiments are done when every 2000 more users are available, we randomly assign 1000 users to arm A, and the other 1000 users in arm B. We expect our test will not reject $H _ { 0 }$ in A/A experiments, since the articles being recommended are the same. Then, we conduct A/B experiment to test the QTE of these two articles. The test statistics and their corresponding critical values are plotted in Figure 3. On average it takes several seconds to implement our test. It can be seen that our test is able to be reject $H _ { 0 }$ after obtaining the first one-third of the observations, in the A/B experiment. In the A/A experiments, we fail to reject $H _ { 0 }$ , as expected. ",
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+ "type": "text",
+ "text": "5 DISCUSSION ",
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+ {
+ "type": "text",
+ "text": "In this paper, we propose a new testing procedure for evaluating the performance of technology products in tech companies based on the notion of qualitative treatment effects. Currently, we only focus on comparing two products. It would be practically useful to develop a multiple testing procedure under settings with multiple treatment options. These topics warrant further investigations. ",
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+ "img_path": "images/d07ec0bda641ab04da632fa857aca2761150230607e09e18e0434c03477db667.jpg",
+ "image_caption": [
+ "Figure 3: Critical values and test statistics. "
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+ "type": "text",
+ "text": "REFERENCES ",
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+ {
+ "type": "text",
+ "text": "Charles J. Stone. Optimal global rates of convergence for nonparametric regression. Ann. Statist., 10(4):1040–1053, 1982. ISSN 0090-5364. ",
+ "bbox": [
+ 173,
+ 636,
+ 823,
+ 665
+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "Matt Taddy, Matt Gardner, Liyun Chen, and David Draper. A nonparametric bayesian analysis of heterogenous treatment effects in digital experimentation. Journal of Business & Economic Statistics, 34(4):661–672, 2016. ",
+ "bbox": [
+ 173,
+ 672,
+ 825,
+ 717
+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "Stefan Wager and Susan Athey. Estimation and inference of heterogeneous treatment effects using random forests. J. Amer. Statist. Assoc., 113(523):1228–1242, 2018. ISSN 0162-1459. doi: 10.1080/01621459.2017.1319839. URL https://doi.org/10.1080/01621459. 2017.1319839. ",
+ "bbox": [
+ 173,
+ 726,
+ 825,
+ 782
+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "C.-F. J. Wu. Jackknife, bootstrap and other resampling methods in regression analysis. Ann. Statist., 14(4):1261–1350, 1986. ISSN 0090-5364. doi: 10.1214/aos/1176350142. URL https:// doi.org/10.1214/aos/1176350142. With discussion and a rejoinder by the author. ",
+ "bbox": [
+ 174,
+ 791,
+ 823,
+ 835
+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "Fanny Yang, Aaditya Ramdas, Kevin G Jamieson, and Martin J Wainwright. A framework for multia (rmed)/b (andit) testing with online fdr control. In Advances in Neural Information Processing Systems, pp. 5957–5966, 2017. ",
+ "bbox": [
+ 173,
+ 843,
+ 825,
+ 886
+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "Miao Yu, Wenbin Lu, and Rui Song. A new framework for online testing of heterogeneous treatment effect. In AAAI, pp. 10310–10317, 2020. ",
+ "bbox": [
+ 173,
+ 895,
+ 821,
+ 924
+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "Input: Number of bootstrap samples $B$ , an $\\alpha$ spending function $\\alpha ( \\cdot )$ ",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Initialize: $n = 0$ , $\\widehat { \\Sigma } _ { 0 } = \\widehat { \\Sigma } _ { 1 } = { \\cal O } _ { p + 1 }$ , $\\widehat { \\gamma } _ { 0 } = \\widehat { \\gamma } _ { 1 } = 0 _ { p + 1 }$ , $\\widehat { \\beta } _ { 0 , b } = \\widehat { \\beta } _ { 1 , b } = 0 _ { p + 1 }$ , and a set $\\mathcal { T } = \\{ 1 , \\ldots , B \\}$ . \nFor $k = 1$ to $K$ do ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Initialize: $m = 0$ and $\\widehat { \\Phi } _ { 0 } = \\widehat { \\Phi } _ { 1 } = O _ { p + 1 }$ ",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Step 1: Online update of ${ \\widehat { \\beta } } _ { a }$ \nFor $i = N ( t _ { k - 1 } ) \\bar { { + } 1 }$ to $N ( t _ { k } )$ do $n = n + 1$ and $m = m + 1$ ; $\\widehat { \\Sigma } _ { a } = ( 1 - n ^ { - 1 } ) \\widehat { \\Sigma } _ { a } + n ^ { - 1 } \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\mathbb { I } ( A _ { i } = a ) , a = 0 , 1 ;$ $\\widehat { \\gamma } _ { a } = ( 1 - n ^ { - 1 } ) \\widehat { \\gamma } _ { a } + n ^ { - 1 } \\varphi ( X _ { i } ) Y _ { i } \\mathbb { I } ( A _ { i } = a ) , a = 0 , 1 ;$ ",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "bCompute $\\widehat { \\beta } _ { a } = \\widehat { \\Sigma } _ { a } ^ { - 1 } \\widehat { \\gamma } _ { a }$ for $a \\in \\{ 0 , 1 \\}$ and $S = \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) ( \\widehat { \\beta } _ { 1 } - \\widehat { \\beta } _ { 0 } ) ;$ ",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Step 2: Bootstrap ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Generate two independent ${ \\cal N } ( 0 , I _ { p + 1 } )$ Gaussian vectors $e _ { 0 } , e _ { 1 }$ ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Step 3: Reject or not ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Set $z$ to be the upper $\\{ \\alpha ( t ) - | \\mathcal { T } | ^ { c } / B | \\} / ( 1 - | \\mathcal { T } ^ { c } | / B )$ -th percentile of $\\{ \\widehat { S } _ { b } \\} _ { b \\in \\mathbb { Z } }$ Update $\\mathcal { T }$ as ${ \\mathcal { T } } \\{ b \\in { \\mathcal { T } } : { \\widehat { S } } _ { b } \\leq z \\}$ . ",
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "If $S > z$ : Reject $H _ { 0 }$ and terminate the experiment. ",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Algorithm 1: the Pseudocode that summarizing the online bootstrap testing procedure. ",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Baqun Zhang, Anastasios A Tsiatis, Eric B Laber, and Marie Davidian. Robust estimation of optimal dynamic treatment regimes for sequential treatment decisions. Biometrika, 100(3):681–694, 2013. ",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Li-xin Zhang. Strong approximations of martingale vectors and their applications in Markovchain adaptive designs. Acta Math. Appl. Sin. Engl. Ser., 20(2):337–352, 2004. ISSN 0168-9673. doi: 10.1007/s10255-004-0173-z. URL https://doi.org/10.1007/ $\\mathtt { s 1 0 2 5 5 - 0 0 4 - 0 1 7 3 - z }$ . ",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "A NOTATIONS ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "We introduce some general notations used in the appendix. For any matrix Mat, we use $\\left\\| \\mathbf { M a t } \\right\\| _ { p }$ to denote the matrix norm induced by the corresponding $\\ell _ { p }$ norm of vectors, for $1 \\leq p \\leq + \\infty$ . For two nonnegative sequences $\\{ s _ { 1 , n } \\} _ { n }$ and $\\{ s _ { 2 , n } \\} _ { n }$ , we will use the notation $s _ { 1 , n } \\preceq s _ { 2 , n }$ to represent that $s _ { 1 , n } \\leq \\bar { c } s _ { 2 , n }$ for some universal constant $\\bar { c } > 0$ whose value is allowed to change from place to place. When a matrix Mat is degenerate, $\\mathrm { { M a t } ^ { - 1 } }$ denotes the Moore-Penrose inverse of Mat. For any vector $\\psi$ , we use $\\psi ^ { ( i ) }$ to denote its $i$ -th element. ",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "In Algorithm 1, we use $O _ { p + 1 }$ to denote a $( p + 1 ) \\times ( p + 1 )$ zero matrix and $0 _ { p + 1 }$ to denote a $( p + 1 )$ -dimensional zero vector. ",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "B MORE ON THE BASIS FUNCTION ",
+ "text_level": 1,
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "B.1 CONDITION (A3) ",
+ "text_level": 1,
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "(A3). Assume $\\lambda _ { \\operatorname* { m i n } } [ \\mathrm { E } \\varphi ( X ) \\varphi ^ { \\top } ( X ) ] \\asymp 1 , \\lambda _ { \\operatorname* { m a x } } [ \\mathrm { E } \\varphi ( X ) \\varphi ^ { \\top } ( X ) ] \\asymp 1 , \\operatorname* { s u p } _ { x } \\| \\varphi ( x ) \\| _ { 1 } = O ( q ^ { \\top } ) ,$ , $\\begin{array} { r } { \\operatorname* { l i m } \\operatorname* { i n f } _ { q } \\operatorname* { i n f } _ { x \\in \\mathbb { X } } \\| \\varphi ( x ) \\| _ { 2 } > 0 } \\end{array}$ . In addition, assume ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/bc37a934116573d3dc38916594e2a8986748d69fedf32e57370035f29b24b164.jpg",
+ "text": "$$\n\\operatorname* { s u p } _ { x , y \\in \\mathbb { X } } \\frac { \\| \\varphi ( x ) - \\varphi ( y ) \\| _ { 2 } } { \\| x - y \\| _ { 2 } } \\preceq q ^ { 1 / 2 } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "When a tensor-product B-spline is used (see Section 6 of Chen & Christensen, 2015, for a brief overview of tenor-product B-splines), (A3) is automatically satisfied. Specifically, $\\lambda _ { \\operatorname* { m i n } } [ \\mathrm { E } \\varphi ( X ) \\varphi ^ { \\top } ( X ) ] \\asymp 1$ , $\\dot { \\lambda } _ { \\mathrm { m a x } } [ \\mathrm { E } \\varphi ( X ) \\dot { \\varphi } ^ { \\top } ( X ) ] \\asymp 1$ follow from Theorem 3.3 of (Burman & Chen, 1989). $\\operatorname* { s u p } _ { x } \\| \\varphi ( x ) \\| _ { 1 } = O ( q ^ { 1 / 2 } )$ follows by noting that the absolute value of each element in $\\varphi ( x )$ is bounded by some universal constant. $\\begin{array} { r } { \\operatorname* { l i m } \\operatorname* { i n f } _ { q } \\operatorname* { i n f } _ { x \\in \\mathbb { X } } \\| \\varphi ( x ) \\| _ { 2 } > 0 } \\end{array}$ follows from the arguments used in the proof of Lemma E.4 of Shi et al. (2020b). The last condition in equation 12 holds by noting that each function in the vector $\\varphi ( \\cdot )$ is Lipschitz continuous when a tensor-product B-spline is used. ",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "B.2 ON THE APPROXIMATION ERROR ",
+ "text_level": 1,
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "The proposed test remains valid as long as the approximation error satisfies ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/6430aad6083eb26df949d353ffe25fae3d6ebf25c588fd090be324281029d7e1.jpg",
+ "text": "$$\n\\operatorname* { i n f } _ { \\beta _ { 0 } , \\beta _ { 1 } \\in \\mathbb { R } ^ { p } } \\operatorname* { s u p } _ { x \\in \\mathbb { X } , a \\in \\{ 0 , 1 \\} } | Q _ { 0 } ( x , a ) - Q ( x , a ; \\beta _ { 0 } , \\beta _ { 1 } ) | = o ( \\{ N ( T ) \\} ^ { - 1 / 2 } ) ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "with probability tending to 1. In the following, we introduce some sufficient conditions for equation 13. ",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Suppose the Q-function $Q _ { 0 } ( \\cdot , a )$ is $p$ -smooth (see the definition of $p$ -smoothness in Stone, 1982), for $a \\in [ 0 , 1 ]$ . When a tensor-product B-spline or Wavelet basis is used for $\\varphi ( \\cdot )$ , then there exist some $\\beta _ { 0 } ^ { \\ast }$ and $\\beta _ { 1 } ^ { * }$ that satisfy ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/6f0c8d27a0832b66adf8ddbdacb13e7232ac3886671974707e0449ff1a0c8b98.jpg",
+ "text": "$$\n\\operatorname* { i n f } _ { \\beta _ { 0 } , \\beta _ { 1 } \\in \\mathbb { R } ^ { p } } \\operatorname* { s u p } _ { x \\in \\mathbb { X } , a \\in \\{ 0 , 1 \\} } | Q _ { 0 } ( x , a ) - Q ( x , a ; \\beta _ { 0 } , \\beta _ { 1 } ) | = O ( q ^ { - p / d } ) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "See Section 2.2 of Huang (1998) for detailed discussions on the approximation power of these basis functions. Condition equation 13 is thus automatically satisfied when ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/889508cae5e1cb04f39271104cc0322b564e84f205d454d257c722e9fb152fa4.jpg",
+ "text": "$$\nq \\gg \\{ N ( T ) \\} ^ { d / ( 2 p ) } ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "with probability tending to 1. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "C ADAPTIVE RANDOMIZATION ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "In practice, the company might want to allocate more traffic to a better treatment based on the observed data stream. The $\\epsilon$ -greedy strategy is commonly used to balance the trade-off between exploration and exploitation. For a given $0 < \\varepsilon _ { 0 } < 1$ , consider the following randomization procedure: for some integer $N _ { 0 } > 0$ and any $j \\geq N _ { 0 }$ , $a \\in \\{ 0 , 1 \\}$ , $x \\in \\mathbb { X }$ , we set ",
+ "bbox": [
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+ },
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+ "type": "text",
+ "text": "π $\\begin{array} { r } { \\tau _ { j - 1 } ( a , x ) = ( 1 - \\varepsilon _ { 0 } ) a \\mathbb { I } \\{ \\varphi ^ { \\top } ( x ) ( \\widehat { \\beta } _ { 1 , j - 1 } - \\widehat { \\beta } _ { 0 , j - 1 } ) > 0 \\} + \\varepsilon _ { 0 } ( 1 - a ) \\mathbb { I } \\{ \\varphi ^ { \\top } ( x ) ( \\widehat { \\beta } _ { 1 , j - 1 } - \\widehat { \\beta } _ { 0 , j - 1 } ) \\leq 0 } \\end{array}$ }, where ",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "img_path": "images/51ce5207529e43fe8b1a8407d42df5c52d861bba4c23d54c7ffc6cf999fa8809.jpg",
+ "text": "$$\n\\widehat { \\beta } _ { a , j } = \\widehat { \\Sigma } _ { a , j } ^ { - 1 } \\frac { 1 } { j } \\sum _ { i = 1 } ^ { j } \\{ \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) Y _ { i } \\} \\mathrm { a n d } \\widehat { \\Sigma } _ { a , j } = \\frac { 1 } { j } \\sum _ { i = 1 } ^ { j } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "It is immediate to see that $\\widehat { \\Sigma } _ { a } ( t ) = \\widehat { \\Sigma } _ { a , n ( t ) }$ and $\\widehat { \\beta } _ { a } ( t ) = \\widehat { \\beta } _ { a , n ( t ) }$ . Define ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/8dcafb92bd47026ee739b96321c0f6d8c86aeb71f23d2e2c317e40e581414d03.jpg",
+ "text": "$$\n\\pi ^ { * } ( a , x ) = ( 1 - \\varepsilon _ { 0 } ) a \\mathbb { I } \\{ \\varphi ^ { \\top } ( x ) ( \\beta _ { 1 } - \\beta _ { 0 } ) > 0 \\} + \\varepsilon _ { 0 } ( 1 - a ) \\mathbb { I } \\{ \\varphi ^ { \\top } ( x ) ( \\beta _ { 1 } - \\beta _ { 0 } ) \\leq 0 \\}\n$$",
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+ "bbox": [
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+ {
+ "type": "text",
+ "text": "for any $a \\in \\{ 0 , 1 \\}$ and $x \\in \\mathbb { X }$ . ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Lemma 2 Assume (A1)-(A3) hold. Assume $\\operatorname* { i n f } _ { x \\in \\mathbb { X } } \\pi ^ { * } ( a , x ) > 0$ and $\\left| Y ^ { * } ( a ) \\right|$ is bounded almost surely, for $a \\in \\{ 0 , 1 \\}$ . Assume $P r ( | \\varphi ^ { \\top } ( X ) ( \\beta _ { 1 } - \\bar { \\beta } _ { 0 } ) | \\le \\epsilon ) \\ \\le L _ { 0 } \\mathfrak { e }$ \u000f, for some constant $L _ { 0 } > 0$ and any $\\epsilon > 0$ . Then for any $\\{ j _ { n } \\} _ { n }$ that satisfies $\\sqrt { j _ { n } } / \\sqrt { \\log j _ { n } } \\gg q ^ { 2 }$ , the following event occurs with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ , ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/861230d88193766b7e64c8aa4c8a9354e7d1962583cb02ceab07107e1fb29111.jpg",
+ "text": "$$\n \\sum _ { a \\in \\{ 0 , 1 \\} } E ^ { \\mathcal { F } _ { i - 1 } } | \\sum _ { i = 1 } ^ { k } \\{ \\pi _ { i - 1 } ( a , X ) - \\pi ^ { * } ( a , X ) \\} | \\preceq q \\sqrt { k \\log k } , \\forall k \\geq j _ { n } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Lemma 2 implies that Condition equation 5 in Theorem 1 automatically holds with $\\alpha _ { 0 } = 1 / 2$ , when the epsilon-greedy strategy is used. When $\\varphi ^ { \\top } ( X ) ( \\beta _ { 1 } - \\beta _ { 0 } )$ is a continuous random variable, the assumption $\\bar { \\operatorname* { P r } } ( | \\varphi ^ { \\top } ( X ) ( \\bar { \\beta _ { 1 } } - \\beta _ { 0 } ) | \\leq \\epsilon ) \\leq { \\dot { L _ { 0 } } } \\epsilon$ for any $\\epsilon > 0$ in Lemma 2 is satisfied if $\\varphi ^ { \\top } ( X ) ( \\beta _ { 1 } -$ $\\beta _ { 0 , }$ ) has a bounded probability density function. When $\\varphi ^ { \\top } ( X ) ( \\beta _ { 1 } - \\beta _ { 0 } )$ is discrete, this assumption is satisfied if $\\begin{array} { r } { \\operatorname* { i n f } _ { x \\in \\bar { \\mathbb { X } } } | \\varphi ^ { \\top } ( x ) \\dot { ( \\beta _ { 1 } - \\beta _ { 0 } ) } | > 0 } \\end{array}$ . ",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "D TESTING THE AVERAGE TREATMENT EFFECTS ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "D.1 THE ALGORITHM ",
+ "text_level": 1,
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+ },
+ {
+ "type": "text",
+ "text": "We focus on testing the following hypothesis, ",
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+ "type": "equation",
+ "img_path": "images/cd2d5b140d865ccb4ec9daf6fd35f1e2fe3068f80c03443af72c60966d2eeb85.jpg",
+ "text": "$$\nH _ { 0 } : { \\mathrm { E } } Y _ { i } ^ { * } ( 1 ) \\leq { \\mathrm { E } } Y _ { i } ^ { * } ( 0 ) { \\mathrm { ~ v e r s u s ~ } } H _ { 1 } : { \\mathrm { E } } Y _ { i } ^ { * } ( 1 ) > { \\mathrm { E } } Y _ { i } ^ { * } ( 0 ) .\n$$",
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+ "type": "text",
+ "text": "Under (A1) and (A2), it suffices to test ",
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+ "text": "$$\nH _ { 0 } : { \\mathrm { E } } Q ( X _ { i } , 1 ) \\leq { \\mathrm { E } } Q ( X _ { i } , 0 ) { \\mathrm { ~ v e r s u s ~ } } H _ { 1 } : { \\mathrm { E } } Q ( X _ { i } , 1 ) > { \\mathrm { E } } Q ( X _ { i } , 0 ) .\n$$",
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+ "type": "text",
+ "text": "We similarly use basis approximations to model the Q-function. Our proposal is summarized in the following algorithm. We next conduct simulation studies to evaluate this algorithm. ",
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+ "type": "text",
+ "text": "Input: Number of bootstrap samples $B$ , an $\\alpha$ spending function $\\alpha ( \\cdot )$ ",
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+ "text": "Initialize: $n = 0$ , $\\widehat { \\Sigma } _ { 0 } = \\widehat { \\Sigma } _ { 1 } = O _ { p + 1 }$ , $\\widehat { \\gamma } _ { 0 } = \\widehat { \\gamma } _ { 1 } = 0 _ { p + 1 }$ , $\\widehat { \\beta } _ { 0 , b } = \\widehat { \\beta } _ { 1 , b } = 0 _ { p + 1 }$ , $\\bar { \\varphi } = 0$ and a set $\\mathcal { T } = \\{ 1 , \\ldots , B \\}$ . \nFor $k = 1$ to $K$ do ",
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+ "type": "text",
+ "text": "Initialize: $m = 0$ , $\\widehat { \\phi } = 0$ and $\\widehat { \\Phi } _ { 0 } = \\widehat { \\Phi } _ { 1 } = O _ { p + 1 }$ ",
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+ "type": "text",
+ "text": "For $i = N ( t _ { k - 1 } ) + 1$ to $N ( t _ { k } )$ do ",
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+ "type": "text",
+ "text": "bCompute $\\widehat { \\beta } _ { a } = \\widehat { \\Sigma } _ { a } ^ { - 1 } \\widehat { \\gamma } _ { a }$ for $a \\in \\{ 0 , 1 \\}$ and $S = \\bar { \\varphi } ^ { \\top } ( \\widehat { \\beta } _ { 1 } - \\widehat { \\beta } _ { 0 } )$ ; ",
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+ "type": "text",
+ "text": "For $b = 1 , \\dots , B$ do ",
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+ "type": "text",
+ "text": "Generate two independent $N ( 0 , I _ { p + 1 } )$ Gaussian vectors e0, e1, $N ( 0 , 1 )$ random variable $e _ { 2 }$ $\\widehat { \\beta } _ { a , b } = ( 1 - m n ^ { - 1 } ) \\widehat { \\beta } _ { a , b } + n ^ { - 1 } \\widehat { \\Phi } _ { a } ^ { 1 / 2 } e _ { a } + n ^ { - 1 } \\widehat { \\phi } ^ { 1 / 2 } e _ { 2 } , a = 0 , 1$ ; \nCompute $\\widehat { S } _ { b } = \\bar { \\varphi } ^ { \\top } ( \\widehat { \\beta } _ { 1 , b } - \\widehat { \\beta } _ { 0 , b } )$ ; Set $z$ to be the upper $\\{ \\alpha ( t ) - | \\mathcal { T } | ^ { c } / B | \\} / ( 1 - | \\mathcal { T } ^ { c } | / B )$ -th percentile of $\\{ \\widehat { S } _ { b } \\} _ { b \\in \\mathbb { Z } }$ ; Update $\\mathcal { T }$ as ${ \\mathcal { T } } \\{ b \\in { \\mathcal { T } } : { \\widehat { S } } _ { b } \\leq z \\}$ . \nIf $S > z$ : ",
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+ "type": "text",
+ "text": "Reject $H _ { 0 }$ and terminate the experiment; ",
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+ "type": "text",
+ "text": "D.2 NUMERICAL STUDIES ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "In this section, we compare our procedure with the always valid test for testing ATE (Johari et al., 2015). We generate the potential outcomes with the same model, except that $\\varepsilon _ { i }$ ’s are i.i.d $N ( 0 , 1 )$ . However, we set $N ( T _ { 1 } ) \\stackrel { - } { = } 1 0 0 0$ and $N ( T _ { j } ) - N ( T _ { j - 1 } ) = 2 n$ for $2 \\leq j \\leq K$ and some $n > 0$ . We consider two combinations of $( n , K )$ , corresponding to $( n , K ) = ( 1 0 0 , 5 )$ and (10, 50). For all settings, we use a linear function to approximate $Q$ . ",
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+ "type": "text",
+ "text": "In Table 2 (see Appendix G) and Figure 4, we show the rejection probabilities and average stopping times of the proposed test aggregated over 400 simulations, when $\\alpha _ { 1 } ( \\cdot )$ is chosen as the spending function. It can be seen that our method behaves better than the always valid test when the effect size is small, and comparable when the effect size is large. The always valid test fails in the adaptive randomization settings, as the type-I error rates are around $50 \\%$ under the null hypothesis. ",
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+ "type": "text",
+ "text": "E PROOFS ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Note that we require $\\mathbb { X }$ to be a compact set. To simplify the proof, we assume $\\mathbb { X } = [ 0 , 1 ] ^ { d }$ . ",
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+ "type": "image",
+ "img_path": "images/190f52fa2835cb10c09afb066ce68d7c745ed90ba954fd3930ae64a0529ced2e.jpg",
+ "image_caption": [
+ "Figure 4: Rejection probabilities and average stopping times of the proposed test when $\\alpha _ { 1 } ( \\cdot )$ is chosen as the spending function. "
+ ],
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+ "type": "text",
+ "text": "E.1 PROOF OF LEMMA 1 ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Set $\\mathcal { F } _ { 0 } = \\emptyset$ . We state the following lemma before proving Lemma 1. ",
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+ "text": "Lemma 3 For any $j \\geq 1$ $, ( X _ { j } , Y _ { j } ^ { * } ( 0 ) , Y _ { j } ^ { * } ( 1 ) ) \\perp \\mathcal { F } _ { j - 1 } .$ ",
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+ "type": "text",
+ "text": "For any $a \\in \\{ 0 , 1 \\} , i \\geq 1$ , notice that ",
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+ "img_path": "images/730e082aa5bcd7ccd7ef1a389e2354703f0f25ee81cfbe8c4b5669672572797b.jpg",
+ "text": "$$\n\\begin{array} { r } { \\mathrm { E I } ( A _ { i } = a ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} = \\mathrm { E I } ( A _ { i } = a ) \\{ Y _ { i } ^ { \\ast } ( a ) - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} } \\\\ { = \\mathrm { E E } ^ { X _ { i } , F _ { i - 1 } } [ \\mathbb { I } ( A _ { i } = a ) \\{ Y _ { i } ^ { \\ast } ( a ) - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} ] , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "where the first equation is due to Assumption (A1) and $\\mathrm { E } ^ { X _ { i } , \\mathcal { F } _ { i - 1 } }$ denotes the conditional expectation given $\\mathcal { F } _ { i - 1 }$ and $X _ { i }$ . By Assumption (A2), we have ",
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+ "img_path": "images/677db7c669ac5bb198a0f6cd4e41da9bafa76cfe6ddaad6407a4095eb4c7afcb.jpg",
+ "text": "$$\n{ } _ { \\mathfrak { X } ^ { \\lambda _ { i } , F _ { i - 1 } } [ \\mathbb { I } ( A _ { i } = a ) \\{ Y _ { i } ^ { * } ( a ) - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} ] } = \\{ \\mathrm { E } ^ { X _ { i } , F _ { i - 1 } } \\mathbb { I } ( A _ { i } = a ) \\} [ \\mathrm { E } ^ { X _ { i } , F _ { i - 1 } } \\{ Y _ { i } ^ { * } ( a ) - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} ]\n$$",
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+ "type": "text",
+ "text": "The second term on the RHS equals zero due to Lemma 3 and our model assumption $\\operatorname { E } \\{ Y _ { i } ^ { * } ( a ) | X _ { i } \\} = \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a }$ . The proof is hence completed. ",
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+ {
+ "type": "text",
+ "text": "E.2 PROOF OF THEOREM 1 ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "Let $n ( \\cdot )$ be the realization of the counting process $N ( \\cdot )$ . We will show the assertion in Theorem 1 holds for any such realizations that satisfy $n ( t _ { 1 } ) < n ( t _ { 2 } ) < \\cdot \\cdot \\cdot < n ( t _ { K } )$ . The case where some of the $n ( t _ { k } )$ ’s are the same can be similarly discussed. ",
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+ "type": "text",
+ "text": "For any $j \\geq 1$ , define $\\sigma ( \\mathcal { F } _ { j } )$ to be the $\\sigma$ -algebra generated by ${ \\mathcal { F } } _ { j }$ . For $a \\in \\{ 0 , 1 \\}$ , define ",
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+ "type": "equation",
+ "img_path": "images/678e7b08efa7470f9b2f2800f0d96ad6c5f215135bf4ebd3f8fde5b9cef4fcee.jpg",
+ "text": "$$\n\\widehat { \\Sigma } _ { a , j } = \\frac { 1 } { j } \\sum _ { i = 1 } ^ { j } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\ \\mathrm { ~ a n d ~ } \\widehat { \\beta } _ { a , j } = \\widehat { \\Sigma } _ { a , j } ^ { - 1 } \\left( \\frac { 1 } { j } \\sum _ { i = 1 } ^ { j } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) Y _ { i } \\right) .\n$$",
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+ "type": "text",
+ "text": "It is immediate to see that $\\widehat { \\Sigma } _ { a } ( t ) = \\widehat { \\Sigma } _ { a , n ( t ) }$ and $\\widehat { \\beta } _ { a } ( t ) = \\widehat { \\beta } _ { a , n ( t ) }$ . Define $\\delta _ { n } = q n ^ { - \\alpha _ { 0 } } \\log ^ { \\alpha _ { 0 } } n$ . We state the following lemmas before proving Theorem 1. ",
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+ "type": "text",
+ "text": "Lemma 4 There exists some constant $0 ~ < ~ \\epsilon _ { 0 } ~ < ~ 1$ such that $\\lambda _ { \\operatorname* { m i n } } [ E \\varphi ( X ) \\varphi ^ { \\top } ( X ) ] ~ \\geq ~ \\epsilon _ { 0 } ,$ , $\\begin{array} { r } { \\operatorname* { m a x } [ E \\varphi ( X ) \\varphi ^ { \\top } ( X ) ] \\leq \\epsilon _ { 0 } ^ { - 1 } , \\operatorname* { s u p } _ { x } \\| \\varphi ( x ) \\| _ { 2 } \\leq \\operatorname* { s u p } _ { x } \\| \\varphi ( x ) \\| _ { 1 } \\leq \\epsilon _ { 0 } ^ { - 1 } \\sqrt { q } , \\operatorname* { m i n } _ { a \\in \\{ 0 , 1 \\} } \\lambda _ { \\operatorname* { m i n } } [ \\Sigma _ { a } ] \\geqslant \\epsilon _ { 0 } ^ { - 1 } } \\end{array}$ \u000f0, $\\begin{array} { r } { \\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\| \\beta _ { a } \\| _ { 2 } \\leq \\epsilon _ { 0 } ^ { - 1 } , } \\end{array}$ , $\\begin{array} { r } { \\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } | Y ^ { * } ( a ) | \\leq \\epsilon _ { 0 } ^ { - 1 } } \\end{array}$ and $\\begin{array} { r } { \\operatorname* { s u p } _ { x } \\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } | \\varphi ^ { \\top } ( x ) \\beta _ { a } | \\leq \\epsilon _ { 0 } ^ { - 1 } } \\end{array}$ . ",
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+ "text": "Lemma 5 Assume the conditions in Theorem $I$ hold. Then for any sequence $\\{ j _ { n } \\} _ { n }$ that satisfies $j _ { n } ^ { \\alpha _ { 0 } } / \\log ^ { \\alpha _ { 0 } } ( j _ { n } ) \\gg q ^ { 2 }$ , we have with probability at least $1 - O ( j _ { n } ^ { - \\alpha _ { 0 } } )$ that for any $a \\in \\{ 0 , 1 \\}$ and ",
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+ "type": "text",
+ "text": "any $k \\geq j _ { n }$ ",
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+ "text": "$$\n\\begin{array} { r } { \\| ( \\widehat { \\Sigma } _ { a , k } - \\Sigma _ { a } ) \\| _ { 2 } \\preceq q \\delta _ { k } + \\sqrt { q k ^ { - 1 } \\log k } , } \\\\ { \\| ( \\widehat { \\Sigma } _ { a , k } ^ { - 1 } - \\Sigma _ { a } ^ { - 1 } ) \\| _ { 2 } \\preceq q \\delta _ { k } + \\sqrt { q k ^ { - 1 } \\log k } . } \\end{array}\n$$",
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+ "text": "Lemma 6 Assume the conditions in Theorem $^ { l }$ hold. The for any sequence $\\{ j _ { n } \\} _ { n }$ that satisfies $j _ { n } / \\log ( j _ { n } ) \\gg q ,$ , we have with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ that for any $a \\in \\{ 0 , 1 \\}$ and any $k \\geq j _ { n } ,$ , ",
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+ "text": "$$\n\\left\\| \\sum _ { i = 1 } ^ { k } \\varphi ( X _ { i } ) \\mathbb { I } ( A _ { i } = a ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} \\right\\| _ { 2 } \\preceq \\sqrt { q k \\log k } .\n$$",
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+ "type": "text",
+ "text": "For $a \\in \\{ 0 , 1 \\}$ , ",
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+ "text": "$$\n\\widehat { \\beta } _ { a , k } - \\beta _ { a } = \\widehat { \\Sigma } _ { a } ^ { - 1 } \\left[ \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} \\right] ,\n$$",
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+ "type": "text",
+ "text": "and hence ",
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+ "text": "$$\n\\begin{array} { r l } & { \\Bigg \\| \\widehat { \\beta } _ { a , k } - \\beta _ { a } - \\Sigma _ { a } ^ { - 1 } \\left[ \\frac { 1 } { k } \\displaystyle \\sum _ { i = 1 } ^ { k } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} \\right] \\Bigg \\| _ { 2 } } \\\\ { \\leq } & { \\| \\widehat { \\Sigma } _ { a , k } ^ { - 1 } - \\Sigma _ { a } ^ { - 1 } \\| _ { 2 } \\Bigg \\| \\frac { 1 } { k } \\displaystyle \\sum _ { i = 1 } ^ { k } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} \\Bigg \\| _ { 2 } } \\\\ { \\leq } & { ( q \\delta _ { k } + \\sqrt { q k ^ { - 1 } \\log k } ) q ^ { 1 / 2 } k ^ { - 1 / 2 } \\log ^ { 1 / 2 } k , ~ \\forall k \\geq j _ { n } , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "with probability at least $1 - O \\big ( j _ { n } ^ { - \\alpha _ { 0 } } \\big )$ , by Lemma 5 and Lemma 6. Define ",
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+ "text": "$$\nB ^ { * } ( t ) = { \\frac { 1 } { \\sqrt { n ( t ) } } } \\sum _ { i = 1 } ^ { n ( t ) } [ \\Sigma _ { 1 } ^ { - 1 } \\varphi ( X _ { i } ) A _ { i } \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 1 } \\} - \\Sigma _ { 0 } ^ { - 1 } \\varphi ( X _ { i } ) ( 1 - A _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 0 } \\} ] .\n$$",
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+ "text": "It follows that ",
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+ "text": "$$\n\\begin{array} { r } { \\| B ^ { * } ( t _ { k } ) - B ( t _ { k } ) \\| _ { 2 } \\preceq \\{ q ^ { 3 / 2 } \\delta _ { n ( t _ { k } ) } + q \\sqrt { n ^ { - 1 } ( t _ { k } ) \\log n ( t _ { k } ) } \\} n ^ { - 1 / 2 } ( t _ { k } ) \\log ^ { 1 / 2 } n ( t _ { k } ) , \\forall k \\geq 1 , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "with probability at least $1 - O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ , and hence ",
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+ "type": "equation",
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+ "text": "$$\n\\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) B ( t _ { k } ) \\| _ { 2 } \\leq \\bar { c } \\{ q ^ { 2 } \\delta _ { n ( t _ { k } ) } + q ^ { 3 / 2 } \\sqrt { n ^ { - 1 } ( t _ { k } ) \\log n ( t _ { k } ) } \\} \\sqrt { n ^ { - 1 } ( t _ { k } ) \\log n ( t _ { k } ) } .\n$$",
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+ "text": "with probability at least $1 - O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ , for some constant $\\bar { c } > 0$ , by equation 41. Under the given conditions on $q$ and $n ( t _ { 1 } )$ , we have ",
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+ "text": "$$\nq \\sqrt { n ^ { - 1 } ( t _ { k } ) \\log n ( t _ { k } ) } = o ( 1 ) , ~ \\forall k \\geq 1 ,\n$$",
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+ "text": "and hence ",
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+ "text": "$$\n\\| \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) B ( t _ { k } ) \\| _ { 2 } \\leq { \\bar { c } } \\{ q \\delta _ { n ( t _ { k } ) } + \\sqrt { q n ^ { - 1 } ( t _ { k } ) \\log n ( t _ { k } ) } \\} , \\forall k \\geq 1 ,\n$$",
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+ "type": "text",
+ "text": "Thus, for any given $z _ { 1 } , z _ { 2 } , \\dotsc , z _ { K }$ , we obtain ",
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+ "img_path": "images/421db15433f0508b092fd376de9810dd95d8d03150262dfaeabf6910a4162801.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - z _ { k , - } \\right) \\leq 0 \\right\\} - O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) ) } \\\\ { \\leq } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ( t _ { k } ) - z _ { k } \\right) \\leq 0 \\right\\} } \\\\ { \\leq } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - z _ { k , + } \\right) \\leq 0 \\right\\} + O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) ) , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "where ",
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+ "text": "$$\n\\begin{array} { r } { z _ { k , - } = z _ { k } - \\bar { c } \\{ q \\delta _ { n ( t _ { k } ) } + \\sqrt { q n ^ { - 1 } ( t _ { k } ) \\log n ( t _ { k } ) } \\} , } \\\\ { z _ { k , + } = z _ { k } + \\bar { c } \\{ q \\delta _ { n ( t _ { k } ) } + \\sqrt { q n ^ { - 1 } ( t _ { k } ) \\log n ( t _ { k } ) } \\} . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "For any $i \\geq 1 , 1 \\leq k \\leq K$ , define a $q$ -dimensional vector ",
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+ "img_path": "images/d762be66d00bf7c1ab2768fe4344df82a8342bb1706cc82bce32cf944c0d89a7.jpg",
+ "text": "$$\n\\dot { \\iota } _ { i , k } = \\frac { 1 } { \\sqrt { n ( t _ { k } ) } } [ \\Sigma _ { 1 } ^ { - 1 } \\varphi ( X _ { i } ) A _ { i } \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 1 } \\} - \\Sigma _ { 0 } ^ { - 1 } \\varphi ( X _ { i } ) ( 1 - A _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 0 } \\} ] \\mathbb { I } ( i \\leq n ( t _ { k } ) , 1 ) .\n$$",
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+ "type": "text",
+ "text": "or equivalently, ",
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+ "img_path": "images/2ca906cb0ea6c713fd744ac32e5b5117a5e5d828346e40b741c5e0fd06bd36ec.jpg",
+ "text": "$$\n\\dot { \\iota } _ { i , k } = \\frac { 1 } { \\sqrt { n ( t _ { k } ) } } [ \\Sigma _ { 1 } ^ { - 1 } \\varphi ( X _ { i } ) A _ { i } \\{ Y _ { i } ^ { * } ( 1 ) - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 1 } \\} - \\Sigma _ { 0 } ^ { - 1 } \\varphi ( X _ { i } ) ( 1 - A _ { i } ) \\{ Y _ { i } ^ { * } ( 0 ) - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 0 } \\} ] \\mathbb { I } ( \\dot { \\iota } _ { i } ) \\mathbb { 1 } - \\mu _ { 0 } \\mathbb { 1 } / \\mathbb { 1 } \\mu _ { 0 } ]\n$$",
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+ "type": "text",
+ "text": "by Condition (A1). Let $\\pmb { \\xi } _ { i } = ( \\xi _ { i , 1 } ^ { \\top } , \\xi _ { i , 2 } ^ { \\top } , \\cdot \\cdot \\cdot , \\xi _ { i , K } ^ { \\top } ) ^ { \\top }$ and $\\begin{array} { r } { \\mathcal { M } _ { j } = \\sum _ { i = 1 } ^ { j } \\pmb { \\xi } _ { i } } \\end{array}$ . The sequence $\\{ \\mathcal { M } _ { i } \\} _ { i \\ge 1 }$ forms a multivariate martingale with respect to the filtration $\\{ \\sigma ( \\mathcal { F } _ { i } ) : i \\ge 1 \\}$ , since ",
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+ "text": "$$\n\\begin{array} { r } { \\mathrm { E } ( \\xi _ { i , k } | \\mathcal { F } _ { i } ) = [ \\{ \\mathrm { E } ( \\xi _ { i , k } | A _ { i } , X _ { i } , \\mathcal { F } _ { i } ) \\} | \\mathcal { F } _ { i } ] = 0 , } \\end{array}\n$$",
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+ "text": "by (A2). Let $n ( t _ { 0 } ) = 0$ . For any $i$ such that $n ( t _ { k - 1 } ) < i \\leq n ( t _ { k } )$ for some $1 \\leq k \\leq K$ , we have ",
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+ "img_path": "images/85b7ff30c1a5a942165b107fd1062b8f5fd64f87562a685a86340928aa06b4fb.jpg",
+ "text": "$$\n\\xi _ { i } | | _ { \\infty } \\leq \\frac { 1 } { \\sqrt { n ( t _ { k } ) } } \\{ \\| \\Sigma _ { 1 } ^ { - 1 } \\varphi ( X _ { i } ) \\{ Y _ { i } ^ { * } ( 1 ) - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 1 } \\} \\| _ { 2 } + \\| \\Sigma _ { 0 } ^ { - 1 } \\varphi ( X _ { i } ) \\{ Y _ { i } ^ { * } ( 0 ) - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 0 } \\} \\| _ { 2 } \\}\n$$",
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+ "type": "text",
+ "text": "where the second inequality is due to Lemma 4. Therefore, ",
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+ "text": "$$\n\\mathbb { E } \\| \\pmb { \\xi } _ { i } \\| _ { \\infty } ^ { 3 } \\preceq \\frac { q ^ { 3 / 2 } } { n ^ { 3 / 2 } ( t _ { k } ) } .\n$$",
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+ "type": "text",
+ "text": "It follows that ",
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+ "img_path": "images/fc7e9df0d372bc3db2ccd0e79e89ffc42c40931801e2aa28fee8f4b1053e4cca.jpg",
+ "text": "$$\n\\begin{array} { r l } { { } } & { \\underset { i = 1 } { \\overset { n ( t _ { k } ) } { \\sum } } \\ \\mathrm { E } \\| \\xi _ { i } \\| _ { \\infty } ^ { 3 } = \\underset { k = 1 } { \\overset { K } { \\sum } } \\ \\underset { i = n ( t _ { k - 1 } ) + 1 } { \\overset { n ( t _ { k } ) } { \\sum } } \\ \\mathrm { E } \\| \\xi _ { i } \\| _ { \\infty } ^ { 3 } \\preceq q ^ { 3 / 2 } \\underset { k = 1 } { \\overset { K } { \\sum } } \\ \\frac { n ( t _ { k } ) - n ( t _ { k - 1 } ) } { n ^ { 3 / 2 } ( t _ { k } ) } } \\\\ { { \\leq } } & { \\frac { q ^ { 3 / 2 } } { \\sqrt { n ( t _ { 1 } ) } } + q ^ { 3 / 2 } \\underset { k = 2 } { \\overset { K } { \\sum } } \\ \\frac { n ( t _ { k } ) - n ( t _ { k - 1 } ) } { n ^ { 3 / 2 } ( t _ { k } ) } \\leq q ^ { 3 / 2 } n ^ { - 1 / 2 } ( t _ { 1 } ) + q ^ { 3 / 2 } \\int _ { n ( t _ { 1 } ) } ^ { + \\infty } x ^ { - 3 / 2 } d x = 3 q ^ { 3 / 2 } n ^ { - 1 / 2 } ( t _ { k } ) } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Define a sequence of independent Gaussian vectors $\\{ \\eta _ { i } \\} _ { i \\ge 1 }$ that satisfy $\\pmb { \\eta _ { i } } \\sim N ( 0 , \\mathbb { E } ( \\pmb { \\xi } _ { i } \\pmb { \\xi } _ { i } ^ { \\top } | \\mathcal { F } _ { i - 1 } ) )$ for any $i \\geq 1$ . Then the distribution of $\\eta _ { i }$ is the same as ",
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+ "text": "$$\n\\left( \\frac { \\mathbb { I } ( i \\leq n ( t _ { 1 } ) ) } { \\sqrt { n ( t _ { 1 } ) } } Z ^ { \\top } , \\frac { \\mathbb { I } ( i \\leq n ( t _ { 2 } ) ) } { \\sqrt { n ( t _ { 2 } ) } } Z ^ { \\top } , \\cdots , \\frac { \\mathbb { I } ( i \\leq n ( t _ { K } ) ) } { \\sqrt { n ( t _ { K } ) } } Z ^ { \\top } \\right) ,\n$$",
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+ {
+ "type": "text",
+ "text": "where $Z$ is a $p$ -dimensional mean-zero Gaussian vector with covariance matrix ",
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+ "img_path": "images/28b92267918a11a9272fd7cbfbf7c8d3621266cd2f948032dd1b4b5548f3d789.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\quad \\mathrm { C o v } [ \\displaystyle \\sum _ { \\alpha \\in \\{ 0 , 1 \\} } \\sum _ { a } ^ { - 1 } \\varphi ( X _ { i } ) ] [ ( A _ { i } = a ) \\{ Y _ { i } ^ { * } ( a ) - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} ] \\mathcal { F } _ { i - 1 } ] } \\\\ { = } & { \\quad \\sum _ { a \\in \\{ 0 , 1 \\} } \\Sigma _ { a } ^ { - 1 } \\mathbb { E } [ \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) ] [ ( A _ { i } = a ) \\{ Y _ { i } ^ { * } ( a ) - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} ^ { 2 } ] \\mathcal { F } _ { i - 1 } ] \\Sigma _ { a } ^ { - 1 } } \\\\ { = } & { \\quad \\sum _ { a \\in \\{ 0 , 1 \\} } \\Sigma _ { a } ^ { - 1 } \\mathbb { E } [ \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) ] [ ( A _ { i } = a ) \\sigma ^ { 2 } ( a , X _ { i } ) | \\mathcal { F } _ { i - 1 } ] \\Sigma _ { a } ^ { - 1 } } \\\\ & { = \\quad \\displaystyle \\sum _ { a \\in \\{ 0 , 1 \\} } \\sum _ { a } ^ { - 1 } \\mathbb { E } \\{ \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\pi _ { i - 1 } ( a , X _ { i } ) \\sigma ^ { 2 } ( a , X _ { i } ) | \\mathcal { F } _ { i - 1 } \\} \\Sigma _ { a } ^ { - 1 } } \\\\ { = } & { \\quad \\displaystyle \\sum _ { a \\in \\{ 0 , 1 \\} } \\Sigma _ { a } ^ { - 1 } \\mathbb { E } \\{ \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\pi _ { i - 1 } ( a , X _ { i } ) \\sigma ^ { 2 } ( a , X _ { i } ) | \\mathcal { F } _ { i - 1 } \\} \\Sigma _ { a } ^ { - 1 } } \\\\ { = } & { \\quad \\displaystyle \\sum _ { a \\in \\{ 0 , 1 \\} } \\Sigma _ { a } ^ { - 1 } \\mathbb { E } ^ { \\mathcal { F } _ { i - 1 } } \\pi _ { i - 1 } ( a , X ) \\sigma ^ { 2 } ( a , X ) \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\Sigma _ { a } ^ { - 1 } , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "where the second equality follows from (A2) and Lemma 3, the third equality is due to the definition of $\\pi _ { i - 1 }$ and the last equality follows from Lemma 3. ",
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+ "type": "text",
+ "text": "Similar to equation 19, we can show that ",
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+ "img_path": "images/7cccfdd817f82d536e29d24053e279a2d5351d6acdb36fccdbeda3d7b2444b63.jpg",
+ "text": "$$\n\\sum _ { i = 1 } ^ { n ( t _ { K } ) } \\operatorname { E } \\lVert \\pmb { \\eta } _ { i } \\rVert _ { \\infty } ^ { 3 } \\preceq q ^ { 3 / 2 } n ^ { - 1 / 2 } ( t _ { 1 } ) .\n$$",
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+ "type": "text",
+ "text": "Using similar arguments in equation 20, we can show that for any $1 \\leq k _ { 1 } \\leq k _ { 2 } \\leq K$ , ",
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+ "text": "$$\n\\sum _ { i = 1 } ^ { \\langle t \\kappa \\rangle } \\mathbf { E } \\{ \\xi _ { i } , _ { k _ { 1 } } \\xi _ { i , k _ { 2 } } ^ { \\top } | \\mathcal { F } _ { i - 1 } \\} = \\frac { 1 } { \\sqrt { n ( t _ { k _ { 1 } } ) n ( t _ { k _ { 2 } } ) } } \\sum _ { i = 1 } ^ { n ( t _ { k _ { 1 } } ) } \\sum _ { a \\in \\{ 0 , 1 \\} } \\Sigma _ { a } ^ { - 1 } \\mathbf { E } ^ { \\mathcal { F } _ { i - 1 } } \\pi _ { i - 1 } ( a , X ) \\sigma ^ { 2 } ( a , X ) \\varphi ( X ) \\varphi ^ { \\top } ( X ) ,\n$$",
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+ "type": "text",
+ "text": "Let ",
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+ "img_path": "images/f5899ff4840a54c280022aa48afb27710306014db22dde8be26d4e9dba7f07b2.jpg",
+ "text": "$$\n\\begin{array} { c } { { V ( k _ { 1 } , k _ { 2 } ) = \\displaystyle \\frac { 1 } { \\sqrt { n ( t _ { k _ { 1 } } ) n ( t _ { k _ { 2 } } ) } } \\sum _ { i = 1 } ^ { n ( t _ { k _ { 1 } } ) } \\sum _ { a \\in \\{ 0 , 1 \\} } \\Sigma _ { a } ^ { - 1 } \\mathbb { E } ^ { \\mathcal { F } _ { i - 1 } } \\pi ^ { * } ( a , X ) \\sigma ^ { 2 } ( a , X ) \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\Sigma _ { a } ^ { - 1 } } } \\\\ { { = \\displaystyle \\frac { 1 } { \\sqrt { n ( t _ { k _ { 1 } } ) n ( t _ { k _ { 2 } } ) } } \\sum _ { i = 1 } ^ { n ( t _ { k _ { 1 } } ) } \\sum _ { a \\in \\{ 0 , 1 \\} } \\Sigma _ { a } ^ { - 1 } \\Phi _ { a } \\Sigma _ { a } ^ { - 1 } = \\displaystyle \\frac { \\sqrt { n ( t _ { k _ { 1 } } ) } } { \\sqrt { n ( t _ { k _ { 2 } } ) } } \\sum _ { a \\in \\{ 0 , 1 \\} } \\Sigma _ { a } ^ { - 1 } \\Phi _ { a } \\Sigma _ { a } ^ { - 1 } . } } \\end{array}\n$$",
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+ },
+ {
+ "type": "text",
+ "text": "Consider an arbitrary sequence of $\\mathbb { R } ^ { p + 1 }$ vectors $\\left\\{ b _ { k } \\right\\} _ { 1 \\leq k \\leq K }$ . Under the given conditions, we have ",
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+ "img_path": "images/0e096c097238e0e6d88f62d25ac67af9085fd1aa3d0fd2f273d82510aeaedd7d.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\left| b _ { k _ { 1 } } ^ { \\top } \\left( \\displaystyle \\sum _ { i = 1 } ^ { n ( t _ { K } ) } \\mathsf { E } ( \\xi _ { i , k _ { 1 } } \\xi _ { 1 , k _ { 2 } } ^ { \\top } | \\mathcal { F } _ { i - 1 } ) - V ( k _ { 1 } , k _ { 2 } ) \\right) b _ { k _ { 2 } } \\right| } \\\\ { \\ \\preceq } & { \\frac { 1 } { n ( t _ { k _ { 1 } } ) } \\displaystyle \\sum _ { a \\in \\{ 0 , 1 \\} } \\left\\| \\displaystyle \\sum _ { i = 1 } ^ { n ( t _ { k _ { 1 } } ) } \\mathsf { E } ^ { \\mathcal { F } _ { i - 1 } } \\left\\{ \\pi _ { i - 1 } ( a , X ) - \\pi ^ { * } ( a , X ) \\right\\} \\sigma ^ { 2 } ( a , X ) \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\right\\| _ { 2 } \\| b _ { k _ { 1 } } \\| _ { 2 } \\| b _ { k _ { 2 } } \\| _ { 2 } . } \\end{array}\n$$",
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+ {
+ "type": "text",
+ "text": "Define a matrix $V$ as ",
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+ "type": "equation",
+ "img_path": "images/7235e14873273852a26c911377352b8d77b7ed0099727646f250964b143159de.jpg",
+ "text": "$$\nV = \\left( \\begin{array} { c c c c } { { V ( 1 , 1 ) } } & { { V ( 1 , 2 ) } } & { { . . . } } & { { V ( 1 , K ) } } \\\\ { { V ( 2 , 1 ) } } & { { V ( 2 , 2 ) } } & { { . . . } } & { { V ( 2 , K ) } } \\\\ { { \\vdots } } & { { \\vdots } } & { { } } & { { \\vdots } } \\\\ { { V ( K , 1 ) } } & { { V ( K , 2 ) } } & { { . . . } } & { { V ( K , K ) } } \\end{array} \\right) .\n$$",
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+ "type": "text",
+ "text": "It follows that ",
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+ "text": "$$\n\\left| \\sum _ { i = 1 } ^ { \\boldsymbol { n } ( t _ { k } ) } \\mathrm { E } ( \\xi _ { i } \\xi _ { i } ^ { \\top } | \\mathcal { F } _ { i - 1 } ) - V \\right| \\Bigg | _ { 2 } \\preceq \\operatorname* { s u p } _ { a \\in \\{ 0 , 1 \\} } \\left\\| \\frac { 1 } { j } \\sum _ { i = 1 } ^ { j } \\mathrm { E } ^ { \\mathcal { F } _ { i - 1 } } \\{ \\pi _ { i - 1 } ( a , X ) - \\pi ^ { * } ( a , X ) \\} \\sigma ^ { 2 } ( a , X ) \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\right\\| ^ { 2 } ,\n$$",
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+ "type": "text",
+ "text": "Using similar arguments in proving equation 14, we can show the RHS of the above equation is upper bounded by ",
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+ "text": "$$\n\\epsilon _ { 0 } ^ { - 2 } q \\operatorname* { s u p } _ { \\stackrel { a \\in \\{ 0 , 1 \\} } { x \\in \\mathbb { X } , j \\geq n ( t _ { 1 } ) } } \\left| \\frac { 1 } { j } \\sum _ { i = 1 } ^ { j } \\{ \\pi _ { i - 1 } ( a , x ) - \\pi ^ { * } ( a , x ) \\} \\right| ,\n$$",
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+ "type": "text",
+ "text": "and hence by $\\epsilon _ { 0 } ^ { - 2 } q \\delta _ { n ( t _ { 1 } ) }$ , with probability at least $1 - O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ . Therefore, we have ",
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+ "text": "$$\n\\lambda _ { \\operatorname* { m i n } } \\left[ { V + \\delta _ { n ( t _ { 1 } ) } I _ { K p \\times K p } - \\sum _ { i = 1 } ^ { n ( t _ { K } ) } \\operatorname { E } ( \\pmb { \\xi } _ { i } \\pmb { \\xi } _ { i } ^ { \\top } | \\mathcal { F } _ { i - 1 } ) } \\right] \\geq 0 ,\n$$",
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+ "text": "with probability at least $1 - O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ , where $I _ { K p \\times K p }$ denotes a $K p \\times K p$ identity matrix. ",
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+ "text": "Moreover, notice that ",
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+ "text": "$$\n\\operatorname* { s u p } _ { a \\in \\{ 0 , 1 \\} } \\left| { \\frac { 1 } { j } } \\sum _ { i = 1 } ^ { j } \\{ \\pi _ { i - 1 } ( a , x ) - \\pi ^ { * } ( a , x ) \\} \\right|\n$$",
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+ "type": "text",
+ "text": "is bounded between 0 and 1. For any $a \\in \\{ 0 , 1 \\}$ and any $z > 0$ , we have ",
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+ "text": "$$\n\\begin{array} { r l } { { } } & { { } \\underset { x \\in \\{ 0 , 1 \\} } { \\operatorname* { s u p } } } \\\\ { { \\leq } } & { { } \\underset { x \\in \\mathcal { X } , j \\geq n ( t _ { 1 } ) } { \\operatorname* { s u p } } | \\frac { 1 } { j } \\sum _ { i = 1 } ^ { j } \\{ \\pi _ { i - 1 } ( a , x ) - \\pi ^ { * } ( a , x ) \\} | } \\\\ { { + } } & { { } \\operatorname* { P r } ( \\underset { x \\in \\mathcal { X } , j \\geq n ( t _ { 1 } ) } { \\operatorname* { s u p } } | \\frac { 1 } { j } \\sum _ { i = 1 } ^ { j } \\{ \\pi _ { i - 1 } ( a , x ) - \\pi ^ { * } ( a , x ) \\} | \\mathbb { 1 } ( \\underset { a \\leq t \\leq \\{ 0 , 1 \\} } { \\operatorname* { s u p } } | \\frac { 1 } { j } \\sum _ { i = 1 } ^ { j } \\{ \\pi _ { i - 1 } ( a , x ) - \\pi ^ { * } ( a , x ) \\} | | } \\\\ { + } & { { } \\operatorname* { P r } ( \\underset { x \\in \\mathcal { X } , j \\geq n ( t _ { 1 } ) } { \\operatorname* { s u p } } | | \\frac { 1 } { j } \\sum _ { i = 1 } ^ { j } \\{ \\pi _ { i - 1 } ( a , x ) - \\pi ^ { * } ( a , x ) \\} | > z ) . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Under the given conditions, we have ",
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+ "text": "$$\n\\left. \\underset { x \\in \\mathbb { X } , j \\geq n ( t _ { 1 } ) } { \\operatorname* { E } } \\right| \\frac { 1 } { j } \\sum _ { i = 1 } ^ { j } \\{ \\pi _ { i - 1 } ( a , x ) - \\pi ^ { * } ( a , x ) \\} \\Bigg | \\preceq \\delta _ { n ( t _ { 1 } ) } + O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) ) .\n$$",
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+ "type": "text",
+ "text": "Therefore, we obtain ",
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+ "img_path": "images/573f80cc74c50021a0b41f82b6b341d4f4c10f983b92ed4373e964d864116f01.jpg",
+ "text": "$$\n\\mathbf { E } \\left\\| \\sum _ { i = 1 } ^ { n ( t _ { K } ) } \\mathrm { E } ( \\pmb { \\xi } _ { i } \\pmb { \\xi } _ { i } ^ { \\top } | \\mathcal { F } _ { i - 1 } ) - V \\right\\| _ { 2 } \\preceq q n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) + q \\delta _ { n ( t _ { 1 } ) } ,\n$$",
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+ "type": "text",
+ "text": "or ",
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+ "type": "equation",
+ "img_path": "images/438773c3fef7b56a505e0d074dc1896cfbd7585d027812ac5de0c667da66c658.jpg",
+ "text": "$$\n\\mathbf { E } \\left\\| \\sum _ { i = 1 } ^ { n ( t _ { K } ) } \\mathbf { E } ( \\pmb { \\xi } _ { i } \\pmb { \\xi } _ { i } ^ { \\top } | \\mathcal { F } _ { i - 1 } ) - V \\right\\| _ { 2 } \\preceq q \\delta _ { n ( t _ { 1 } ) } ,\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "since $n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) \\ll \\delta _ { n ( t _ { 1 } ) }$ . Combining equation 19 with equation 21, equation 23 and equation 24, an application of Theorem 2.1 in Belloni & Oliveira (2018) yields that ",
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+ "img_path": "images/56fcabf292d46d39a587a47c8f460d99923308fef0996f0a1c9c417789df9e3e.jpg",
+ "text": "$$\n\\begin{array} { r l } & { | \\mathrm { E } \\psi ( \\mathcal { M } _ { n ( t _ { K } ) } ) - \\mathrm { E } \\psi ( N ( 0 , { \\pmb V } ) ) | } \\\\ { \\preceq } & { c _ { 0 } ( \\psi ) n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) + c _ { 2 } ( \\psi ) q \\delta _ { n ( t _ { 1 } ) } + c _ { 3 } ( \\psi ) q ^ { 3 / 2 } n ^ { - 1 / 2 } ( t _ { 1 } ) , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "for any thrice differential function $\\psi ( \\cdot )$ , and ",
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+ "img_path": "images/7f5435e98450fb25a40eaafc886429b380896ef5d83951f1c52266c93bc353c7.jpg",
+ "text": "$$\nc _ { 0 } ( \\psi ) = \\operatorname* { s u p } _ { z , z ^ { \\prime } \\in \\mathbb { R } ^ { p K } } | \\psi ( z ) - \\psi ( z ^ { \\prime } ) | \\mathrm { ~ a n d ~ } c _ { i } = \\operatorname* { s u p } _ { z \\in \\mathbb { R } ^ { p K } } \\sum _ { j _ { 1 } , \\cdots , j _ { i } } | \\partial _ { j _ { 1 } } \\partial _ { j _ { 2 } } \\cdots \\partial _ { j _ { i } } \\psi ( z ) | , i = 2 , 3 ,\n$$",
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+ {
+ "type": "text",
+ "text": "where $\\partial _ { j } g ( z )$ denotes the partial derivative $\\partial g ( z ) / \\partial z ^ { ( j ) }$ for any function $\\mathbf { g } ( \\cdot )$ and $z ^ { ( j ) }$ stands for the $j$ -th element of $z$ . ",
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+ "type": "text",
+ "text": "Let ${ \\mathbb X } _ { k , 0 }$ be an $\\varepsilon$ -net of $\\mathbb { X }$ that satisfies the following: for any $x \\in \\mathbb { X }$ , there exists some $x _ { 0 } \\in \\mathbb { X } _ { 0 }$ such that $\\| x - x _ { 0 } \\| _ { 2 } \\leq \\varepsilon$ . Set $\\varepsilon = \\sqrt { d } / n ^ { 4 } ( t _ { 1 } )$ . Since $\\mathbb { X } = [ 0 , 1 ] ^ { d }$ , there exists some ${ \\mathbb X } _ { 0 }$ with ",
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+ "text": "$$\n| \\mathbb { X } _ { 0 } | \\leq n ^ { 4 d } ( t _ { 1 } ) ,\n$$",
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+ {
+ "type": "text",
+ "text": "where $| \\mathbb { X } _ { 0 } |$ denotes the number of elements in ${ \\mathbb X } _ { 0 }$ . Under Condition (A3), we have ",
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+ "img_path": "images/b5f6c27b39a710f3a27c194c6262bc197aefc43185cf428decb7e824843d8e6c.jpg",
+ "text": "$$\n\\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\operatorname* { i n f } _ { x _ { 0 } \\in \\mathbb { X } _ { 0 } } \\| \\varphi ( x ) - \\varphi ( x _ { 0 } ) \\| _ { 2 } \\preceq \\frac { \\sqrt { q } } { n ^ { 4 } ( t _ { 1 } ) } .\n$$",
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+ "text": "It follows that ",
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+ "img_path": "images/8d3f39bbbce5737299289409a80b88a62fde42a3a596baab6bef00056f169c62.jpg",
+ "text": "$$\n\\operatorname* { s u p } _ { \\| \\nu \\| _ { 2 } = 1 } | \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) \\nu - \\operatorname* { s u p } _ { x \\in \\mathbb { X } _ { 0 } } \\varphi ^ { \\top } ( x ) \\nu | \\preceq \\frac { \\sqrt { q } } { n ^ { 4 } ( t _ { 1 } ) } .\n$$",
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+ "type": "text",
+ "text": "Using similar arguments in showing equation 17, we can show the following event occurs with probability at least $1 - O ( n ^ { - 1 } ( t _ { 1 } ) )$ , ",
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+ "text": "$$\n\\| B ^ { * } ( t _ { k } ) \\| _ { 2 } \\preceq q ^ { 1 / 2 } \\log ^ { 1 / 2 } n ( t _ { k } ) , ~ \\forall k \\geq 1 .\n$$",
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+ "text": "This together with equation 27 yields ",
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+ "img_path": "images/06a76f0351a6ac04842346ee2484d51bdb7b3a1850193c25864c722b470f09ff.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\left| \\underset { k \\in \\{ 1 , \\cdots , K \\} } { \\operatorname* { m a x } } \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - \\underset { k \\in \\{ 1 , \\cdots , K \\} } { \\operatorname* { m a x } } \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) \\right| } \\\\ { \\leq } & { \\underset { k \\in \\{ 1 , \\cdots , K \\} } { \\operatorname* { m a x } } \\left| \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) \\right| \\preceq \\frac { q \\log n ^ { 1 / 2 } ( t _ { K } ) } { n ^ { 4 } ( t _ { 1 } ) } , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "with probability at least $\\mathrm { ~ 1 ~ - ~ } O ( n ^ { - 1 } ( t _ { 1 } ) )$ . Under the given conditions, we have $n ( t _ { 1 } ) ~ \\gg$ $\\operatorname* { m a x } ( q , \\log n ( t _ { K } ) )$ . It follows that there exists some constant $\\bar { c } ^ { * } > 0$ such that ",
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+ "img_path": "images/14f734545f4fe0a929eb1ff16ad9e37c8152fed6840a03cfeb3ac252d049ef6a.jpg",
+ "text": "$$\n\\left. \\operatorname* { m a x } _ { k \\in \\{ 1 , \\ldots , K \\} } \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) B ^ { \\ast } ( t _ { k } ) - \\operatorname* { m a x } _ { k \\in \\{ 1 , \\ldots , K \\} } \\operatorname* { s u p } _ { x \\in \\mathbb { X } _ { 0 } } \\varphi ^ { \\top } ( x ) B ^ { \\ast } ( t _ { k } ) \\right. \\leq { \\vec { c } } ^ { \\ast } n ^ { - 2 } ( t _ { 1 } ) ,\n$$",
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+ "type": "text",
+ "text": "with probability at least $1 - O ( n ^ { - 1 } ( t _ { 1 } ) )$ . ",
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+ "type": "text",
+ "text": "Define ",
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+ "img_path": "images/155b86c7e0da21b2776f99df075b14bbdb0e9bbe1f73280a28c7a21f8be52c6e.jpg",
+ "text": "$$\n\\begin{array} { r l } & { z _ { k , - } ^ { * } = z _ { k } - \\bar { c } \\{ q \\delta _ { n ( t _ { k } ) } + \\sqrt { q n ^ { - 1 } ( t _ { k } ) \\log n ( t _ { k } ) } \\} - \\bar { c } ^ { * } n ^ { - 2 } ( t _ { 1 } ) , } \\\\ & { z _ { k , + } ^ { * } = z _ { k } + \\bar { c } \\{ q \\delta _ { n ( t _ { k } ) } + \\sqrt { q n ^ { - 1 } ( t _ { k } ) \\log n ( t _ { k } ) } \\} + \\bar { c } ^ { * } n ^ { - 2 } ( t _ { 1 } ) . } \\end{array}\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "Combining equation 28 with equation 18 yields ",
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+ "type": "equation",
+ "img_path": "images/ed3b67cbac3ea0556546bcf778c7a3683ad1e5dfb864c3d6be172a5b75651845.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - z _ { k , - } ^ { * } \\right) \\leq 0 \\right\\} - O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) ) } \\\\ { \\leq } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ( t _ { k } ) - z _ { k } \\right) \\leq 0 \\right\\} } \\\\ { \\leq } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - z _ { k , + } ^ { * } \\right) \\leq 0 \\right\\} + O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) ) . } \\end{array}\n$$",
+ "text_format": "latex",
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+ },
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+ "type": "text",
+ "text": "Notice that $\\mathcal { M } _ { n ( t _ { K } ) } = \\{ B ^ { * } ( t _ { 1 } ) ^ { \\top } , B ^ { * } ( t _ { 2 } ) ^ { \\top } , \\cdot \\cdot \\cdot , B ^ { * } ( t _ { K } ) ^ { \\top } \\} ^ { \\top }$ . By equation 26 and Lemma 4, there exist a set of vectors $d _ { 1 } , d _ { 2 } , \\dots , d _ { L } \\in \\mathbb { R } ^ { q K }$ with $L \\leq n ^ { 4 d } ( t _ { 1 } ) K$ , $\\mathrm { m a x } _ { j } \\| d _ { j } \\| _ { 1 } \\leq \\epsilon _ { 0 } ^ { - 1 } q ^ { 1 / 2 }$ and a function $k ( \\cdot )$ that maps $\\{ 1 , \\ldots , L \\}$ into $\\{ 1 , \\ldots , K \\}$ such that ",
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+ "text": "$$\n\\operatorname* { m a x } _ { k \\in \\{ 1 , \\ldots , K \\} } \\left\\{ \\operatorname* { s u p } _ { x \\in \\mathbb { X } _ { 0 } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - \\nu _ { k } \\right\\} = \\operatorname* { m a x } _ { 1 \\leq j \\leq L } \\{ d _ { j } ^ { \\top } \\mathcal { M } _ { n ( t _ { K } ) } - \\nu _ { k ( j ) } \\} ,\n$$",
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+ "text": "for any $\\{ \\nu _ { k } \\} _ { k = 1 } ^ { K }$ . For any $\\eta > 0$ , $m \\in \\mathbb { R } ^ { q K }$ , consider the function $\\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } : \\mathbb { R } ^ { q K } \\mathbb { R }$ , defined as ",
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+ "type": "equation",
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+ "text": "$$\n\\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ( m ) = \\frac { 1 } { \\eta } \\log \\left\\{ \\sum _ { j = 1 } ^ { L } \\exp [ \\eta \\{ d _ { j } ^ { \\top } m - \\eta \\nu _ { k ( j ) } \\} ] \\right\\} .\n$$",
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+ "text": "It has the following property: ",
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+ "text": "$$\n\\begin{array} { r l } & { \\underset { 1 \\leq j \\leq L } { \\operatorname* { m a x } } \\{ d _ { j } ^ { \\top } m - \\nu _ { k ( j ) } \\} \\leq \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ( m ) \\leq \\underset { 1 \\leq j \\leq L } { \\operatorname* { m a x } } \\{ d _ { j } ^ { \\top } m - \\nu _ { k ( j ) } \\} + \\eta ^ { - 1 } \\log L } \\\\ { \\leq } & { \\underset { 1 \\leq j \\leq L } { \\operatorname* { m a x } } \\{ d _ { j } ^ { \\top } m - \\nu _ { k ( j ) } \\} + \\eta ^ { - 1 } \\{ \\log K + 4 d \\log n ( t _ { 1 } ) \\} } \\\\ { = } & { \\underset { 1 \\leq j \\leq L } { \\operatorname* { m a x } } [ d _ { j } ^ { \\top } m - \\{ \\nu _ { k ( j ) } - \\eta ^ { - 1 } \\log K - \\eta ^ { - 1 } 4 d \\log n ( t _ { 1 } ) \\} ] . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "It follows that ",
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+ "text": "$$\n\\begin{array} { r l } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - z _ { k , + } ^ { * } \\right) \\leq 0 \\right\\} \\leq \\operatorname* { P r } \\left\\{ \\phi _ { \\eta , \\{ z _ { k , + } ^ { * * } \\} _ { k } ^ { * * } \\left( \\mathcal { M } _ { n ( t _ { K } ) } \\right) \\leq 0 } , \\quad \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\Omega , } } \\\\ & \\right\\{ \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - z _ { k , - } ^ { * } \\right) \\leq 0 \\right\\} } & { \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\Omega ~ } \\\\ { = } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - ( z _ { k , - } ^ { * } - 3 \\delta ) \\right) \\leq 3 \\delta \\right\\} \\geq \\operatorname* { P r } \\left\\{ \\phi _ { \\eta , \\{ z _ { k , - } ^ { * * } \\} _ { k } } \\left( \\mathcal { M } _ { n ( t _ { K } ) } \\right) \\leq 3 \\delta \\right\\} } \\end{array}\n$$",
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+ "type": "text",
+ "text": "where ",
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+ "type": "text",
+ "text": "The value of $\\delta$ will be specified later. In addition, with some calculations, we have ",
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+ "type": "equation",
+ "img_path": "images/caeebe71ea528fa4e1da56a329240f3fb7744f71fb5b1a54ec1516336c33e37f.jpg",
+ "text": "$$\n\\begin{array} { r l } { \\partial _ { t } \\phi _ { \\mathrm { c a i r c } } \\Delta \\phi _ { \\mathrm { c a i r c } } } & { \\to - \\frac { \\sqrt { 2 } - \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\rho \\mathrm { e r f } ( \\phi _ { \\mathrm { c a i r c } } ^ { 2 } - \\phi _ { \\mathrm { c a c i r c } } ^ { 2 } ) , } \\\\ { \\partial _ { t } \\phi _ { \\mathrm { c a i r c } } \\Delta \\phi _ { \\mathrm { c a i r c } } } & { \\to - \\frac { \\sqrt { 2 } - \\alpha ^ { 2 } } { \\sqrt { 2 } } \\rho \\mathrm { e r f } ( \\phi _ { \\mathrm { c a i r c } } ^ { 2 } ) \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\exp ( - \\pi \\phi _ { \\mathrm { c a i r c } } ^ { 2 } ) , } \\\\ { \\partial _ { t } \\phi _ { \\mathrm { c a i r c } } \\Delta \\phi _ { \\mathrm { c a i r c } } } & { \\to \\frac { \\sqrt { 2 } - \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\exp ( \\exp ( - \\pi \\phi _ { \\mathrm { c a i r c } } ^ { 2 } ) ) , } \\\\ { \\phantom { = } } & { \\to \\frac { \\sqrt { 2 } - \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\exp ( \\exp ( - \\pi \\phi _ { \\mathrm { c a i r c } } ^ { 2 } ) ) , } \\\\ { \\phantom { = } } & { \\to \\frac { \\sqrt { 2 } - \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\exp ( \\exp ( - \\pi \\phi _ { \\mathrm { c a c c } } ^ { 2 } ) ) ^ { \\frac { 1 } { \\sqrt { 2 } } } , } \\\\ \\frac { \\sqrt { 2 } - \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac { \\alpha ^ { 2 } } { \\sqrt { 2 } } \\frac \\end{array}\n$$",
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+ "text": "Since $\\operatorname* { m a x } _ { i } \\| d _ { i } \\| _ { 1 } \\leq \\epsilon _ { 0 } ^ { - 1 } q ^ { 1 / 2 }$ , we obtain that ",
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+ "img_path": "images/8470b59a097e03c6a0cad5fd174cc11f15d8f6e0774f502c7ce567ce991afb0d.jpg",
+ "text": "$$\n\\begin{array} { r l r } { { \\sum _ { j } | \\partial _ { j } \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ( m ) | \\le \\epsilon _ { 0 } ^ { - 1 } q ^ { 1 / 2 } , \\sum _ { j _ { 1 } , j _ { 2 } } | \\partial _ { j _ { 1 } } \\partial _ { j _ { 2 } } \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ( m ) | \\le 2 \\eta \\epsilon _ { 0 } ^ { - 2 } q , } } \\\\ & { } & { \\ \\sum _ { j _ { 1 } , j _ { 2 } , j _ { 3 } } | \\partial _ { j _ { 1 } } \\partial _ { j _ { 2 } } \\partial _ { j _ { 3 } } \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ( m ) | \\le 6 \\eta ^ { 2 } \\epsilon _ { 0 } ^ { - 3 } q ^ { 3 / 2 } . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "By Lemma 5.1 of Chernozhukov et al. (2016), for any $\\delta > 0$ , there exists some function $\\mathbf { g } _ { \\delta } ( \\cdot ) : \\mathbb { R } \\to$ $\\mathbb { R }$ with $\\| \\mathbf { g } _ { \\delta } ^ { \\prime } \\| _ { \\infty } \\leq \\delta ^ { - 1 } , \\| \\mathbf { g } _ { \\delta } ^ { \\prime \\prime } \\| _ { \\infty } \\leq K _ { 0 } \\delta ^ { - 2 } , \\| \\mathbf { g } _ { \\delta } ^ { \\prime \\prime \\prime } \\| _ { \\infty } \\leq K _ { 0 } \\delta ^ { - 3 } \\mathbf { f }$ or some constant $K _ { 0 } > 0$ such that ",
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+ "type": "text",
+ "text": "It follows that ",
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+ "text": "$$\n\\mathbb { I } ( \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ( m ) \\le 0 ) \\le \\mathtt { g o } \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ( m ) \\le \\mathbb { I } ( \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ( m ) \\le 3 \\delta ) ,\n$$",
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+ "type": "text",
+ "text": "for any $m \\in \\mathbb { R } ^ { q K }$ . Combining this together with equation 30, equation 31 and equation 32, we obtain that ",
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+ "text": "$$\n\\begin{array} { r l } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - z _ { k , + } ^ { * } \\right) \\leq 0 \\right\\} \\leq \\mathrm { E } \\mathrm { g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ z _ { k , + } ^ { * * } \\} _ { k } } \\big ( \\mathcal { M } _ { n ( t _ { K } ) } \\big ) , } \\\\ & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - z _ { k , - } ^ { * } \\right) \\leq 0 \\right\\} \\geq \\mathrm { E } \\mathrm { g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ z _ { k , - } ^ { * * } \\} _ { k } } \\big ( \\mathcal { M } _ { n ( t _ { K } ) } \\big ) . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Consider the function $\\mathbf { g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } }$ . Apparently, we have ",
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+ "img_path": "images/094a877e5d6e3f659d5f4ab3057d5f6839dfd5ed209e3ec8378bcc77448a589e.jpg",
+ "text": "$$\n\\operatorname* { s u p } _ { \\delta , \\eta , \\{ \\nu _ { k } \\} _ { k } } c _ { 0 } ( \\mathbf { g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ) \\leq 1 .\n$$",
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+ "type": "text",
+ "text": "By equation 33, we can show that ",
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+ "text": "$$\n\\begin{array} { r l r } & { } & { \\underset { \\delta , \\eta , \\{ \\nu _ { k } \\} _ { k } } { \\operatorname* { s u p } } c _ { 2 } ( \\mathbf { g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ) \\preceq \\delta ^ { - 2 } q + \\delta ^ { - 1 } \\eta q , } \\\\ & { } & { \\underset { \\delta , \\eta , \\{ \\nu _ { k } \\} _ { k } } { \\operatorname* { s u p } } c _ { 3 } ( \\mathbf { g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ) \\preceq \\delta ^ { - 3 } q ^ { 3 / 2 } + \\delta ^ { - 2 } \\eta q ^ { 3 / 2 } + \\delta ^ { - 1 } \\eta ^ { 2 } q ^ { 3 / 2 } . } \\end{array}\n$$",
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+ "text": "Set $\\delta = \\eta ^ { - 1 } \\{ \\log K + 4 d \\log n ( t _ { 1 } ) \\}$ , we obtain ",
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+ "text": "$$\n\\operatorname* { s u p } _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } c _ { i } ( \\mathbf { g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ) \\preceq q ^ { i / 2 } \\eta ^ { i } \\{ \\log ^ { i } K + \\log ^ { i } n ( t _ { 1 } ) \\} , i = 2 , 3 .\n$$",
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+ "type": "text",
+ "text": "Combining equation 37 together with equation 25 and equation 36 yields ",
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+ "text": "$$\n\\begin{array} { r l } & { \\underset { \\delta , \\eta , \\{ \\nu _ { k } \\} _ { k } } { \\operatorname* { s u p } } \\vert \\mathrm { E g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ( \\mathcal { M } _ { n ( t _ { K } ) } ) - \\mathrm { E g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ( N ( 0 , V ) ) \\vert } \\\\ { \\overset { \\mathrm { } } { \\to } } & { n ^ { - 1 / 2 } ( t _ { 1 } ) q ^ { 3 } \\eta ^ { 3 } \\{ \\log ^ { 3 } N + \\log ^ { 3 } n ( t _ { 1 } ) \\} + q ^ { 2 } \\eta ^ { 2 } \\{ \\log ^ { 2 } N + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\delta _ { n ( t _ { 1 } ) } + n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) . } \\end{array}\n$$",
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+ "text": "This together with equation 34 and equation 35 yields ",
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+ "img_path": "images/9abc401e8d8d32cf2d15f3b65b268568e0271e235b7abd237b0453b8ef3db119.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - z _ { k , + } ^ { * } \\right) \\leq 0 \\right\\} - \\mathrm { E g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ z _ { k , + } ^ { * } \\} _ { k } } ( N ( 0 , V ) ) \\quad ( 3 ) } \\\\ { \\preceq } & { n ^ { - 1 / 2 } ( t _ { 1 } ) q ^ { 3 } \\eta ^ { 3 } \\{ \\log ^ { 3 } K + \\log ^ { 3 } n ( t _ { 1 } ) \\} + q ^ { 2 } \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\delta _ { n ( t _ { 1 } ) } + n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) , } \\\\ & { \\mathrm { E g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ z _ { k , - } ^ { * } \\} _ { k } } ( N ( 0 , V ) ) - \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - z _ { k , - } ^ { * } \\right) \\leq 0 \\right\\} \\quad ( 3 ) } \\\\ { \\preceq } & { n ^ { - 1 / 2 } ( t _ { 1 } ) q ^ { 3 } \\eta ^ { 3 } \\{ \\log ^ { 3 } K + \\log ^ { 3 } n ( t _ { 1 } ) \\} + q ^ { 2 } \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\delta _ { n ( t _ { 1 } ) } + n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Similar to equation 31-equation 35, we can show ",
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+ "type": "equation",
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+ "text": "$$\n\\begin{array} { r l } & { { \\mathrm { E g } } _ { \\delta } \\circ \\phi _ { \\eta , \\{ z _ { k , + } ^ { * * } \\} _ { k } } ( N ( 0 , V ) ) \\leq \\operatorname* { P r } \\left( \\phi _ { \\eta , \\{ z _ { k , + } ^ { * * } \\} _ { k } } ( N ( 0 , V ) ) \\leq 3 \\delta \\right) } \\\\ { \\leq } & { \\operatorname* { P r } \\left( \\underset { 1 \\leq j \\leq L } { \\operatorname* { m a x } } \\{ d _ { j } ^ { \\top } N ( 0 , V ) - z _ { k ( j ) , + } ^ { * * } \\} \\leq 3 \\delta \\right) = \\operatorname* { P r } \\left( \\underset { 1 \\leq j \\leq L } { \\operatorname* { m a x } } \\{ d _ { j } ^ { \\top } N ( 0 , V ) - z _ { k ( j ) , + } ^ { * * * } \\} \\leq 0 \\right) , } \\\\ & { { \\mathrm { E g } } _ { \\delta } \\circ \\phi _ { \\eta , \\{ z _ { k , - } ^ { * * } \\} _ { k } } ( N ( 0 , V ) ) \\geq \\operatorname* { P r } \\left( \\phi _ { \\eta , \\{ z _ { k , - } ^ { * * } \\} _ { k } } ( N ( 0 , V ) ) \\leq 0 \\right) } \\\\ { \\geq } & { \\operatorname* { P r } \\left( \\underset { 1 \\leq j \\leq L } { \\operatorname* { m a x } } \\{ d _ { j } ^ { \\top } N ( 0 , V ) - z _ { k ( j ) , - } ^ { * * * } \\} \\leq 0 \\right) , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "where ",
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+ "text": "$= z _ { k , + } ^ { * } + \\eta ^ { - 1 } \\{ \\log K + 4 d \\log n ( t _ { 1 } ) \\} + 3 \\delta \\mathrm { ~ a n d ~ } z _ { k , - } ^ { * * * } = z _ { k , - } ^ { * } - \\eta ^ { - 1 } \\{ \\log K + 4 d \\log n ( t _ { 1 } ) \\} - 3 \\delta ,$ for each $k$ . Notice that for any $\\{ \\nu _ { k } \\} _ { k }$ , we have ",
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+ "img_path": "images/89cf405c52b003e24b8ec1070abcc10a8137a413a973d904356e5eba3dfd5b55.jpg",
+ "text": "$$\n\\operatorname* { P r } \\left\\{ \\operatorname* { m a x } _ { k \\in \\{ 1 , \\ldots , K \\} } \\left( \\operatorname* { s u p } _ { x \\in \\Xi _ { 0 } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - \\nu _ { k } \\right) \\leq 0 \\right\\} = \\operatorname* { P r } \\left( \\operatorname* { m a x } _ { 1 \\leq j \\leq L } \\{ d _ { j } ^ { \\top } N ( 0 , V ) - \\nu _ { k ( j ) } \\} \\leq 0 \\right) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "This together with equation 38 and equation 39 yields ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/0d3947b1be72ef2768f083311df693501e972ff85a5a696510cdc5b8bc280c17.jpg",
+ "text": "$$\n\\begin{array} { r l r } & { } & { \\mathrm { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { R } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - z _ { k , + } ^ { * } \\right) \\leq 0 \\right\\} - \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { R } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - z _ { k , + } ^ { * * } \\right) \\leq 0 \\right\\} } \\\\ & { } & { \\preceq \\ 3 n ^ { - 1 / 2 } ( t _ { 1 } ) q ^ { 3 } \\eta ^ { 3 } \\{ \\log ^ { 3 } K + \\log ^ { 3 } n ( t _ { 1 } ) \\} + q ^ { 2 } \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\delta _ { n ( t _ { 1 } ) } + n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) \\delta _ { n ( t _ { 1 } ) } } \\\\ & { } & { \\mathrm { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { R } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - z _ { k , - } ^ { * * * } \\right) \\leq 0 \\right\\} - \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { R } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ^ { * } ( t _ { k } ) - z _ { k , - } ^ { * } \\right) \\leq 0 \\right\\} } \\\\ & { } & { \\preceq \\lambda n ^ { - 1 / 2 } ( t _ { 1 } ) q ^ { 3 } \\eta ^ { 3 } \\{ \\log ^ { 3 } K + \\log ^ { 3 } n ( t _ { 1 } ) \\} + q ^ { 2 } \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\delta _ { n ( t _ { 1 } ) } + n ^ { - \\alpha _ { 0 } } , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 20
+ },
+ {
+ "type": "text",
+ "text": "In view of equation 29, we have shown that ",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "img_path": "images/f3cbf46334ffb88e7d7895b71d58ec13067029780a6e7e07d0edc49df17269e6.jpg",
+ "text": "$$\n\\begin{array} { r l r } & { } & { \\mathrm { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\dots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) B ( t _ { k } ) - z _ { k } \\right) \\leq 0 \\right\\} - \\mathrm { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\dots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - z _ { k , + } ^ { * * } \\right) \\leq 0 \\right\\} } \\\\ & { } & { \\qquad \\preceq \\lambda \\mathrm { \\Lambda } ^ { - 1 / 2 } ( t _ { 1 } ) q ^ { 3 } \\eta ^ { 3 } \\{ \\log ^ { 3 } K + \\log ^ { 3 } n ( t _ { 1 } ) \\} + q ^ { 2 } \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\delta _ { n ( t _ { 1 } ) } + n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) , } \\\\ & { } & { \\mathrm { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\dots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - z _ { k , - } ^ { * * * } \\right) \\leq 0 \\right\\} - \\mathrm { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\dots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } } { \\operatorname* { m p } } \\varphi ^ { \\top } ( x ) B ( t _ { k } ) - z _ { k } \\right) \\leq 0 \\right\\} } \\\\ & { } & { \\qquad \\preceq \\lambda \\mathrm { \\Lambda } ^ { - 1 / 2 } ( t _ { 1 } ) q ^ { 3 } \\eta ^ { 3 } \\{ \\log ^ { 3 } K + \\log ^ { 3 } n ( t _ { 1 } ) \\} + q ^ { 2 } \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\delta _ { n ( t _ { 1 } ) } + n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "The covariance matrix $\\operatorname { C o v } ( G ( t _ { k } ) )$ is given by $\\begin{array} { r } { \\sum _ { a \\in \\{ 0 , 1 \\} } \\sum _ { a } ^ { - 1 } \\Phi _ { a } \\Sigma _ { a } ^ { - 1 } } \\end{array}$ and is nonsingular by Lemma 4. In addition, we have $\\| \\varphi ( x ) \\| _ { 2 } \\geq \\bar { c } , \\forall x \\in \\mathbb { X } _ { 0 }$ , by Condition A3. Thus, there exists some constant $c _ { * } > 0$ such that ",
+ "bbox": [
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+ "type": "equation",
+ "img_path": "images/f2e147a1164f6f226793440b2c9e630859aa23e409bb5f4cc4f330de0d376ddf.jpg",
+ "text": "$$\nc _ { * } \\leq \\varphi ^ { \\top } ( x ) \\left( \\sum _ { a \\in \\{ 0 , 1 \\} } \\Sigma _ { a } ^ { - 1 } \\Phi _ { a } \\Sigma _ { a } ^ { - 1 } \\right) ^ { 1 / 2 } \\varphi ( x ) , \\forall x \\in \\mathbb { X } _ { 0 } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "By Theorem 1 of Chernozhukov et al. (2017), we obtain that ",
+ "bbox": [
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+ "page_idx": 21
+ },
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+ "type": "equation",
+ "img_path": "images/80bb55b8595d580f8a724cd3ea854ca803a5ce984e04d39df56334fd877c55d3.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\operatorname* { p } _ { \\mathrm { { r } } } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - z _ { k , + } ^ { * * * } \\right) \\leq 0 \\right\\} - \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - z _ { k , - } ^ { * * * } \\right) \\leq 0 \\right\\} } \\\\ & { \\ l ^ { - 1 } \\{ \\log K + \\log n ( t _ { 1 } ) \\} ^ { 3 / 2 } + q \\delta _ { n ( t _ { 1 } ) } \\{ \\log K + \\log n ( t _ { 1 } ) \\} ^ { 1 / 2 } + \\sqrt { q n ^ { - 1 } ( t _ { 1 } ) \\log n ( t _ { 1 } ) } \\{ \\log K + \\log n ( t _ { 1 } ) \\} } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "Thus, we obtain ",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "img_path": "images/baffea1771cd54328b90e59df4f5c3314d00ce0db24212055e1c6a2264160b9e.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathfrak { X } } { \\operatorname* { m a x } } \\varphi ^ { \\top } ( x ) B ( t _ { k } ) - z _ { k } \\right) \\leq 0 \\right\\} - \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathfrak { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - z _ { k } \\right) \\leq 0 \\right\\} \\bigg | } \\\\ & { \\ i ^ { - 1 / 2 } ( t _ { 1 } ) q ^ { 3 } \\eta ^ { 3 } \\{ \\log ^ { 3 } K + \\log ^ { 3 } n ( t _ { 1 } ) \\} + q ^ { 2 } \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\delta _ { n ( t _ { 1 } ) } + n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) } \\\\ & { \\ j ^ { - 1 } \\{ \\log K + \\log n ( t _ { 1 } ) \\} ^ { 3 / 2 } + q \\delta _ { n ( t _ { 1 } ) } \\{ \\log K + \\log n ( t _ { 1 } ) \\} ^ { 1 / 2 } + \\sqrt { q n ^ { - 1 } ( t _ { 1 } ) \\log n ( t _ { 1 } ) } \\{ \\log K + \\log n ( t _ { 1 } ) \\} } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "Setting $\\eta = \\operatorname* { m i n } ( q ^ { - 3 / 4 } n ^ { 1 / 8 } ( t _ { 1 } ) \\log ^ { - 3 / 8 } \\{ K n ( t _ { 1 } ) \\} , q ^ { - 1 } n ^ { - \\alpha _ { 0 } / 3 } ( t _ { 1 } ) \\log ^ { - \\alpha _ { 0 } / 3 - 1 / 6 } \\{ K n ( t _ { 1 } ) \\} )$ yields the desired results. The proof is hence completed. ",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "E.3 PROOF OF LEMMA 3 ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "The assertion trivially holds for $j = 1$ . We prove it holds for any $j \\geq 2$ , by induction. By (A2), we have $( X _ { j } , Y _ { j } ^ { * } ( 0 ) , \\overset { \\cdot } { Y _ { j } ^ { * } } ( 1 ) ) \\overset { } { \\bot } \\overset { } { A _ { 1 } } | X _ { 1 }$ . Since $( X _ { j } , Y _ { j } ^ { * } ( 0 ) , Y _ { j } ^ { * } ( 1 ) ) \\perp ( X _ { 1 } , Y _ { 1 } ^ { * } ( 0 ) , Y _ { 1 } ^ { * } ( 1 ) )$ , this further implies $( X _ { j } , Y _ { j } ^ { * } ( 0 ) , Y _ { j } ^ { * } ( 1 ) ) \\perp { \\cal A } _ { 1 }$ and hence $( X _ { j } , Y _ { j } ^ { * } ( 0 ) , Y _ { j } ^ { * } ( 1 ) ) \\perp ( X _ { 1 } , A _ { 1 } , Y _ { 1 } ^ { * } ( 0 ) , Y _ { 1 } ^ { * } ( 1 ) )$ . By (A1), $Y _ { 1 }$ is completely determined by $A _ { 1 }$ , $Y _ { 1 } ^ { * } ( \\bar { 0 } )$ and $Y _ { 1 } ^ { * } ( 1 )$ . Therefore, we obtain $( X _ { j } , Y _ { j } ^ { * } ( 0 ) , Y _ { j } ^ { * } ( 1 ) ) \\perp F _ { 1 }$ . ",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "Suppose we have shown that $( X _ { j } , Y _ { j } ^ { * } ( 0 ) , Y _ { j } ^ { * } ( 1 ) ) ~ \\bot ~ \\mathcal { F } _ { k }$ for some $k \\ < \\ j \\ - \\ 1$ . To prove $( X _ { j } , Y _ { j } ^ { * } ( 0 ) , Y _ { j } ^ { * } ( 1 ) ) \\ \\perp \\ \\mathcal { F } _ { k + 1 } ,$ , it suffices to show $( X _ { j } , Y _ { j } ^ { * } ( 0 ) , Y _ { j } ^ { * } ( 1 ) ) ~ .$ ⊥⊥ $( X _ { k + 1 } , A _ { k + 1 } , Y _ { k + 1 } )$ . By (A1), $Y _ { k + 1 }$ is determined by $A _ { k + 1 }$ , $Y _ { k + 1 } ^ { * } ( 0 )$ and $Y _ { k + 1 } ^ { * } ( 1 )$ . Since $( X _ { j } , Y _ { j } ^ { * } ( 0 ) , Y _ { j } ^ { * } ( 1 ) ) ~ .$ ⊥⊥ $( X _ { k + 1 } , Y _ { k + 1 } ^ { * } ( 0 ) , Y _ { k + 1 } ^ { * } ( 1 ) )$ , it suffices to show $( X _ { j } , Y _ { j } ^ { * } ( 0 ) , Y _ { j } ^ { * } ( 1 ) ) \\ \\perp \\ A _ { k + 1 }$ . This is implied by $( X _ { j } , Y _ { j } ^ { * } ( 0 ) , Y _ { j } ^ { * } ( 1 ) ) \\perp A _ { k + 1 } | X _ { k + 1 } , \\mathcal { F } _ { k }$ and that $( X _ { j } , Y _ { j } ^ { * } ( 0 ) , Y _ { j } ^ { * } ( 1 ) ) \\perp \\perp X _ { k + 1 } , \\mathcal { F } _ { k }$ . The proof is hence completed. ",
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+ },
+ {
+ "type": "text",
+ "text": "E.4 PROOF OF LEMMA 4 ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "The assertions ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/d21643e0583df8df80a14fe044fa5ab5b43bb154005ae132693e1786375a89c4.jpg",
+ "text": "$$\n\\epsilon _ { 0 } \\le \\lambda _ { \\operatorname* { m i n } } [ \\mathsf { E } \\varphi ( X ) \\varphi ^ { \\top } ( X ) ] \\le \\lambda _ { \\operatorname* { m a x } } [ \\mathsf { E } \\varphi ( X ) \\varphi ^ { \\top } ( X ) ] \\le \\epsilon _ { 0 } ^ { - 1 } ,\n$$",
+ "text_format": "latex",
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+ {
+ "type": "text",
+ "text": "and ",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "img_path": "images/21dca32db5d6b3d1473aa3872b61249cb75d8c1b73c7592800bbf58a5d44874e.jpg",
+ "text": "$$\n\\operatorname* { s u p } _ { x } \\| \\varphi ( x ) \\| _ { 1 } \\leq \\epsilon _ { 0 } ^ { - 1 } \\sqrt { q } ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "for some $0 ~ < ~ \\epsilon _ { 0 } ~ < ~ 1$ are directly implied by the conditions that $\\lambda _ { \\mathrm { m i n } } [ \\mathrm { E } \\varphi ( X ) \\varphi ^ { \\top } ( X ) ] \\ \\asymp \\ 1$ , $\\lambda _ { \\mathrm { m a x } } [ \\mathrm { E } \\varphi ( X ) \\varphi ^ { \\top } ( X ) ] \\ \\asymp \\ 1$ , $\\begin{array} { r } { \\operatorname* { s u p } _ { x } \\| \\varphi ( x ) \\| _ { 1 } \\ \\leq \\ \\epsilon _ { 0 } ^ { - 1 } \\sqrt { q } } \\end{array}$ . Since $\\| \\varphi ( x ) \\| _ { 2 } ~ \\leq ~ \\| \\varphi ( x ) \\| _ { 1 }$ , we obtain $\\begin{array} { r } { \\operatorname* { s u p } _ { x } \\| \\varphi ( x ) \\| _ { 2 } \\leq \\operatorname* { s u p } _ { x } \\| \\varphi ( x ) \\| _ { 1 } \\leq \\epsilon _ { 0 } ^ { - 1 } \\sqrt { q } } \\end{array}$ . ",
+ "bbox": [
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+ },
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+ "type": "text",
+ "text": "Under the condition $\\begin{array} { r } { \\operatorname* { i n f } _ { a , x } \\pi ^ { * } ( a , x ) > 0 } \\end{array}$ , we can similarly show that $\\lambda _ { \\operatorname* { m i n } } [ \\Sigma _ { a } ] \\geq \\epsilon _ { 0 }$ for some $\\epsilon _ { 0 } > 0$ . ",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "Since $\\lvert Y ^ { * } ( 0 ) \\rvert$ and $\\lvert Y ^ { * } ( 1 ) \\rvert$ are bounded, there exists some constant $0 ~ < ~ \\epsilon _ { 0 } ~ < ~ 1$ that satisfies $\\begin{array} { r } { \\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } | Y ^ { * } ( a ) | \\leq \\epsilon _ { 0 } ^ { - 1 } } \\end{array}$ . Notice that $\\varphi ^ { \\top } ( x ) \\beta _ { a } = \\operatorname { E } \\{ Y ^ { * } ( a ) | X = x \\}$ . Boundedness of $\\left| Y ^ { * } ( a ) \\right|$ implies that the conditional mean $\\operatorname { E } \\{ Y ^ { * } ( a ) | X \\}$ is a bounded random variable as well. As a result, we obtain $\\begin{array} { r } { \\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } | \\varphi ^ { \\top } ( x ) \\beta _ { a } | \\leq \\epsilon _ { 0 } ^ { - 1 } } \\end{array}$ . ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Notice that $\\beta _ { a } = \\Sigma _ { a } ^ { - 1 } \\mathrm { E } \\varphi ^ { \\top } ( X ) Y ^ { \\ast } ( a )$ . Since $\\lambda _ { \\operatorname* { m i n } } [ \\Sigma _ { a } ]$ is bounded away from 0, it suffices to show $\\| \\mathrm { E } \\varphi ^ { \\top } ( X ) Y ^ { * } ( a ) \\| _ { 2 } \\stackrel { } { = } { \\cal O } ( 1 )$ , or equivalently, ",
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+ },
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+ "type": "equation",
+ "img_path": "images/a7d84c86e5191f977437b9ecc6569e8e5c1842267420860f4197db43d2343f46.jpg",
+ "text": "$$\n\\operatorname* { s u p } _ { \\nu \\in \\mathbb { R } ^ { p } , \\| \\nu \\| _ { 2 } = 1 } | \\mathrm { E } \\nu ^ { \\top } \\varphi ( X ) Y ^ { \\ast } ( a ) | = O ( 1 ) .\n$$",
+ "text_format": "latex",
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "By Cauchy-Schwarz inequality, it suffices to show ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/ee4bcfd33c5212998a365ff71d4b94b109577dca9f8b485fbbfa0ac090343ae9.jpg",
+ "text": "$$\n\\operatorname* { s u p } _ { \\nu \\in \\mathbb { R } ^ { p } , \\| \\nu \\| _ { 2 } = 1 } \\mathrm { E } | Y ^ { * } ( a ) | ^ { 2 } \\mathrm { E } | \\nu ^ { \\top } \\varphi ( X ) | ^ { 2 } = O ( 1 ) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "Since $| Y ^ { * } ( a ) | = O ( 1 )$ almost surely, we have by the condition $\\lambda _ { \\operatorname* { m a x } } [ \\mathrm { E } \\varphi ( X ) \\varphi ^ { \\top } ( X ) ] = O ( 1 )$ that ",
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+ {
+ "type": "equation",
+ "img_path": "images/bcb7b78433d4f322159451cd85d683526257b6de59dc82f3d064f6da69badd75.jpg",
+ "text": "$$\n\\operatorname* { s u p } _ { \\nu \\in \\mathbb { R } ^ { p } , \\| \\nu \\| _ { 2 } = 1 } \\mathbf { E } | \\nu ^ { \\top } \\varphi ( X ) | ^ { 2 } = \\operatorname* { s u p } _ { \\nu \\in \\mathbb { R } ^ { p } , \\| \\nu \\| _ { 2 } = 1 } \\nu ^ { \\top } \\mathbf { E } \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\nu \\leq \\lambda _ { \\operatorname* { m a x } } [ \\mathbf { E } \\varphi ( X ) \\varphi ^ { \\top } ( X ) ] = O ( 1 ) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "The proof is hence completed. ",
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+ },
+ {
+ "type": "text",
+ "text": "E.5 PROOF OF LEMMA 5 ",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "E.5.1 PROOF OF EQUATION 14 ",
+ "text_level": 1,
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+ },
+ {
+ "type": "text",
+ "text": "Notice that ",
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+ {
+ "type": "equation",
+ "img_path": "images/b5d16af9003d3307b18d4125c3666b87af4ff921dcd66e88053e106750201436.jpg",
+ "text": "$$\n\\begin{array} { r } { \\| j ( \\widehat { \\Sigma } _ { 1 , j } - \\Sigma _ { 1 } ) \\| _ { 2 } = \\left\\| \\displaystyle \\sum _ { i = 1 } ^ { j } \\{ A _ { i } \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) - \\mathrm { \\mathrm { E } } ^ { \\mathcal { F } _ { i - 1 } } \\pi _ { i - 1 } ( 1 , X ) \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\} \\right\\| _ { 2 } } \\\\ { + j \\left\\| \\mathrm { E } ^ { \\mathcal { F } _ { i - 1 } } \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\left( \\displaystyle \\frac { 1 } { j } \\displaystyle \\sum _ { i = 1 } ^ { j } \\pi _ { i - 1 } ( 1 , X ) - \\pi ^ { * } ( 1 , X ) \\right) \\right\\| _ { 2 } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "By Lemma 4, we have ",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "img_path": "images/2b53ed79d5085e9d09f0a35e76b5b57ab7e04bda5fd67703aa639c9985fc3b6d.jpg",
+ "text": "$$\n\\begin{array} { r } { \\left| { \\mathrm { E } } ^ { \\mathcal { F } _ { i - 1 } } \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\left( \\frac { 1 } { j } \\displaystyle \\sum _ { i = 1 } ^ { j } \\pi _ { i - 1 } ( 1 , X ) - \\pi ^ { * } ( 1 , X ) \\right) \\right| \\Big | _ { 2 } \\leq \\varepsilon _ { 0 } ^ { - 2 } q { \\mathrm { E } } ^ { \\mathcal { F } _ { i - 1 } } \\left| \\frac { 1 } { j } \\displaystyle \\sum _ { i = 1 } ^ { j } \\pi _ { i - 1 } ( 1 , X ) - \\pi ^ { * } ( 1 , X ) \\right| } \\\\ { \\leq \\varepsilon _ { 0 } ^ { - 2 } q ^ { 2 } j ^ { - \\alpha _ { 0 } } \\log ^ { \\alpha _ { 0 } } j , \\forall j \\geq . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O \\big ( j _ { n } ^ { - \\alpha _ { 0 } } \\big )$ ",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "Consider the first term on the RHS of equation 42. For any $i \\geq 1$ , define $M _ { i } = \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ A _ { i } -$ $\\pi _ { i - 1 } ( 1 , X _ { i } ) \\}$ . Notice that $\\{ M _ { i } \\} _ { i \\geq 1 }$ forms a martingale difference sequence with respect to the filtration $\\{ \\sigma ( \\mathcal { F } _ { i - 1 } ) : i \\ge 2 \\}$ , since ",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "img_path": "images/c89454f92bf83e2c0afe3a43e15ae954002dd9b83f6f2f326195ca6d0e28facf.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } [ \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ A _ { i } - \\pi _ { i - 1 } ( X _ { i } ) \\} \\vert \\mathcal { F } _ { i - 1 } ] } \\\\ { = } & { \\mathbb { E } ^ { \\mathcal { F } _ { i - 1 } } [ \\mathbb { E } ( \\varphi ( X _ { i } ) \\varphi ( X _ { i } ) ^ { \\top } \\{ A _ { i } - \\pi _ { i - 1 } ( X _ { i } ) \\} \\vert \\mathcal { F } _ { i - 1 } , X _ { i } ) ] = 0 , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "where $\\mathrm { E } ^ { \\mathcal { F } _ { i } , { X } _ { i } }$ denotes the conditional expectation given $X _ { i }$ and ${ \\mathcal { F } } _ { i }$ . Here, the first equality is due to that $X _ { i } \\perp \\perp F _ { i - 1 }$ , implied by Lemma 3. Under the given conditions on the basis function $\\varphi ( \\cdot )$ , using similar arguments in proving Equation (C.15) of Shi et al. (2020b), we can show that the following event occurs with probability at least $1 - O ( j ^ { - 2 } )$ , ",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "img_path": "images/645f1e1015e9bf71e392b2f562cbe14ef39c72b29365961c7dc919b5213137c3.jpg",
+ "text": "$$\n\\left\\| \\sum _ { i = 1 } ^ { j } M _ { i } \\right\\| _ { 2 } \\preceq \\sqrt { q j \\log ( j ) } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "Notice that $\\textstyle \\sum _ { k \\geq j } k ^ { - 2 } \\leq j ^ { - 2 } + \\sum _ { k > j } \\{ k ( k - 1 ) \\} ^ { - 1 } = j ^ { - 2 } + j ^ { - 1 }$ . Thus, the following occurs with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ , ",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "img_path": "images/6241a704f222b25d15ae194e74796ea83258131f4c4cae94832855e4410e6cf2.jpg",
+ "text": "$$\n\\left\\| \\sum _ { i = 1 } ^ { j } \\{ A _ { i } \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) - \\mathbf { E } ^ { \\mathcal { F } _ { i - 1 } } \\pi _ { i - 1 } ( 1 , X ) \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\} \\right\\| _ { 2 } \\preceq \\sqrt { q j \\log j } , ~ \\forall j \\geq j _ { n } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "It follows that ",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "img_path": "images/ad6c52e28f6a115fed42965cfbf7d138fbd7d28c9f558c15cac7a557fdf6d6bb.jpg",
+ "text": "$$\n\\begin{array} { r } { \\| \\big ( \\widehat { \\Sigma } _ { 1 , k } - \\Sigma _ { 1 } \\big ) \\| _ { 2 } \\preceq q \\delta _ { k } + \\sqrt { q k ^ { - 1 } \\log k } , \\forall k \\geq j _ { n } , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( j ^ { - \\alpha _ { 0 } } )$ . Similarly, we can show ",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "img_path": "images/fde34ebf4f47073d3164c6a5257cb8a246dd6257a3cbfdf7724669d99317d6f7.jpg",
+ "text": "$$\n\\begin{array} { r } { \\| \\big ( \\widehat { \\Sigma } _ { 0 , k } - \\Sigma _ { 0 } \\big ) \\| _ { 2 } \\preceq q \\delta _ { k } + \\sqrt { q k ^ { - 1 } \\log k } , \\forall k \\geq j _ { n } , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O \\big ( j _ { n } ^ { - \\alpha _ { 0 } } \\big )$ . The proof is hence completed. ",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "E.5.2 PROOF OF EQUATION 15 ",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "When $j _ { n }$ satisfies $j _ { n } ^ { \\alpha _ { 0 } } / \\log ^ { \\alpha _ { 0 } } ( j _ { n } ) \\gg q ^ { 2 }$ , it follows from equation 14 and equation 40 that ",
+ "bbox": [
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+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "img_path": "images/79b6f79d22ec377a11c83d33f9c48044c876be3563ecefcfd922caa681f49a39.jpg",
+ "text": "$$\n\\begin{array} { r } { \\lambda _ { \\operatorname* { m i n } } [ \\widehat { \\Sigma } _ { a , k } ] \\geq \\lambda _ { \\operatorname* { m i n } } [ \\Sigma _ { a } ] - \\Vert \\widehat { \\Sigma } _ { a , k } - \\Sigma _ { a } \\Vert _ { 2 } \\geq 2 ^ { - 1 } \\varepsilon _ { 0 } , \\forall k \\geq j _ { n } , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O \\big ( j _ { n } ^ { - \\alpha _ { 0 } } \\big )$ . Combining equation 40 with equation 45 and equation 14, we obtain ",
+ "bbox": [
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+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "img_path": "images/af142631fb71f4c152b63b9a18a2d933b099045e731f6b520324a74acd971c42.jpg",
+ "text": "$$\n\\begin{array} { r l r } & { } & { \\Vert \\widehat { \\Sigma } _ { a , k } ^ { - 1 } - \\Sigma _ { a } ^ { - 1 } \\Vert _ { 2 } = \\Vert \\widehat { \\Sigma } _ { a , k } ^ { - 1 } ( \\widehat { \\Sigma } _ { a , k } - \\Sigma _ { a } ) \\Sigma _ { a } ^ { - 1 } \\Vert _ { 2 } \\leq \\lambda _ { \\operatorname* { m i n } } [ \\Sigma _ { a } ] \\lambda _ { \\operatorname* { m i n } } [ \\widehat { \\Sigma } _ { a , k } ) \\Vert \\widehat { \\Sigma } _ { a , k } - \\Sigma _ { a } \\Vert _ { 2 } } \\\\ & { } & { \\preceq q \\delta _ { k } + \\sqrt { q k ^ { - 1 } \\log k } , \\forall k \\geq j _ { n } , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O \\big ( j _ { n } ^ { - \\alpha _ { 0 } } \\big )$ . The proof is hence completed. ",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "E.6 PROOF OF LEMMA 6 ",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "For any $l \\in \\{ 1 , \\ldots , q \\}$ and $i \\geq 1$ , define $M _ { i } ( l ) = \\varphi ^ { ( l ) } ( X _ { i } ) A _ { i } \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 1 } \\}$ . Here, $\\varphi ^ { ( l ) } ( X _ { i } )$ corresponds to the $l$ -th element of $\\varphi ( X _ { i } )$ . Similar to equation 43, we can show $\\{ \\dot { M } _ { i } ( l ) \\} _ { i \\geq 1 }$ forms a martingale difference sequence with respect to the filtration $\\{ \\sigma ( \\mathcal { F } _ { i - 1 } ) : i \\ge 1 \\}$ . By equation 43, we have for any $l$ , ",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "img_path": "images/afe12fc10ee3c227df777d4acef5193eee9cdd96ed4cf07fcf1bfcea0bcb2f97.jpg",
+ "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\{ \\varphi ^ { ( l ) } ( X _ { i } ) \\} ^ { 2 } \\le \\lambda _ { \\operatorname* { m a x } } [ \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) ] \\le \\epsilon _ { 0 } ^ { - 1 } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "Notice that ",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "img_path": "images/2668bb434e3b56e16a896864ab70ca250759cabc5da7815e1ed564d40b89ef96.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } \\{ M _ { i } ^ { 2 } ( l ) | \\mathcal { F } _ { i - 1 } \\} = \\mathbb { E } [ \\{ \\varphi ^ { ( l ) } ( X _ { i } ) \\} ^ { 2 } A _ { i } \\{ Y _ { i } ^ { * } ( 1 ) - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 1 } \\} ^ { 2 } | \\mathcal { F } _ { i - 1 } ] } \\\\ { \\le } & { \\mathbb { E } [ \\{ \\varphi ^ { ( l ) } ( X _ { i } ) \\} ^ { 2 } \\{ Y _ { i } ^ { * } ( 1 ) - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 1 } \\} ^ { 2 } | \\mathcal { F } _ { i - 1 } ] = \\mathbb { E } \\sigma ^ { 2 } ( 1 , X _ { i } ) \\{ \\varphi ^ { ( l ) } ( X _ { i } ) \\} ^ { 2 } } \\\\ { \\le } & { 4 \\epsilon _ { 0 } ^ { - 2 } \\mathbb { E } \\{ \\varphi ^ { ( l ) } ( X _ { i } ) \\} ^ { 2 } \\le 4 \\epsilon _ { 0 } ^ { - 3 } , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "where the first equality is due to (A1), the first inequality is due to that $A$ is bounded between 0 and 1, the second equality follows from Lemma 3, the second inequality follows from Lemma 4, and the last inequality is due to equation 46. It follows that ",
+ "bbox": [
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+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "img_path": "images/23e67584f8cc36883e9a3b4e29af308ce55d06d8088a487c30d7cff53d3a9776.jpg",
+ "text": "$$\n\\sum _ { i = 1 } ^ { k } \\operatorname { E } \\{ M _ { i } ^ { 2 } ( l ) | \\mathcal { F } _ { i - 1 } \\} \\leq 4 k \\epsilon _ { 0 } ^ { - 3 } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "Similarly, by (A1) and Lemma 4, we have ",
+ "bbox": [
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+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "img_path": "images/48406becbd6ba4ea4f0fc07394d67ea5376f890f7006ef3c39e89e2a74975f18.jpg",
+ "text": "$$\n\\sum _ { i = 1 } ^ { k } M _ { i } ^ { 2 } ( l ) \\leq 4 \\epsilon _ { 0 } ^ { - 2 } \\sum _ { i = 1 } ^ { k } \\{ \\psi ^ { ( l ) } ( X _ { i } ) \\} ^ { 2 } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "Similar to equation 44, we can show with probability at least $1 - O ( j ^ { - 1 } )$ that ",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "img_path": "images/f8fd90bd8a2e51301e49cefc88bb4296fcdcb9e20235726a8a800a8940838f57.jpg",
+ "text": "$$\n\\sum _ { i = 1 } ^ { k } [ M _ { i } ^ { 2 } ( l ) - \\mathtt { E } \\{ M _ { i } ^ { 2 } ( l ) | \\mathcal { F } _ { i - 1 } \\} ] \\preceq \\sqrt { q k \\log k } , \\forall k \\ge j .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "Thus, for any sequence $j _ { n }$ that satisfies $j _ { n } / \\log ( j _ { n } ) \\gg q$ , we have by equation 47 that ",
+ "bbox": [
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+ "page_idx": 24
+ },
+ {
+ "type": "equation",
+ "img_path": "images/b41093d819950a1f95b6e5633c4c569faa132acbf11e3da3e39890934d14c894.jpg",
+ "text": "$$\n\\sum _ { i = 1 } ^ { k } M _ { i } ^ { 2 } ( l ) + \\sum _ { i = 1 } ^ { k } \\mathrm { E } \\{ M _ { i } ^ { 2 } ( l ) | \\mathcal { F } _ { i - 1 } \\} \\leq \\bar { c } k , \\forall k \\geq j _ { n } ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "for some constant $\\bar { c } > 0$ , with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ . It follows that ",
+ "bbox": [
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+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "equation",
+ "img_path": "images/838907517dcd3561377156e25d0bd971e23583574cd972c086444e6f60d46ce2.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } _ { \\mathbb { P } } ( \\underset { \\mathrm { S t } } { \\bigcap } \\underset { s = 1 } { \\overset { k } { \\sum } } M _ { i } ( \\ell ) \\leq 2 \\sqrt { \\ 6 \\mathbb { k } } \\log k ) \\Bigg \\} } \\\\ & \\mathbb { P } _ { \\mathbb { P } } ( \\{ \\underset { s = 2 } { \\bigcap } \\underset { s = 1 } { \\overset { k } { \\prod } } \\frac { \\lambda } { \\underset { s = 1 } { \\prod } } ( \\lambda ) \\leq 2 \\sqrt { 6 \\mathbb { k } \\mathbb { k } \\log k } \\} \\prod _ { s = 2 } \\Biggl \\{ \\underset { s = 1 } { \\overset { k } { \\prod } } \\underset { s = 1 } { \\overset { k } { \\sum } } \\big [ M _ { i } ^ { 2 } ( \\ell ) + \\frac { 4 \\lambda _ { i } ^ { 2 } ( \\ell ) ( \\ell ) ( \\mathcal { F } _ { i - 1 } ) } { \\underset { s = 1 } { \\overset { k } { \\prod } } } ] \\leq \\mathbb { e } \\mathbb { k } \\} \\Bigg \\} \\\\ & { \\mathcal { O } _ { \\mathbb { Q } ^ { ( - 1 ) } } ^ { - 1 } \\succeq \\mathbb { P r } ( \\{ \\underset { \\mathrm { S t } \\geq \\delta _ { i } } { \\bigcap } \\underset { s = 1 } { \\overset { k } { \\sum } } ( \\lambda ) ^ { \\frac { \\ell } { \\ell } } ( \\ell ) + \\underset { s = 1 } { \\overset { k } { \\prod } } ( \\ell ) ( \\mathcal { F } _ { i } ) \\underset { s = 1 } { \\overset { k } { \\sum } } \\} \\leq \\mathbb { E } \\mathbb { k } \\} \\Bigg ) - \\mathcal { O } ( \\underset { \\mathbb { P } } { \\longrightarrow } ^ { - 1 } ) } \\\\ & \\mathbb { P } _ { \\mathbb { P } } ( \\{ \\underset { \\mathrm { S t } \\geq \\delta _ { i } } { \\bigcup } \\underset { s = 1 } { \\overset { k } { \\sum } } \\frac { \\lambda } { \\underset { s = 1 } { \\overset { k } { \\prod } } ( \\ell ) \\leq 2 \\sqrt { 6 \\mathbb { k } \\log \\ell } } \\} \\int \\prod _ { s = \\underset { i = 1 } { \\overset { k } { \\sum } } \\sum _ { i } \\mathcal { M } _ { i } ^ { 2 } ( \\ell ) + \\underset { s = 1 } { \\overset { k } { \\sum } } ( \\lambda ) ^ { \\frac { \\ell } { \\sum } } ( \\ell ) \\mathcal { F } _ { i - 1 } \\mathbb { I } \\} \\leq \\mathbb { e } \\mathbb { k } \\Bigg \\} ) } \\\\ & 1 - \\mathbb { P } _ \\mathbb \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "By Bonferroni’s inequality and Theorem 2.1 of Bercu & Touati (2008), we have ",
+ "bbox": [
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+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "equation",
+ "img_path": "images/b1509510102a43bcf4a0c195e06f4307afea19d0869d9161e6467f7744b813a1.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\operatorname* { P r } ( \\displaystyle \\bigcap _ { k \\geq j _ { n } } \\{ \\displaystyle \\sum _ { i = 1 } ^ { k } { M _ { i } ( l ) } | \\leq 2 \\sqrt { c k \\log { k } } \\} ) \\geq 1 - O ( j _ { n } ^ { - 1 } ) } \\\\ { - } & { \\displaystyle \\sum _ { k \\geq j _ { n } } \\operatorname* { P r } _ { \\epsilon } ( \\{ | \\displaystyle \\sum _ { i = 1 } ^ { k } { M _ { i } ( l ) } | > 2 \\sqrt { c k \\log { k } } \\} \\bigcap _ { k ^ { \\prime } \\geq j _ { n } } \\{ \\displaystyle \\sum _ { i = 1 } ^ { k ^ { \\prime } } [ \\displaystyle M _ { i } ^ { 2 } ( l ) + \\{ M _ { i } ^ { 2 } ( l ) | \\mathcal { F } _ { i - 1 } \\} ] \\leq \\bar { \\alpha } k ^ { \\prime } \\} \\} ) } \\\\ { \\geq } & { 1 - O ( j _ { n } ^ { - 1 } ) - \\displaystyle \\sum _ { k \\geq j _ { n } } \\operatorname* { P r } ( \\{ | \\displaystyle \\sum _ { i = 1 } ^ { k } { M _ { i } ( l ) } | > 2 \\sqrt { c k \\log { k } } \\} \\bigcap \\{ \\displaystyle \\sum _ { i = 1 } ^ { k } [ \\displaystyle M _ { i } ^ { 2 } ( l ) + \\{ M _ { i } ^ { 2 } ( l ) | \\mathcal { F } _ { i - 1 } \\} ] \\leq \\bar { \\alpha } k \\} } \\\\ { \\geq } & { 1 - O ( j _ { n } ^ { - 1 } ) - 2 \\displaystyle \\sum _ { k \\geq j _ { n } } \\exp ( - \\frac { 4 \\bar { c } k \\log { k } } { 2 \\bar { c k } } ) = 1 - O ( j _ { n } ^ { - 1 } ) - \\displaystyle \\sum _ { k \\geq j _ { n } } 2 k ^ { - 2 } . } \\end{array}\n$$",
+ "text_format": "latex",
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+ "text": "The last term on the RHS of equation 49 is $1 - O ( j _ { n } ^ { - 1 } )$ . To summarize, we have shown that the following event occurs with probability at least $1 - \\dot { O } ( j _ { n } ^ { - 1 } )$ , ",
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+ "text": "$$\n\\bigcap _ { k \\geq j _ { n } } \\left\\{ | \\sum _ { i = 1 } ^ { k } M _ { i } ( l ) | \\leq 2 { \\sqrt { { \\bar { c } } k \\log k } } \\right\\} .\n$$",
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+ "type": "text",
+ "text": "By Bonferroni’s inequality, we have ",
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+ "text": "$$\n\\bigcap _ { k \\geq j _ { n } } \\left\\{ \\left. \\sum _ { i = 1 } ^ { k } \\varphi ( X _ { i } ) A _ { i } \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 1 } \\} \\right. _ { 2 } \\leq 2 \\sqrt { \\bar { c } q k \\log k } \\right\\} ,\n$$",
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+ "type": "text",
+ "text": "with probability at least $1 - O ( j _ { n } ^ { - 1 / 2 } )$ . Similarly, we can show ",
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+ "text": "$$\n\\bigcap _ { k \\geq j _ { n } } \\left\\{ \\left. \\sum _ { i = 1 } ^ { k } \\varphi ( X _ { i } ) ( 1 - A _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 0 } \\} \\right. _ { 2 } \\leq c \\sqrt { q k \\log k } \\right\\} ,\n$$",
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+ "type": "text",
+ "text": "for some constant $c > 0$ , with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ . The proof is hence completed. ",
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+ "type": "text",
+ "text": "E.7 PROOF OF THEOREM 3 ",
+ "text_level": 1,
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+ "text": "We state the following lemmas before presenting the proof. ",
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+ "text": "Lemma 7 Assume the conditions in Theorem 3 hold. Then for any sequence $\\{ j _ { n } \\} _ { n }$ that satisfies $j _ { n } ^ { \\alpha _ { 0 } } / \\log ^ { \\alpha _ { 0 } } j _ { n } \\gg q ^ { 2 }$ , we have with probability at least $1 - O \\big ( j _ { n } ^ { - \\alpha _ { 0 } } \\big )$ that ",
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+ "text": "$$\n\\| \\widehat { \\beta } _ { a , k } - \\beta _ { a } \\| _ { 2 } \\preceq q ^ { 1 / 2 } k ^ { - 1 / 2 } \\sqrt { \\log k } , \\quad \\forall a \\in \\{ 0 , 1 \\} , \\forall k \\geq j _ { n } .\n$$",
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+ "text": "Lemma 8 Assume the conditions in Theorem 3 hold. Then for any sequence $\\{ j _ { n } \\} _ { n }$ that satisfies $j _ { n } ^ { \\alpha _ { 0 } } / \\log ^ { \\alpha _ { 0 } } j _ { n } \\gg q ^ { 2 }$ , we have with probability at least $1 - O \\big ( j _ { n } ^ { - \\alpha _ { 0 } } \\big )$ that ",
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+ "text": "$$\n\\begin{array} { r l r } & { } & { \\left| \\left| \\frac { 1 } { k } \\displaystyle \\sum _ { i = 1 } ^ { k } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} ^ { 2 } - \\Phi _ { a } \\right| \\right| _ { 2 } \\preceq q \\delta _ { k } + q ^ { 1 / 2 } k ^ { - 1 / 2 } \\sqrt { \\log k } , } \\\\ & { } & { \\forall a \\in \\{ 0 , 1 \\} , k \\geq j _ { n } . } \\end{array}\n$$",
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+ "text": "Similar to the proof of Theorem 1, we will show the assertion in Theorem 3 holds for any $n ( \\cdot )$ that correspond to the realizations of $N ( \\cdot )$ that satisfy $n ( t _ { 1 } ) < n ( t _ { 2 } ) < \\cdots < n ( t _ { K } )$ . For any $1 \\leq k _ { 1 } \\leq k _ { 2 } \\leq K$ , define ",
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+ "text": "$$\n\\begin{array} { l } { { \\displaystyle \\widehat { V } ( k _ { 1 } , k _ { 2 } ) = \\sqrt { n ( t _ { k _ { 1 } } ) n ( t _ { k _ { 2 } } ) } \\mathrm { C o v } \\left( \\widehat { \\beta } _ { 1 } ^ { \\mathrm { M B } * } ( t _ { k 1 } ) - \\widehat { \\beta } _ { 0 } ^ { \\mathrm { M B } * } ( t _ { k 1 } ) , \\widehat { \\beta } _ { 1 } ^ { \\mathrm { M B } * } ( t _ { k 2 } ) - \\widehat { \\beta } _ { 0 } ^ { \\mathrm { M B } * } ( t _ { k 2 } ) | \\{ ( X _ { i } , A _ { i } , Y _ { i } ) \\} _ { i = 1 , \\ldots , n } ^ { + \\infty } \\right) } } \\\\ { { \\displaystyle = \\frac { 1 } { \\sqrt { n ( t _ { k _ { 1 } } ) n ( t _ { k _ { 2 } } ) } } \\sum _ { a = 0 } ^ { 1 } \\sum _ { j = 1 } ^ { k _ { 1 } } \\sum _ { i = n ( t _ { j - 1 } ) + 1 } ^ { n ( t _ { j } ) } \\widehat { \\Sigma } _ { a } ^ { - 1 } ( t _ { j } ) \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\widehat { \\beta } _ { a } ( t _ { j } ) \\} ^ { 2 } \\widehat { \\Sigma } _ { a } ^ { - 1 } ( t _ { k 1 } ) , } } \\end{array}\n$$",
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+ "type": "text",
+ "text": "and ",
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+ "img_path": "images/6902a0d7fbe73dd237203693e21633bb68d0a466d1e777ab92d781df410e5708.jpg",
+ "text": "$$\n\\widehat { \\pmb { V } } = \\left( \\begin{array} { c c c c } { \\widehat { V } ( 1 , 1 ) } & { \\widehat { V } ( 1 , 2 ) } & { \\ldots } & { \\widehat { V } ( 1 , K ) } \\\\ { \\widehat { V } ( 2 , 1 ) } & { \\widehat { V } ( 2 , 2 ) } & { \\ldots } & { \\widehat { V } ( 2 , K ) } \\\\ { \\vdots } & { \\vdots } & { \\vdots } \\\\ { \\widehat { V } ( K , 1 ) } & { \\widehat { V } ( K , 2 ) } & { \\ldots } & { \\widehat { V } ( K , K ) } \\end{array} \\right) .\n$$",
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+ "text": "We aim to bound the entrywise $\\ell _ { \\infty }$ norm of $\\widehat { V } - V$ where $V$ is defined in equation 22. It suffices to bound $\\begin{array} { r l } { \\operatorname* { m a x } _ { 1 \\leq k _ { 1 } \\leq k _ { 2 } \\leq K } \\operatorname* { s u p } _ { b _ { 1 } , b _ { 2 } \\in \\mathbb { R } ^ { p + 1 } , \\| b _ { 1 } \\| _ { 2 } = \\| b _ { 2 } \\| _ { 2 } = 1 } | b _ { 1 } ^ { T } \\{ \\widehat { V } ( k _ { 1 } , k _ { 2 } ) - V ( k _ { 1 } , k _ { 2 } ) \\} b _ { 2 } | } & { = \\| b _ { 2 } ^ { T } \\{ ( k _ { 2 } , k _ { 2 } ) - V ( k _ { 2 } , k _ { 2 } ) \\} b _ { 1 } \\| _ { 2 } = \\| b _ { 2 } ^ { T } \\{ ( k _ { 1 } , k _ { 2 } ) - V ( k _ { 1 } , k _ { 2 } ) \\} b _ { 2 } \\| } \\end{array}$ $\\begin{array} { r } { \\operatorname* { m a x } _ { 1 \\leq k _ { 1 } \\leq k _ { 2 } \\leq K } \\| \\widehat { V } ( k _ { 1 } , k _ { 2 } ) - V ( k _ { 1 } , k _ { 2 } ) \\| _ { 2 } } \\end{array}$ . For any $k _ { 1 } , k _ { 2 }$ , we decompose $\\widehat { V } ( k _ { 1 } , k _ { 2 } ) - V ( k _ { 1 } , k _ { 2 } )$ as ",
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+ "type": "text",
+ "text": "Vb (k1, k ${ \\mathrm { ~ \\psi ~ } } _ { 2 } ) - V ( k _ { 1 } , k _ { 2 } ) = \\widehat { V } ( k _ { 1 } , k _ { 2 } ) - \\widehat { V } ^ { * } ( k _ { 1 } , k _ { 2 } ) + \\widehat { V } ^ { * } ( k _ { 1 } , k _ { 2 } ) - \\widehat { V } ^ { * * } ( k _ { 1 } , k _ { 2 } ) + \\widehat { V } ^ { * * } ( k _ { 1 } , k _ { 2 } ) - V ( k _ { 1 } , k _ { 2 } ) ,$ where ",
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+ "type": "equation",
+ "img_path": "images/9500720c045a11a397bc15fc16cdbd0aba89cb0288678f16d538d6f68fc10711.jpg",
+ "text": "$$\n\\begin{array} { c } { { k _ { 1 } , k _ { 2 } ) = \\displaystyle { \\frac { 1 } { \\sqrt { n ( t _ { k _ { 1 } } ) n ( t _ { k _ { 2 } } ) } } \\sum _ { a = 0 } ^ { 1 } \\sum _ { j = 1 } ^ { k _ { 1 } } \\sum _ { i = n ( t _ { j - 1 } ) + 1 } ^ { n ( t _ { j } ) } \\Sigma _ { a } ^ { - 1 } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\widehat \\beta _ { a } ( t _ { j } ) \\} ^ { 2 } \\Sigma _ { a } ^ { - 1 } \\mathbb { I } ( X _ { i } ) \\varphi ( X _ { i } ) } } } \\\\ { { \\widehat { V } ^ { \\ast \\ast } ( k _ { 1 } , k _ { 2 } ) = \\displaystyle { \\frac { 1 } { \\sqrt { n ( t _ { k _ { 1 } } ) n ( t _ { k _ { 2 } } ) } } \\sum _ { a = 0 } ^ { 1 } \\sum _ { j = 1 } ^ { n ( t _ { k _ { 1 } } ) } \\Sigma _ { a } ^ { - 1 } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} ^ { 2 } \\Sigma _ { a } ^ { - 1 } \\mathbb { I } ( X _ { i } ) \\varphi ( X _ { i } ) \\varphi ( X _ { i } ) \\} ^ { 2 } \\Sigma _ { a } ^ { - 1 } \\mathbb { I } ( X _ { i } ) \\varphi ( X _ { i } ) \\varphi ( X _ { j } ) } } } \\end{array}\n$$",
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+ "type": "text",
+ "text": "By Lemma 4 and Lemma 8, we obtain that ",
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+ "img_path": "images/4014d66a2a14ed0ec4a2585119ce6df54abed7f17e4b78768c111f8153240aac.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\underset { 1 \\leq k _ { 1 } \\leq k _ { 2 } \\leq K } { \\operatorname* { m a x } } \\| \\hat { V } ^ { * * } ( k _ { 1 } , k _ { 2 } ) - V ( k _ { 1 } , k _ { 2 } ) \\| _ { 2 } } \\\\ { \\leq } & { \\underset { 1 \\leq k _ { 1 } \\leq k _ { 1 } \\leq K } { \\operatorname* { m a x } } \\underset { a = 0 } { \\overset { 1 } { \\sum } } \\Bigg \\| \\frac { 1 } { n ( t _ { k _ { 1 } } ) } \\underset { j = 1 } { \\overset { n ( t _ { k _ { 1 } } ) } { \\sum } } \\underset { 2 = 1 } { \\overset { n ( t _ { k } ) } { \\sum } } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} ^ { 2 } \\Sigma _ { a } ^ { - 1 } - \\Sigma _ { a } ^ { - 1 } \\Phi _ { a } \\Sigma _ { a } ^ { - } } \\\\ { \\leq } & { \\underset { 1 \\leq k _ { 1 } \\leq K } { \\operatorname* { m a x } } \\frac { 1 } { \\epsilon _ { 0 } ^ { 2 } } \\underset { a = 0 } { \\overset { 1 } { \\sum } } \\Bigg \\| \\frac { 1 } { n ( t _ { k _ { 1 } } ) } \\underset { j = 1 } { \\overset { n ( t _ { k _ { 1 } } ) } { \\sum } } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} ^ { 2 } - \\Phi _ { a } \\Bigg \\| _ { 2 } } \\\\ { \\overset { \\leq } } & { q \\delta _ { n ( t _ { 1 } ) } + q ^ { 1 / 2 } n ^ { - 1 / 2 } ( t _ { 1 } ) \\sqrt { \\log n ( t _ { 1 } ) } , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "with probability at least $1 - O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ . ",
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+ "type": "text",
+ "text": "Notice that ",
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+ "img_path": "images/f7ad89a33ccf60cca3eb02d2bcb406f4053c0c971fd42f814903cb5dc5c592ad.jpg",
+ "text": "$$\n\\begin{array} { r l r } { { \\sum _ { i = 1 } ^ { n } \\operatorname { n u f f } _ { k \\to i } \\hat { \\mathcal { V } } ^ { * } ( k _ { 1 } , k _ { 2 } ) = \\sum _ { s = i } ^ { 1 } \\sum _ { i = 1 } ^ { k } \\sum _ { m \\in \\{ i , j \\} } \\Sigma _ { a } ^ { - 1 } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\xi _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\hat { \\mathcal { B } } _ { a } ( t _ { j } ) \\big ] ^ { 2 } \\Sigma _ { a } ^ { - 1 } } } \\\\ & { } & { \\underset { = 1 } { \\overset { k _ { 1 } } { \\sum } } \\underset { = 1 } { \\overset { n ( \\cdot ) } { \\sum } } \\sum _ { i = 1 } ^ { n } \\sum _ { ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) } \\xi ^ { \\top } ( X _ { i } ) \\big \\{ ( Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } + \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } - \\varphi ^ { \\top } ( X _ { i } ) \\hat { \\mathcal { B } } _ { a } ( t _ { j } ) \\big ) ^ { 2 } \\Sigma _ { a } ^ { - 1 } } \\\\ & { } & { + \\sum _ { 1 } ^ { k _ { 1 } } \\sum _ { i = 0 } ^ { k } \\sum _ { \\beta = 1 } ^ { k } \\sum _ { \\beta = 1 } ^ { k } \\sum _ { 0 } ^ { k } \\sum _ { 0 } ^ { - 1 } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ \\varphi ^ { \\top } ( X _ { i } ) \\} \\hat { \\varphi } _ { a } - \\varphi ^ { \\top } ( X _ { i } ) \\hat { \\mathcal { B } } _ { a } ( t _ { j } ) \\big \\} ^ { 2 } \\Sigma _ { a } ^ { - 1 } } \\\\ & { } & + 2 \\sum _ { a = 0 } ^ { 1 } \\sum _ { i = 1 } ^ { k _ { 1 } } \\sum _ { \\beta = 1 } ^ { k _ { 2 } } \\sum _ { \\alpha } ^ { - 1 } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { \\alpha } - \\varphi ^ { \\top } ( X _ { i } \\end{array}\n$$",
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+ "type": "text",
+ "text": "It follows that ",
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+ "type": "equation",
+ "img_path": "images/0e255892d7c2ff036db5844834d38f73e159107ecd6115d6f60e86d3cb2289fe.jpg",
+ "text": "$$\n\\begin{array} { r l } { \\displaystyle \\operatorname* { m a x } _ { 1 \\leq k _ { 1 } \\leq k _ { 2 } \\leq K } \\left\\| \\widehat { V } ^ { * } ( k _ { 1 } , k _ { 2 } ) - \\widehat { V } ^ { * * } ( k _ { 1 } , k _ { 2 } ) \\right\\| _ { 2 } } & { { } } \\\\ { \\displaystyle \\operatorname* { m a x } _ { 1 \\leq k _ { 1 } \\leq K } \\frac { 1 } { n ( t _ { k _ { 1 } } ) } \\left\\| \\displaystyle \\sum _ { a = 0 } ^ { 1 } \\sum _ { j = 1 } ^ { k _ { 1 } } \\sum _ { i = n ( t _ { j - 1 } ) + 1 } ^ { n ( t _ { j } ) } \\Sigma _ { a } ^ { - 1 } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } - \\varphi ^ { \\top } ( X _ { i } ) \\widehat { \\beta } _ { a } ( t _ { j } ) \\} ^ { \\top } \\mathbb { I } ( X _ { i } ) \\right\\| _ { 2 } ^ { 2 } } & { { } } \\\\ { \\displaystyle \\operatorname* { m a x } _ { 1 \\leq k _ { 1 } \\leq K } \\frac { 2 } { n ( t _ { k _ { 1 } } ) } \\left\\| \\displaystyle \\sum _ { a = 0 } ^ { 1 } \\sum _ { j = 1 } ^ { k _ { 1 } } \\sum _ { i = n ( t _ { j - 1 } ) + 1 } ^ { n ( t _ { j } ) } \\Sigma _ { a } ^ { - 1 } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} \\varphi ^ { \\top } ( X _ { i } ) ( \\beta _ { a } - \\varphi ^ { \\top } ( X _ { i } ) \\mathbb { I } ( X _ { i } ) \\mathbb { I } ( \\beta _ { a } - \\varphi ^ { \\top } ( X _ { i } ) \\mathbb { I } ( \\beta _ { a } ) ) \\varphi ( X _ { i } ) \\mathbb { I } ( \\beta _ { a } ) \\right\\| _ { 2 } ^ { 2 } } & { { } = 0 } \\end{array}\n$$",
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+ "type": "text",
+ "text": "By Lemma 4, we obtain that ",
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+ "type": "equation",
+ "img_path": "images/68ae25e47cf6d8bf4b9681cff0f465fbb3ec7ea8dc1ae29cec1a223910da0613.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\quad \\underset { 1 \\leq k \\leq k \\leq K } { \\operatorname* { m a x } } \\| \\hat { V } ^ { * } ( k _ { 1 } , k _ { 2 } ) - \\widehat { V } ^ { * * } ( k _ { 1 } , k _ { 2 } ) \\| _ { 2 } } \\\\ & { \\quad \\underset { 1 \\leq k \\leq K } { \\operatorname* { m a x } } \\frac { 1 } { K } \\| \\underset { 2 \\leq - 1 : = n ( \\{ k _ { 1 } , - 1 \\} + 1 ) } { \\overset { k _ { 1 } } { \\sum } } \\mathbb { \\overset { n ( k _ { 2 } ) } { \\sum } } \\mathbb { \\overset { \\eta ( A _ { i } ) } { = } } \\alpha ^ { \\top } ( X _ { i } ) \\boldsymbol { \\varphi } ^ { \\top } ( X _ { i } ) \\boldsymbol { \\{ \\varphi } } ^ { \\top } ( X _ { i } ) \\beta _ { a } - \\boldsymbol { \\varphi } ^ { \\top } ( X _ { i } ) \\beta _ { a } ( t _ { 2 } ) \\boldsymbol { \\} ^ { 2 } \\| _ { 2 } } \\\\ & { \\quad \\underset { \\theta \\leq \\{ 0 , 1 \\} } { \\overset { \\operatorname* { m a x } } { = } } \\frac { 1 } { K } \\| \\underset { 1 \\leq k \\leq K } { \\sum } \\mathbb { \\overset { n ( k _ { 1 } ) } { = } } \\mathbb { \\overset { n ( k _ { 2 } ) } { = } } \\mathbb { \\overset { n ( k _ { 1 } ) } { = } } \\mathbb { \\overset { n ( k _ { 2 } ) } { = } } \\mathbb { \\overset { n ( k _ { 1 } ) } { = } } \\mathbb { \\overset { n ( k _ { 2 } ) } { = } } \\mathbb { \\overset { n ( k _ { 1 } ) } { = } } \\mathbb { \\overset { n ( k _ { 2 } ) } { = } } \\mathbb { \\overset { n ( k _ { 1 } ) } { = } } \\mathbb { \\overset { n ( k _ { 2 } ) } { = } } \\mathbb { \\overset { n ( k _ { 1 } ) } { = } } \\mathbb { \\overset { n ( k _ { 2 } ) } { = } } \\mathbb { \\overset { n ( k _ { 1 } ) } { = } } \\mathbb { \\overset { n ( k _ { 2 } ) } { = } } \\mathbb { \\overset { n ( k _ { 1 } ) } { = } } } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 26
+ },
+ {
+ "type": "text",
+ "text": "By Lemmas 4 and 7, we have with probability at least $1 - O ( n ^ { - 1 } ( t _ { 1 } ) )$ that ",
+ "bbox": [
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+ {
+ "type": "equation",
+ "img_path": "images/2294f172ea3c007f254ce98ce62c336ef5f15c87acbe8f47fa9ce2e079559c89.jpg",
+ "text": "$$\n\\frac { 1 } { n ( t _ { k _ { 1 } } ) } \\| \\Psi _ { 1 , a , k _ { 1 } } \\| _ { 2 } \\preceq q ^ { 2 } n ^ { - 1 } ( t _ { 1 } ) \\log \\{ n ( t _ { 1 } ) \\} \\left\\| \\frac { 1 } { n ( t _ { k _ { 1 } } ) } \\sum _ { i = 1 } ^ { n ( t _ { k _ { 1 } } ) } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\right\\| _ { 2 } ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 26
+ },
+ {
+ "type": "text",
+ "text": "Similar to Lemma 5, we can show there exists some constant $c _ { * } > 0$ that ",
+ "bbox": [
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+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "equation",
+ "img_path": "images/28f7e71fc8de827bc285a3a6b1b39ad721c5331ac8aad8b64a3a759cd747cb5c.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\frac { 1 } { n ( t _ { k _ { 1 } } ) } \\left\\| \\displaystyle \\sum _ { i = 1 } ^ { n ( t _ { k _ { 1 } } ) } \\mathbb { I } \\big ( A _ { i } = a \\big ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) - \\mathrm { E } ^ { \\mathcal { F } _ { i - 1 } } \\{ \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\} \\big ] \\right\\| _ { 2 } } \\\\ & { \\quad \\quad \\leq c _ { * } \\{ q \\delta _ { n ( t _ { k _ { 1 } } ) } + q ^ { 1 / 2 } n ^ { - 1 / 2 } ( t _ { k _ { 1 } } ) \\sqrt { \\log n ( t _ { k _ { 1 } } ) } \\} , \\forall 1 \\leq k _ { 1 } \\leq K , a \\in \\{ 0 , 1 \\} , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 26
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( n ^ { - 1 } ( t _ { 1 } ) )$ . By Lemma 4, we can show with probability at least $1 - \\bar { O } ( n ^ { - 1 } ( t _ { 1 } ) )$ that ",
+ "bbox": [
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+ "type": "equation",
+ "img_path": "images/4c19b795794f1a20727e1549383501561f5a15747452499df0608f6bb5941a90.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { 1 \\leq k _ { 1 } \\leq K } \\frac { 1 } { n ( t _ { k _ { 1 } } ) } \\left\\| \\sum _ { i = 1 } ^ { n ( t _ { k _ { 1 } } ) } \\mathrm { E } ^ { \\mathcal { F } _ { i - 1 } } \\{ \\mathbb I ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\} \\right\\| _ { 2 } = O ( 1 ) .\n$$",
+ "text_format": "latex",
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+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "This together with equation 52 and equation 53 yields ",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( n ^ { - 1 } ( t _ { 1 } ) )$ ",
+ "bbox": [
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+ ],
+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "$\\begin{array} { r } { n ^ { - 1 } ( t _ { k 1 } ) \\| \\Psi _ { 1 , a , k _ { 1 } } \\| _ { 2 } \\preceq q ^ { 2 } n ^ { - 1 } ( t _ { 1 } ) \\log \\{ n ( t _ { 1 } ) \\} , \\forall 1 \\leq k _ { 1 } \\leq K , a \\in \\{ 0 , 1 \\} , } \\end{array}$ . ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Moreover, using similar arguments in proving Equation (C.15) of Shi et al. (2020b), we can show that for any $1 \\leq k _ { 1 } \\leq K$ , the following event occurs with probability at least $1 - O ( n ^ { - 2 } ( t _ { k _ { 1 } } ) )$ , ",
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+ {
+ "type": "equation",
+ "img_path": "images/b528917172b00eaab720b780fa28188acf59e1804b2a40c6bfe59770fe209841.jpg",
+ "text": "$$\n\\frac { 1 } { \\iota ( t _ { k _ { 1 } } ) } \\left\\| \\sum _ { i = 1 } ^ { n ( t _ { k _ { 1 } } ) } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} \\varphi ^ { ( l ) } ( X _ { i } ) \\right\\| _ { 2 } \\preceq q ^ { 1 / 2 } n ^ { - 1 / 2 } ( t _ { k _ { 1 } } ) \\sqrt { \\log n ( t _ { k _ { 1 } } ) } ,\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "$\\begin{array} { r } { \\sum _ { k _ { 1 } = 1 } ^ { K } n ^ { - 2 } ( t _ { k _ { 1 } } ) \\leq n ^ { - 1 } ( t _ { 1 } ) } \\end{array}$ $1 - O ( n ^ { - 1 } ( t _ { 1 } ) )$ ",
+ "bbox": [
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+ {
+ "type": "equation",
+ "img_path": "images/0bc36a084eb2ebcb3b98a93887bc37430402e973e8fb34a708c056581d72d74c.jpg",
+ "text": "$$\n\\frac { 1 } { \\iota ( t _ { k _ { 1 } } ) } \\left\\| \\sum _ { i = 1 } ^ { n ( t _ { k _ { 1 } } ) } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} \\varphi ^ { ( l ) } ( X _ { i } ) \\right\\| _ { 2 } \\preceq q ^ { 1 / 2 } n ^ { - 1 / 2 } ( t _ { k _ { 1 } } ) \\sqrt { \\log n ( t _ { k _ { 1 } } ) } ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "In addition, it follows from Lemma 7 that ",
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+ "page_idx": 27
+ },
+ {
+ "type": "equation",
+ "img_path": "images/71dc3c0168e0045f066825ba8708f70f7cd37955ed2187033322a65322dfdd1a.jpg",
+ "text": "$$\n\\begin{array} { r } { n ^ { - 1 } ( t _ { k _ { 1 } } ) \\| \\Psi _ { 2 , a , k _ { 1 } } \\| _ { 2 } \\preceq q ^ { 3 / 2 } n ^ { - 1 } ( t _ { 1 } ) \\log \\{ n ( t _ { 1 } ) \\} , \\forall 1 \\leq k _ { 1 } \\leq K , a \\in \\{ 0 , 1 \\} . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "This together with equation 54 yields that ",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "equation",
+ "img_path": "images/58fd4a50956a62fae9a7e7805b4c4dc55052f270be671c4713baf11b8630ffcd.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { 1 \\leq k _ { 1 } \\leq k _ { 2 } \\leq K } \\Big \\| \\widehat { V } ^ { * } ( k _ { 1 } , k _ { 2 } ) - \\widehat { V } ^ { * * } ( k _ { 1 } , k _ { 2 } ) \\Big \\| _ { 2 } \\preceq q ^ { 2 } n ^ { - 1 } ( t _ { 1 } ) \\log n ( t _ { 1 } ) ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( n ^ { - 1 } ( t _ { 1 } ) )$ . Under the given conditions, we have ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/6071cef6b0af5bf1ad3eee7741170d33e5c24a6ad76908800171d7740dde65f6.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { 1 \\leq k _ { 1 } \\leq k _ { 2 } \\leq K } \\Big \\| \\widehat { V } ^ { * } ( k _ { 1 } , k _ { 2 } ) - \\widehat { V } ^ { * * } ( k _ { 1 } , k _ { 2 } ) \\Big \\| _ { 2 } \\preceq q ^ { 1 / 2 } n ^ { - 1 / 2 } ( t _ { 1 } ) \\log ^ { 1 / 2 } n ( t _ { 1 } ) ,\n$$",
+ "text_format": "latex",
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+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( n ^ { - 1 } ( t _ { 1 } ) )$ ",
+ "bbox": [
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+ ],
+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "Moreover, with some calculations, we can show that ",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "equation",
+ "img_path": "images/04be3cae4a9688e33606be7ed589be85efc05f4f25d1b2dc1151488d51182b38.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\underset { 1 \\leq k _ { 1 } \\leq k _ { 2 } \\leq K } { \\operatorname* { m a x } } \\left\\| \\widehat { V } ( k _ { 1 } , k _ { 2 } ) - \\widehat { V } ^ { * } ( k _ { 1 } , k _ { 2 } ) \\right\\| _ { 2 } \\leq \\underset { a = 0 } { \\overset { 1 } { \\sum } } \\underset { j \\geq 1 } { \\operatorname* { m a x } } \\left\\| \\Sigma _ { a } ^ { - 1 } - \\widehat { \\Sigma } _ { a } ^ { - 1 } ( l _ { j } ) \\right\\| _ { 2 } } \\\\ { \\times } & { \\underset { 1 \\leq k _ { 1 } \\leq K } { \\operatorname* { m a x } } \\frac { 2 } { n ( t _ { k _ { 1 } } ) } \\left\\| \\underset { j = 1 } { \\overset { k _ { 1 } } { \\sum } } \\underset { i = n ( t _ { j - 1 } ) + 1 } { \\overset { n ( t _ { j } ) } { \\sum } } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\widehat { \\beta } _ { a } ( t _ { j } ) \\} ^ { 2 } \\Sigma _ { a } ^ { - 1 } \\right\\| _ { 2 } } \\\\ { + } & { \\underset { a = 0 } { \\overset { 1 } { \\sum } } \\underset { 1 \\leq k _ { 1 } \\leq K } { \\operatorname* { m a x } } \\frac { 1 } { n ( t _ { k _ { 1 } } ) } \\left\\| \\underset { j = 1 } { \\overset { k _ { 1 } } { \\sum } } \\underset { i = n ( t _ { j - 1 } ) + 1 } { \\overset { n ( t _ { i } ) } { \\sum } } \\Sigma _ { a } ^ { - 1 } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\widehat { \\beta } _ { a } ( t _ { j } ) \\} ^ { 2 } \\Sigma _ { a } ^ { - 1 } \\right\\| } \\\\ { \\times } & { \\underset { j \\geq 1 } { \\operatorname* { m a x } } \\| \\Sigma _ { a } ^ { - 1 } - \\widehat { \\Sigma } _ { a } ^ { - 1 } ( t _ { j } ) \\| _ { 2 } ^ { 2 } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "In view of Lemma 4 and Lemma 5, we have with probability at least $1 - O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ that ",
+ "bbox": [
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+ {
+ "type": "equation",
+ "img_path": "images/298782662824acf531c5b9456cce46c58772056078045001b98db00e60d16d0a.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\underset { 1 \\leq k _ { 1 } \\leq k _ { 2 } \\leq K } { \\operatorname* { m a x } } \\left\\| \\widehat { V } ( k _ { 1 } , k _ { 2 } ) - \\widehat { V } ^ { * } ( k _ { 1 } , k _ { 2 } ) \\right\\| _ { 2 } \\leq O ( 1 ) ( q \\delta _ { n ( t _ { 1 } ) } + \\sqrt { q n ^ { - 1 } ( t _ { 1 } ) \\log n ( t _ { 1 } ) } ) } \\\\ { \\times } & { \\underset { 1 \\leq k _ { 1 } \\leq K } { \\operatorname* { m a x } } \\frac { 1 } { n ( t _ { k _ { 1 } } ) } \\left\\| \\underset { \\underset { j = 1 } { \\overset { k _ { 1 } } { \\sum } } i = n ( t _ { j - 1 } ) + 1 } { \\overset { n ( t _ { j } ) } { \\sum } } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\widehat { \\beta } _ { a } ( t _ { j } ) \\} ^ { 2 } \\right\\| , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "where $O ( 1 )$ denotes some positive constant. Similar to equation 50 and equation 55, we can show with probability at least $1 \\stackrel { - } { - } O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ that ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/1fab5457321bc242941464700f2be8af4c80b454b4340877699ba9d476f9dc26.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\operatorname* { m a x } _ { 1 \\leq k _ { 1 } \\leq K } \\left\\| { \\frac { 1 } { n ( t _ { k _ { 1 } } ) } } \\Psi _ { 3 , a , k _ { 1 } } - \\Psi _ { a } \\right\\| _ { 2 } = o ( 1 ) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "Similar to Lemma 4, we can show $\\begin{array} { r } { \\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\| \\Psi _ { a } \\| _ { 2 } = O ( 1 ) } \\end{array}$ . It follows that ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/a9192a7fcda4c92cb8f75adcab18fd6b499211246abaaa65fa75569d72169ebb.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { 1 \\leq k _ { 1 } \\leq k _ { 2 } \\leq K } \\Big \\| \\widehat { V } ( k _ { 1 } , k _ { 2 } ) - \\widehat { V } ^ { * } ( k _ { 1 } , k _ { 2 } ) \\Big \\| _ { 2 } \\preceq q \\delta _ { n ( t _ { 1 } ) } + \\sqrt { q n ^ { - 1 } ( t _ { 1 } ) \\log n ( t _ { 1 } ) } ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ . Combining this together with equation 50 and equation 55, we obtain with probability at least $1 - \\dot { O } ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ that ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/a417f09523a6add689d0026a3b3578e95fe4ed870caf7ef06efdec331f04c547.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { 1 \\leq k _ { 1 } \\leq k _ { 2 } \\leq K } \\left\\| { \\widehat { V } } ( k _ { 1 } , k _ { 2 } ) - V ( k _ { 1 } , k _ { 2 } ) \\right\\| _ { 2 } \\preceq q \\delta _ { n ( t _ { 1 } ) } + { \\sqrt { q n ^ { - 1 } ( t _ { 1 } ) \\log n ( t _ { 1 } ) } } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "Consider the function $\\mathbf { g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } }$ defined in the proof of Theorem 1. We fix $\\delta = \\eta ^ { - 1 } \\{ \\log K +$ $4 d \\log n ( t _ { 1 } ) \\}$ . Based on Lemma A2 in Belloni $\\&$ Oliveira (2018), we have with probability at least $1 - O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ that ",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "img_path": "images/cfeeb911fb3cd7dcf55e2454f6e1f3c166ae1b764787191c509dfd33ece8097e.jpg",
+ "text": "$$\n\\begin{array} { r l r } & { } & { \\underset { \\{ \\nu _ { k } \\} _ { k } } { \\operatorname* { s u p } } \\left| { \\mathbf E } ^ { * } { \\mathbf g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ( N ( 0 , \\widehat { V } ) ) - { \\mathbf E } { \\mathbf g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k } \\} _ { k } } ( N ( 0 , { \\pmb V } ) ) \\right| } \\\\ & { } & { \\preceq q \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\left( q \\delta _ { n ( t _ { 1 } ) } + \\sqrt { q n ^ { - 1 } ( t _ { 1 } ) \\log n ( t _ { 1 } ) } \\right) , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "where $\\mathbf { E } ^ { \\ast }$ denotes the expectation conditional on the observed data. For a given set of thresholds $\\{ \\nu _ { k } \\} _ { k }$ , using similar arguments in proving equation 31, equation 32, equation 34 and equation 35, we can show with probability at least $1 - O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ that ",
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+ "type": "equation",
+ "img_path": "images/cc351a59d858ee7490babaaeb95a55a89c6746faf6069c29b942d06882afd573.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\operatorname* { P r } ^ { * } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\sqrt { n ( t _ { k } ) } \\widehat { S } ^ { \\scriptscriptstyle { \\mathrm { M B } * } } - \\nu _ { k } \\right) \\le 0 \\right\\} \\le \\mathrm { E } ^ { * } \\mathtt { g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k , + } \\} _ { k } } ( N ( 0 , \\widehat { V } ) ) } \\\\ { \\le } & { \\mathrm { E } \\underline { { \\mathtt { g } } } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k , + } \\} _ { k } } ( N ( 0 , V ) ) + O ( 1 ) q \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\left( q \\delta _ { n ( t _ { 1 } ) } + \\sqrt { q n ^ { - 1 } ( t _ { 1 } ) \\log n ( t _ { 1 } ) } \\right) } \\\\ { \\le } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\overline { { \\mathtt { X } } } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - \\nu _ { k , + } ^ { * } \\right) \\le 0 \\right\\} } \\\\ { + } & { O ( 1 ) q \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\left( q \\delta _ { n ( t _ { 1 } ) } + \\sqrt { q n ^ { - 1 } ( t _ { 1 } ) \\log n ( t _ { 1 } ) } \\right) , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "and ",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "img_path": "images/93ad3108d19767a0e7bdebc66e4c173502d86665bbb8bce23498ead2eeefc581.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\operatorname* { P r } ^ { * } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\sqrt { n ( t _ { k } ) } \\widehat { S } ^ { \\scriptscriptstyle { \\mathrm { M B } * } } - \\nu _ { k } \\right) \\leq 0 \\right\\} \\geq \\mathrm { E } ^ { * } \\mathrm { g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k , - } \\} _ { k } } ( N ( 0 , \\widehat { V } ) ) } \\\\ { \\geq } & { \\mathrm { E } \\mathrm { g } _ { \\delta } \\circ \\phi _ { \\eta , \\{ \\nu _ { k , - } \\} _ { k } } ( N ( 0 , V ) ) - O ( 1 ) q \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\left( q \\delta _ { n ( t _ { 1 } ) } + \\sqrt { q n ^ { - 1 } ( t _ { 1 } ) \\log n ( t _ { 1 } ) } \\right) } \\\\ { \\geq } & { \\operatorname* { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - \\nu _ { k , - } ^ { * } \\right) \\leq 0 \\right\\} } \\\\ { - } & { O ( 1 ) q \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} \\left( q \\delta _ { n ( t _ { 1 } ) } + \\sqrt { q n ^ { - 1 } ( t _ { 1 } ) \\log n ( t _ { 1 } ) } \\right) , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "where $O ( 1 )$ denotes some positive constant, and ",
+ "bbox": [
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "img_path": "images/dc99c18c562c2884047ccad7efef2eb4838bd16083b3ca6d71e93d3c0cc346c6.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\nu _ { k , + } = \\nu _ { k } + \\eta ^ { - 1 } \\{ 4 d \\log n ( t _ { 1 } ) + \\log K \\} + \\bar { c } ^ { * } n ^ { - 2 } ( t _ { 1 } ) , \\nu _ { k , + } ^ { * } = \\nu _ { k , + } + 3 \\eta ^ { - 1 } \\{ 4 d \\log n ( t _ { 1 } ) + \\log K \\} } \\\\ & { \\nu _ { k , - } = \\nu _ { k } - 3 \\eta ^ { - 1 } \\{ 4 d \\log n ( t _ { 1 } ) + \\log K \\} - \\bar { c } ^ { * } n ^ { - 2 } ( t _ { 1 } ) , \\nu _ { k , - } ^ { * } = \\nu _ { k , - } - \\eta ^ { - 1 } \\{ 4 d \\log n ( t _ { 1 } ) + \\log K \\} } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ 843
+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "By Theorem 2 of Chernozhukov et al. (2017), we obtain that ",
+ "bbox": [
+ 173,
+ 845,
+ 573,
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "img_path": "images/035cd86cc5282b44c2b747c06c0961c61522ced32361a9725cacb548c0b94124.jpg",
+ "text": "$$\n\\begin{array} { r } { \\mathtt { p } _ { \\mathrm { T } } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - \\nu _ { k , + } ^ { * } \\right) \\le 0 \\right\\} - \\mathtt { P r } \\left\\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } \\left( \\underset { x \\in \\mathbb { X } _ { 0 } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - \\nu _ { k , - } ^ { * } \\right) \\le 0 \\right\\} } \\\\ { \\preceq \\eta ^ { - 1 } \\{ \\log ^ { 3 / 2 } n ( t _ { 1 } ) + \\log ^ { 3 / 2 } K \\} + \\bar { c } ^ { * } n ^ { - 2 } ( t _ { 1 } ) \\{ \\log ^ { 1 / 2 } n ( t _ { 1 } ) + \\log ^ { 1 / 2 } K \\} . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 181,
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+ 839,
+ 929
+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "It follows that ",
+ "bbox": [
+ 173,
+ 103,
+ 267,
+ 118
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "img_path": "images/14011b32e0863773df48fc26ca76e79e1487225d9103c264dcc95cb23749c27f.jpg",
+ "text": "$$\n\\begin{array} { r } { \\underset { \\nu _ { k } \\not \\mathrm { J } _ { k } } { \\operatorname* { s u p } } | { \\operatorname* { P r } } ^ { * } \\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } ( \\sqrt { n ( t _ { k } ) } \\widehat { S } ^ { \\mathrm { M B } * } - \\nu _ { k } ) \\leq 0 \\} - { \\operatorname* { P r } } \\{ \\underset { k \\in \\{ 1 , \\ldots , K \\} } { \\operatorname* { m a x } } ( \\underset { x \\in \\mathbb { X } } { \\operatorname* { s u p } } \\varphi ^ { \\top } ( x ) G ( t _ { k } ) - \\nu _ { k } ) \\leq 0 \\} } \\\\ { \\preceq q \\eta ^ { 2 } \\{ \\log ^ { 2 } K + \\log ^ { 2 } n ( t _ { 1 } ) \\} ( q \\delta _ { n ( t _ { 1 } ) } + \\sqrt { q n ^ { - 1 } ( t _ { 1 } ) \\log n ( t _ { 1 } ) } ) } \\\\ { + \\eta ^ { - 1 } \\{ \\log ^ { 3 / 2 } n ( t _ { 1 } ) + \\log ^ { 3 / 2 } K \\} + \\bar { c } ^ { * } n ^ { - 2 } ( t _ { 1 } ) \\{ \\log ^ { 1 / 2 } n ( t _ { 1 } ) + \\log ^ { 1 / 2 } K \\} } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 181,
+ 125,
+ 839,
+ 213
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( n ^ { - \\alpha _ { 0 } } ( t _ { 1 } ) )$ . Set ",
+ "bbox": [
+ 173,
+ 218,
+ 482,
+ 234
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "img_path": "images/6896a4985723f61306460a5100ec6aa098d4224277481af733881c1bc66b21a1.jpg",
+ "text": "$$\n\\eta = \\operatorname* { m i n } [ q ^ { - 1 } n ^ { \\alpha _ { 0 } / 3 } ( t _ { 1 } ) \\log ^ { - ( 1 + 2 \\alpha _ { 0 } ) / 6 } \\{ K n ( t _ { 1 } ) \\} , q ^ { - 1 / 2 } n ^ { 1 / 6 } ( t _ { 1 } ) \\log ^ { - 1 / 3 } \\{ K n ( t _ { 1 } ) \\} ] ,\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 223,
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+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "we obtain the desired result. ",
+ "bbox": [
+ 173,
+ 272,
+ 359,
+ 287
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "E.8 PROOF OF LEMMA 7 ",
+ "text_level": 1,
+ "bbox": [
+ 174,
+ 308,
+ 357,
+ 321
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "Combining Lemma 6 with Lemma 4 yields that ",
+ "bbox": [
+ 173,
+ 334,
+ 486,
+ 349
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "img_path": "images/b0a6063c35d1e3122f587337a9ba3c78e38c7bb832a251582d9aa652c294aa43.jpg",
+ "text": "$$\n\\left. \\Sigma _ { a } ^ { - 1 } \\left( \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} \\right) \\right. _ { 2 } \\preceq q ^ { 1 / 2 } k ^ { - 1 / 2 } \\sqrt { \\log k } , \\forall k \\geq j _ { n } , a \\in \\{ 0 , 1 \\} ,\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 181,
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+ 826,
+ 404
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ . Combining this together with equation 16 yields that ",
+ "bbox": [
+ 171,
+ 411,
+ 766,
+ 429
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "img_path": "images/bc2cf4ae409cbfad26663ccbbefa14adf028030f3840b8a4cd1fe6fc98112b3b.jpg",
+ "text": "$$\n\\begin{array} { r } { \\| \\widehat { \\beta } _ { a , k } - \\beta _ { a } \\| _ { 2 } \\preceq q ^ { 1 / 2 } k ^ { - 1 / 2 } \\sqrt { \\log k } , \\forall k \\geq j _ { n } , a \\in \\{ 0 , 1 \\} , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ 436,
+ 674,
+ 458
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ . The proof is hence completed. ",
+ "bbox": [
+ 171,
+ 467,
+ 620,
+ 483
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "E.9 PROOF OF LEMMA 8 ",
+ "text_level": 1,
+ "bbox": [
+ 173,
+ 502,
+ 359,
+ 516
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "Notice that ",
+ "bbox": [
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+ 250,
+ 544
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "img_path": "images/8475210fe69162edb06d280f3b6535563fbb9e59422470507dbea1829fb922e0.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\left\\| \\frac { 1 } { k } \\displaystyle \\sum _ { i = 1 } ^ { k } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) ( Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } ) ^ { 2 } - \\Phi _ { a } \\right\\| _ { 2 } } \\\\ { \\le } & { \\left\\| \\frac { 1 } { k } \\displaystyle \\sum _ { i = 1 } ^ { k } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) [ \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} ^ { 2 } - \\sigma ^ { 2 } ( a , X _ { i } ) ] \\right\\| _ { 2 } } \\\\ { + } & { \\left\\| \\frac { 1 } { k } \\displaystyle \\sum _ { i = 1 } ^ { k } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\sigma ^ { 2 } ( a , X _ { i } ) - \\Phi _ { a } \\right\\| _ { 2 } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 261,
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+ 750,
+ 686
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "Similar to the proof of Lemma 5, we can show that the second term on the RHS of equation 56 is of the order $O ( q \\delta _ { k } + \\sqrt { q k ^ { - 1 } \\log k } )$ , for any $a \\in \\{ 0 , 1 \\}$ and any $k \\geq j _ { n }$ , with probability at least $1 - O \\big ( j _ { n } ^ { - \\alpha _ { 0 } } \\big )$ . As for the first term, notice that each element in the matrix ",
+ "bbox": [
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+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "img_path": "images/c1eefd4a3f072fb139798ea9dd495c7b3d5de3bb86fdb5361c8f334000ae3261.jpg",
+ "text": "$$\n\\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) \\{ ( Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } ) ^ { 2 } - \\sigma ^ { 2 } ( a , X _ { i } ) \\}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 264,
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+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "corresponds to a martingale with respect to the filtration $\\{ \\sigma ( \\mathcal { F } _ { i - 1 } ) : i \\ge 1 \\}$ , under (A1) and (A2). Using similar arguments in proving Equation (C.15) of Shi et al. (2020b), we can show that ",
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "img_path": "images/c5ef5ba1bf838d63226ef6e70d156c9f5306b602bda9c7bc55db3c50fcc52e94.jpg",
+ "text": "$$\n\\left. \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) [ \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} ^ { 2 } - \\sigma ^ { 2 } ( a , X _ { i } ) ] \\right. _ { 2 } \\preceq q ^ { 1 / 2 } k ^ { - 1 / 2 } \\sqrt { \\log k } ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ 898
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ . The proof is hence completed. ",
+ "bbox": [
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+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "E.10 PROOF OF LEMMA 2 ",
+ "text_level": 1,
+ "bbox": [
+ 174,
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+ 369,
+ 118
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "We begin by providing an upper bound for $\\begin{array} { r } { \\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\| \\widehat { \\beta } _ { a , k } - \\beta _ { a } \\| _ { 2 } } \\end{array}$ . With some calculations, we have ",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "img_path": "images/f99d49c76ea24f57de770704be3f6da67df34d4a6c067d1dc2ff547e09d87889.jpg",
+ "text": "$$\n\\begin{array} { r } { \\displaystyle \\underset { a \\in \\{ 0 , 1 \\} } { \\operatorname* { m a x } } \\| \\widehat { \\beta } _ { a , k } - \\beta _ { a } \\| _ { 2 } = \\displaystyle \\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\frac { 1 } { k } \\left\\| \\widehat { \\boldsymbol { \\Sigma } } _ { a , k } ^ { - 1 } \\left( \\displaystyle \\sum _ { i = 1 } ^ { k } \\varphi ( X _ { i } ) \\mathbb { I } ( A _ { i } = a ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} \\right) \\right\\| _ { 2 } } \\\\ { \\leq \\displaystyle \\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\left\\| \\widehat { \\boldsymbol { \\Sigma } } _ { a , k } ^ { - 1 } \\right\\| _ { 2 a \\in \\{ 0 , 1 \\} } \\displaystyle \\operatorname* { m a x } _ { k } \\left\\| \\displaystyle \\sum _ { i = 1 } ^ { k } \\varphi ( X _ { i } ) \\mathbb { I } ( A _ { i } = a ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} \\right\\| _ { 2 } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "By Lemma 6, we obtain with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ that ",
+ "bbox": [
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+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "img_path": "images/8e02308c99fdcb5824d3e79b41fea44dfc335ef10ff4c22e3e84cb6f54f98c20.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\frac { 1 } { k } \\left. \\sum _ { i = 1 } ^ { k } \\varphi ( X _ { i } ) \\mathbb { I } ( A _ { i } = a ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} \\right. _ { 2 } \\preceq q ^ { 1 / 2 } k ^ { - 1 / 2 } \\sqrt { \\log k } , \\forall k \\geq j _ { n } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "Similarly, we can show with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ that ",
+ "bbox": [
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+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "img_path": "images/c7495ade52eb7d5c794ef9d165dd659ff162e83a0d29075f2fe50e77b3ecae2a.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\frac { 1 } { k } \\left\\| \\sum _ { i = 1 } ^ { k } \\varphi ( X _ { i } ) \\mathbb { I } ( A _ { i } = a ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { a } \\} \\right\\| _ { 2 } \\preceq q ^ { 1 / 2 } k ^ { - 1 / 2 } \\sqrt { \\log j _ { n } } ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ 732,
+ 391
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "Similar to equation 42, we have ",
+ "bbox": [
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+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "img_path": "images/112658f5f18d71dff63814ee8aeac48fb40109561b305adca62497dac74cea42.jpg",
+ "text": "$$\n\\begin{array} { r l r } & { } & { \\underset { a \\in \\{ 0 , 1 \\} } { \\operatorname* { m a x } } ~ \\lambda _ { \\mathrm { m i n } } [ \\widehat { \\Sigma } _ { a , k } ] \\geq \\underset { a \\in \\{ 0 , 1 \\} } { \\operatorname* { m i n } } ~ \\lambda _ { \\mathrm { m i n } } \\left( \\mathrm { E } ^ { \\mathcal { F } _ { i - 1 } } \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\frac { 1 } { k } \\underset { i = 1 } { \\overset { k } { \\sum } } \\pi _ { i - 1 } ( a , X ) \\right) } \\\\ & { } & { - \\underset { a \\in \\{ 0 , 1 \\} } { \\operatorname* { m a x } } ~ \\frac { 1 } { k } \\left\\| \\sum _ { i = 1 } ^ { k } \\{ \\mathbb { I } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) - \\mathrm { E } ^ { \\mathcal { F } _ { i - 1 } } \\pi _ { i - 1 } ( a , X ) \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\} \\right\\| _ { 2 } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "Using similar arguments in proving equation 44, we can show that ",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "img_path": "images/8243d3944f2a103dde0cdd28b11c4f9bd983f4d5c87a4c5166ab7795bc02f78c.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\left\\| \\sum _ { i = 1 } ^ { k } \\{ { \\mathbb { I } } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) - \\mathbf { E } ^ { \\mathcal { F } _ { i - 1 } } \\pi _ { i - 1 } ( a , X ) \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\} \\right\\| _ { 2 } \\preceq \\sqrt { q k \\log k } ,\n$$",
+ "text_format": "latex",
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+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ . Similarly, we can show ",
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+ "type": "equation",
+ "img_path": "images/8f0e5115f02954dee91dfc36586b5fd172651251319fea0d46280c2833ebf7dc.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\left\\| \\sum _ { i = 1 } ^ { k } \\{ { \\mathbb { I } } ( A _ { i } = a ) \\varphi ( X _ { i } ) \\varphi ^ { \\top } ( X _ { i } ) - \\mathbf { E } ^ { \\mathcal { F } _ { i - 1 } } \\pi _ { i - 1 } ( a , X ) \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\} \\right\\| _ { 2 } \\preceq \\sqrt { q k \\log j _ { n } } ,\n$$",
+ "text_format": "latex",
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+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ ",
+ "bbox": [
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+ },
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+ "type": "text",
+ "text": "Without loss of generality, assume $\\varepsilon _ { 0 } \\leq 1 / 2$ . Notice that we have $\\pi _ { i - 1 } ( a , x ) \\geq \\varepsilon _ { 0 }$ , for any $a \\in$ $\\{ 0 , 1 \\} , x \\in \\mathbb { X }$ and $i \\geq N _ { 0 }$ . This together with Lemma equation 4 implies that ",
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+ "type": "equation",
+ "img_path": "images/8583626ddc7a477a67a134fba07ab17a194e0c368d18cfe5ee3160f6e996dfa1.jpg",
+ "text": "$$\n\\operatorname* { i n f } _ { a \\in \\{ 0 , 1 \\} , n \\geq j _ { n } } \\lambda _ { \\operatorname* { m i n } } \\left( \\mathrm { E } ^ { \\mathcal { F } _ { i - 1 } } \\varphi ( X ) \\varphi ^ { \\top } ( X ) \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\pi _ { i - 1 } ( a , X ) \\right) \\geq \\frac { n - N _ { 0 } } { n } \\varepsilon _ { 0 } \\geq \\frac { j - N _ { 0 } } { j } \\varepsilon _ { 0 } .\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "Combining this together with equation 60 and equation 61 yields ",
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+ "type": "equation",
+ "img_path": "images/85070b0f79dc8a2678bd4332fdfcf8fcc1b4282538fe6ebac2a7baa9794971d8.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\lambda _ { \\operatorname* { m i n } } [ \\widehat { \\Sigma } _ { a , k } ] \\geq \\frac { \\varepsilon _ { 0 } } { 2 } , \\forall k \\geq L ^ { * } \\sqrt { q \\log j _ { n } } ,\n$$",
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+ "type": "text",
+ "text": "for some constant $L ^ { * } \\geq 1$ , with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ . This together with equation 58 and equation 59 yields that ",
+ "bbox": [
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+ "type": "equation",
+ "img_path": "images/32eae6a2cd006bc94c93939067fb95113a411988e498c1f8593cf8d26868d21a.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { a \\in \\{ 0 , 1 \\} } \\| \\widehat { \\beta } _ { a , k } - \\beta _ { a } \\| _ { 2 } \\preceq q ^ { 1 / 2 } k ^ { - 1 / 2 } \\sqrt { \\log \\operatorname* { m a x } ( k , j _ { n } ) } , \\forall k \\geq L ^ { * } \\sqrt { q \\log j _ { n } } ,\n$$",
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ ",
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "By Condition (A3), we have ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/ddb1d90e27280361dbab6ce36fd255d2d4f0e6ead7bf20cb281fd61fc751f619.jpg",
+ "text": "$$\n\\begin{array} { r } { | \\varphi ^ { \\mathsf { T } } ( X ) ( \\widehat { \\beta } _ { 1 , k } - \\widehat { \\beta } _ { 0 , k } - \\beta _ { 1 } + \\beta _ { 0 } ) | \\leq \\bar { L } q k ^ { - 1 / 2 } \\log ^ { 1 / 2 } \\operatorname* { m a x } ( k , j _ { n } ) , \\forall k \\geq L ^ { * } \\sqrt { q \\log j _ { n } } , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "for some constant $\\bar { L } > 0$ , with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ . ",
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "For any $z _ { 1 } , z _ { 2 } \\in \\mathbb { R }$ , we have $\\mathbb { I } ( z _ { 1 } > 0 ) \\neq \\mathbb { I } ( z _ { 2 } > 0 )$ only when $| z _ { 1 } - z _ { 2 } | \\geq | z _ { 2 } |$ . Hence, under the event defined in equation 62, the event $\\mathbb { I } \\{ \\varphi ^ { \\top } ( X ) ( \\widehat { \\beta } _ { 1 , k } - \\widehat { \\beta } _ { 0 , k } ) > 0 \\} \\neq \\mathbb { I } \\{ \\varphi ^ { \\top } ( X ) ( \\beta _ { 1 } - \\beta _ { 0 } ) > 0 \\}$ occurs only when ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/241af2b941e8433242d3376619be37c7a81a0e63dd6597742ecf5c6a1ef35b97.jpg",
+ "text": "$$\n\\begin{array} { r } { | \\varphi ^ { \\top } ( X ) ( \\beta _ { 1 } - \\beta _ { 0 } ) | \\le | \\varphi ^ { \\top } ( X ) ( \\widehat \\beta _ { 1 , k } - \\widehat \\beta _ { 0 , k } - \\beta _ { 1 } + \\beta _ { 0 } ) | \\le \\bar { L } q k ^ { - 1 / 2 } \\sqrt { \\log \\operatorname* { m a x } ( k , j _ { n } ) } , } \\end{array}\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "for any $k \\geq j _ { n }$ . Under the given conditions, we have ",
+ "bbox": [
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+ },
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+ "type": "equation",
+ "img_path": "images/64facdf10834de275f42513b2ddc9f6855f03f3bc3fac35d07fd7b06ee94e6c9.jpg",
+ "text": "$$\n\\begin{array} { r } { \\operatorname* { P r } \\Big ( | \\varphi ^ { \\top } ( X ) ( \\beta _ { 1 } - \\beta _ { 0 } ) | \\leq \\bar { L } q k ^ { - 1 / 2 } \\log ^ { 1 / 2 } \\operatorname* { m a x } ( k , j _ { n } ) \\Big ) \\leq \\bar { L } L _ { 0 } q k ^ { - 1 / 2 } \\log ^ { 1 / 2 } \\operatorname* { m a x } ( k , j _ { n } ) . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "Notice that when $\\mathbb { I } \\{ \\varphi ^ { \\top } ( X ) ( \\widehat { \\beta } _ { 1 , k } \\ - \\ \\widehat { \\beta } _ { 0 , k } ) \\ > \\ \\ 0 \\} = \\mathbb { I } \\{ \\varphi ^ { \\top } ( X ) ( \\beta _ { 1 } \\ - \\ \\beta _ { 0 } ) \\ > \\ \\ 0 \\}$ , we have $\\pi _ { k } ( a , X ) = \\pi ^ { * } ( a , X ) $ . Thus, we obtain $\\pi _ { k } ( a , X ) ~ = ~ \\pi ^ { * } ( a , X )$ if $\\begin{array} { r l } { \\vert \\varphi ^ { \\top } ( X ) ( \\beta _ { 1 } - \\beta _ { 0 } ) \\vert } & { { } > } \\end{array}$ $\\bar { L } q k ^ { - 1 / 2 } \\sqrt { \\log \\operatorname* { m a x } ( k , j _ { n } ) }$ , for any $k \\geq L ^ { * } { \\sqrt { q \\log j _ { n } } }$ . Set $k _ { 0 } = L ^ { * } { \\sqrt { q \\log j _ { n } } }$ . By equation 62 and equation 63, we have with probability at least $1 - O ( j _ { n } ^ { - 1 } )$ that ",
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+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "img_path": "images/a467558b115027eac90b8bfd57ea2595bbf517c18d0cc91e2f5c782ce7abefa8.jpg",
+ "text": "$$\n\\begin{array} { r l } { { \\sum _ { \\alpha \\in \\{ 0 , 1 \\} } \\sum _ { i = 1 } ^ { K } \\Big \\{ \\underset { i = 1 } { \\overset { K } { \\sum } } \\{ \\alpha _ { i - 1 } ( \\alpha , X ) - \\pi ^ { * } ( \\alpha , X ) \\} \\Big \\} \\Bigg \\{ \\underset { \\alpha \\in \\{ 0 , 1 \\} } { \\sum } \\sum _ { i = 1 } ^ { K } \\sum _ { i = 1 } ^ { K } \\big \\lvert \\pi _ { i - 1 } ( \\alpha , X ) - \\pi ^ { * } ( \\alpha , X ) \\big \\rvert } } \\\\ { + \\ } & { \\sum _ { \\alpha \\in \\{ 0 , 1 \\} } \\sum _ { i = K + 1 } ^ { K } \\sum _ { i = 1 } ^ { K } \\big \\lvert \\pi _ { i - 1 } ( \\alpha , X ) - \\pi ^ { * } ( \\alpha , X ) \\big \\rvert \\leq 2 L ^ { * } \\sqrt { \\| \\Phi \\| _ { 2 } } } \\\\ { + \\ } & { \\mathrm { a s t ~ } \\big \\{ \\underset { i = 1 } { \\overset { K } { \\sum } } \\big \\} \\sum _ { i = 1 } ^ { K } \\mathbb { E } ^ { I _ { i - 1 } } \\big \\lvert \\pi _ { i - 1 } ( \\alpha , X ) - \\pi ^ { * } ( \\alpha , X ) \\big \\rvert \\mathbb { E } \\big \\| \\mathcal { E } ^ { T } ( X ) ( \\beta _ { 1 } - \\beta _ { 0 } ) \\big \\rvert > L \\bar { q } ^ { 1 - 1 / 2 } \\log ^ { 1 / 2 } \\delta \\big \\} } \\\\ { + \\ } & { \\underset { \\mathrm { a s t ~ } \\{ 0 , 1 \\} + \\kappa = + 1 } { \\overset { K } { \\sum } } \\mathbb { E } ^ { I _ { i } } \\big \\lvert \\pi _ { i - 1 } ( \\alpha , X ) - \\pi ^ { * } ( \\alpha , X ) \\big \\rvert \\mathbb { E } \\big \\| \\mathcal { E } ^ { T } ( X ) ( \\beta _ { 1 } - \\beta _ { 0 } ) \\big \\rvert \\leq L \\bar { q } ^ { 1 - 1 / 2 } \\log ^ { 1 / 2 } \\delta \\big \\} } \\\\ { + \\ } & \\underset { \\mathrm { a s t ~ } \\{ 0 , 1 \\} + \\kappa = + 1 } { \\overset { K } { \\sum } } \\mathbb { E } ^ { I _ { i } } \\cdot \\big \\| \\pi _ { i - 1 } ( \\alpha , X ) - \\pi ^ { * } ( \\alpha , X ) \\big \\| \\mathbb { E } \\big \\| \\mathcal { E } ^ { T } ( X ) ( \\beta _ { 1 } - \\beta _ { 0 } ) \\big \\| \\leq L \\bar { q } ^ { 1 - 1 / 2 } \\log ^ { 1 / 2 } \\delta \\big \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "The proof is hence completed. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "F COMPARISON OF THE BASELINE",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "Consider our test statistic $S ( t )$ . Under $H _ { 0 }$ , it can be bounded from above by ",
+ "bbox": [
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+ },
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+ "type": "equation",
+ "img_path": "images/4e4f08c3e1765f2867133075a8f17d560b165aebd99f647fa2b66a4d309ebc48.jpg",
+ "text": "$$\n\\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\varphi ^ { \\top } ( x ) \\{ { \\widehat { \\beta } } _ { 1 } ( t ) - \\beta _ { 1 } ^ { * } - { \\widehat { \\beta } } _ { 0 } ( t ) + \\beta _ { 0 } ^ { * } \\} .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "It suffices to provide an upper bound for the above expression. By Cauchy-Schwarz inequality, equation 64 can be upper bounded by ",
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "img_path": "images/bcfb5c2f16ca4d4b73ee4a5fa8da9e9ee6f1ba036d8fb061746f168c789d29e6.jpg",
+ "text": "$$\n\\operatorname* { s u p } _ { x \\in \\mathbb { X } } \\| \\varphi ( x ) \\| _ { 2 } \\| { \\widehat { \\beta } } _ { 1 } ( t ) - \\beta _ { 1 } ^ { * } - { \\widehat { \\beta } } _ { 0 } ( t ) + \\beta _ { 0 } ^ { * } \\| _ { 2 } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "It suffices to provide anytime upper bound for $\\| \\widehat { \\beta } _ { 1 } ( t ) - \\beta _ { 1 } ^ { * } - \\widehat { \\beta } _ { 0 } ( t ) - \\beta _ { 0 } ^ { * } \\| _ { 2 }$ . ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Recall that ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/320f0e25225397b8780814639c9dbf20b2aa71664c0d627b8c71741d221ac190.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\widehat { \\beta } _ { 1 } ( t ) - \\beta _ { 1 } ^ { * } - \\widehat { \\beta } _ { 0 } ( t ) + \\beta _ { 0 } ^ { * } } \\\\ { = } & { \\frac { 1 } { N ( t ) } \\displaystyle \\sum _ { i = 1 } ^ { N ( t ) } [ \\mathbb { I } ( A _ { i } = 1 ) \\widehat { \\Sigma } _ { 1 } ^ { - 1 } ( t ) \\varphi ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 1 } ^ { * } \\} - \\mathbb { I } ( A _ { i } = 0 ) \\widehat { \\Sigma } _ { 0 } ^ { - 1 } ( t ) \\varphi ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\} ] } \\end{array}\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "The above expression is asymptotically equivalent to ",
+ "bbox": [
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+ "page_idx": 32
+ },
+ {
+ "type": "equation",
+ "img_path": "images/7be9828d8f1930eb8d95936fbb1a58b809648e88b97a4597dc0c45697b8f013e.jpg",
+ "text": "$$\n\\frac { 1 } { N ( t ) } \\sum _ { i = 1 } ^ { N ( t ) } [ \\mathbb { I } ( A _ { i } = 1 ) \\Sigma _ { 1 } ^ { - 1 } \\varphi ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 1 } ^ { * } \\} - \\mathbb { I } ( A _ { i } = 0 ) \\Sigma _ { 0 } ^ { - 1 } \\varphi ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\beta _ { 0 } ^ { * } \\} ] .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 32
+ },
+ {
+ "type": "text",
+ "text": "By the law of iterated logarithm, the $\\ell$ -th dimension of the above expression can be upper bounded by ",
+ "bbox": [
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+ ],
+ "page_idx": 32
+ },
+ {
+ "type": "equation",
+ "img_path": "images/4f345e4d113717d4323a9b5eb14c21ed44fc574d0ab1281aaa4620a0248705c1.jpg",
+ "text": "$$\nN ^ { - 1 / 2 } ( t ) \\sqrt { 2 \\sigma _ { \\ell } ^ { 2 } \\log \\log \\{ N ( t ) \\} } ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 32
+ },
+ {
+ "type": "text",
+ "text": "where $\\textstyle \\sum _ { \\ell } { \\widehat { \\sigma } } _ { \\ell } ^ { 2 }$ can be consistently estimated by ",
+ "bbox": [
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+ "page_idx": 32
+ },
+ {
+ "type": "equation",
+ "img_path": "images/00f33e25aa2d94ac77a786b1cb1b44b00034b9e5ab291679ee697ed278c07605.jpg",
+ "text": "$$\n{ \\frac { 1 } { \\mathrm { V } ( t ) } } \\sum _ { i = 1 } ^ { N ( t ) } \\| \\mathbb { I } ( A _ { i } = 1 ) { \\widehat { \\Sigma } } _ { 1 } ^ { - 1 } ( t ) \\varphi ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) { \\widehat { \\beta } } _ { 1 } ( t ) \\} - \\mathbb { I } ( A _ { i } = 0 ) { \\widehat { \\Sigma } } _ { 0 } ^ { - 1 } ( t ) \\varphi ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) { \\widehat { \\beta } } _ { 0 } ( t ) \\} \\varphi ( X _ { i } ) \\| \\varphi ( X _ { i } ) \\| _ { L ^ { \\infty } ( L ^ { \\infty } ( H ) ) } ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 32
+ },
+ {
+ "type": "text",
+ "text": "As such, the finite error bound is given by ",
+ "bbox": [
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+ ],
+ "page_idx": 32
+ },
+ {
+ "type": "equation",
+ "img_path": "images/5db848cc0d7fb62565b39d4e94d050a4488a84ecfbddd46547f20699c18adde7.jpg",
+ "text": "$$\n\\begin{array} { r l r } { \\displaystyle } & { } & { \\displaystyle \\operatorname* { s u p } _ { x \\in \\widetilde { \\mathbb { X } } } \\| \\varphi ( x ) \\| _ { 2 } \\frac { \\sqrt { 2 \\log \\log \\{ 1 \\} } } { \\sqrt { N ( t ) } } } \\\\ { \\displaystyle } & { } & { \\displaystyle \\sqrt { \\frac { 1 } { N ( t ) } \\sum _ { i = 1 } ^ { N ( t ) } \\| \\mathbb { I } ( A _ { i } = 1 ) \\widehat { \\Sigma } _ { 1 } ^ { - 1 } ( t ) \\varphi ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\widehat { \\beta } _ { 1 } ( t ) \\} - \\mathbb { I } ( A _ { i } = 0 ) \\widehat { \\Sigma } _ { 0 } ^ { - 1 } ( t ) \\varphi ( X _ { i } ) \\{ Y _ { i } - \\varphi ^ { \\top } ( X _ { i } ) \\} } } \\end{array}\n$$",
+ "text_format": "latex",
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+ "page_idx": 32
+ },
+ {
+ "type": "text",
+ "text": "G ADDITIONAL TABLES AND FIGURES ",
+ "text_level": 1,
+ "bbox": [
+ 173,
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+ 508,
+ 463
+ ],
+ "page_idx": 32
+ },
+ {
+ "type": "table",
+ "img_path": "images/83b323dea972b5109fec273889ad3e87eb415945052b5a99367ae5e971167498.jpg",
+ "table_caption": [],
+ "table_footnote": [
+ "Table 1: QTE: rejection probabilities (multiplied by 100) and average stopping times under Scenarios 1 and 2 when $\\alpha _ { 1 } ( \\cdot )$ is chosen as the spending function. Standard errors are reported in the parentheses. "
+ ],
+ "table_body": " | | method | BAT | AVT |
| Random | Adaptive | Random | Adaptive |
| (n,K) | 8 | rej probs | E[stop] | rej probs | E[stop] | rej probs | E[stop] | rej probs | E[stop] |
| S1 | (200,5) | 0.000.100.150.200.250.30 | 5.2(1.1)27.5(2.2)45.5(2.5)62.5(2.4)80.2(2.0)88.2(1.6) | 1763(8) | 6.2(1.2) | 1762(8) | 0.2(0.2) | 1800(0) | 51.2(2.5) | 1586(11) |
| 1644(15) | 26.0(2.2) | 1647(14) | 6.8(1.3) | 1771(6) | 89.0(1.6) | 1344（9) |
| 1511(17)1391(18)1263(17)1176(14) | 44.2(2.5) | 1527(17) | 19.0(2.0) | 1718(10) | 98.0(0.7) | 1250(6) |
| 64.2(2.4) | 1383(18) | 42.8(2.5) | 1581(15) | 99.5(0.4) | 1196(6) |
| 78.8(2.0) | 1266(17) | 72.8(2.2) | 1394(17) | 99.8(0.2) | 1145(5)1091(5) |
| 88.8(1.6) | 1179(14) | 89.2(1.5) | 1216(14) | 100.0(0.0) |
| (20,50) | 0.000.100.150.200.250.30 | 5.8(1.2)27.5(2.2)45.5(2.5)67.0(2.4)83.8(1.8)92.0(1.4) | 1933(9)1771(18)1617(22)1446(22)1287(19)1182(16) | 5.0(1.1) | 1936(9) | 0.2(0.2) | 1978(1) | 51.5(2.5) | 1621(17) |
| 27.5(2.2) | 1771(18) | 8.0(1.4) | 1929(9) | 89.2(1.5) | 1276(13) |
| 45.8(2.5) | 1630(21) | 25.2(2.2) | 1826(15) | 97.8(0.7) | 1166(7) |
| 65.5(2.4)82.5(1.9)91.5(1.4) | 1459(22) | 53.8(2.5) | 1617(20) | 99.0(0.5) | 1105(6) |
| 1288(19) | 79.0(2.0) | 1379(19) | 99.8(0.2) | 1061(4)1030(2) |
| 1193(16) | 94.0(1.2) | 1187(15) | 100.0(0.0) |
| S2 | (200,5) | 0.000.100.150.200.250.30 | 5.2(1.1)18.2(1.9)29.0(2.3)40.5(2.5)50.5(2.5)62.5(2.4) | 1763(8)1692(12) | 6.2(1.2)16.8(1.9) | 1762(8)1699(12) | 0.2(0.2) | 1800(0) | 51.2(2.5)82.2(1.9) | 1586(11)1406(10) |
| 3.0(0.9) | 1788(3) |
| 1633(15)1559(17)1489(18)1407(18) | 25.2(2.2) | 1642(15) | 8.5(1.4) | 1762（7) | 91.8(1.4) | 1323（9) |
| 42.0(2.5)49.8(2.5)62.5(2.4) | 1548(17)1492(18)1413(18) | 17.2(1.9) | 1724(10) | 97.8(0.7) | 1257(7) |
| 33.0(2.4)53.5(2.5) | 1641(14) | 99.0(0.5)99.5(0.4) | 1218(6)1181(6) |
| 62.5(2.4) | 1522(16) |
| (20,50) | 0.000.100.150.200.250.30 | 5.8(1.2)19.0(2.0)28.5(2.3)39.0(2.4)50.7(2.5)65.2(2.4) | 1933(9)1839(16) | 5.0(1.1) | 1936(9) | 0.2(0.2) | 1978(1) | 51.5(2.5) | 1621(17)1360(15) |
| 19.0(2.0) | 1837(16) | 3.5(0.9) | 1961(5) | 81.0(2.0) |
| 1763(19)1680(21) | 28.0(2.2) | 1771(18) | 11.5(1.6) | 1911(10) | 90.8(1.4) | 1256(12)1171(8)1124(6)1092(5) |
| 1680(21) | 41.8(2.5) | 1685(20) | 24.0(2.1) | 1830(15) | 97.5(0.8) |
| 1592(22)1479(22) | 52.5(2.5) | 1568(22) | 44.2(2.5) | 1688(19)1539(20) | 99.0(0.5)99.2(0.4) |
| 63.7(2.4) | 1481(22) | 63.5(2.4) |
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| Random | Adaptive | Random | Adaptive |
| (n,K) | 8 | rej probs | E[stop] | rej probs | E[stop] | rej probs | E[stop] | rej probs | E[stop] |
| S1 | (200,5) | 0.000.100.150.200.250.30 | 5.2(1.1)27.5(2.2)45.5(2.5)62.5(2.4)80.2(2.0)88.2(1.6) | 1763(8) | 6.2(1.2) | 1762(8) | 0.2(0.2) | 1800(0) | 51.2(2.5) | 1586(11) |
| 1644(15) | 26.0(2.2) | 1647(14) | 6.8(1.3) | 1771(6) | 89.0(1.6) | 1344（9) |
| 1511(17)1391(18)1263(17)1176(14) | 44.2(2.5) | 1527(17) | 19.0(2.0) | 1718(10) | 98.0(0.7) | 1250(6) |
| 64.2(2.4) | 1383(18) | 42.8(2.5) | 1581(15) | 99.5(0.4) | 1196(6) |
| 78.8(2.0) | 1266(17) | 72.8(2.2) | 1394(17) | 99.8(0.2) | 1145(5)1091(5) |
| 88.8(1.6) | 1179(14) | 89.2(1.5) | 1216(14) | 100.0(0.0) |
| (20,50) | 0.000.100.150.200.250.30 | 5.8(1.2)27.5(2.2)45.5(2.5)67.0(2.4)83.8(1.8)92.0(1.4) | 1933(9)1771(18)1617(22)1446(22)1287(19)1182(16) | 5.0(1.1) | 1936(9) | 0.2(0.2) | 1978(1) | 51.5(2.5) | 1621(17) |
| 27.5(2.2) | 1771(18) | 8.0(1.4) | 1929(9) | 89.2(1.5) | 1276(13) |
| 45.8(2.5) | 1630(21) | 25.2(2.2) | 1826(15) | 97.8(0.7) | 1166(7) |
| 65.5(2.4)82.5(1.9)91.5(1.4) | 1459(22) | 53.8(2.5) | 1617(20) | 99.0(0.5) | 1105(6) |
| 1288(19) | 79.0(2.0) | 1379(19) | 99.8(0.2) | 1061(4)1030(2) |
| 1193(16) | 94.0(1.2) | 1187(15) | 100.0(0.0) |
| S2 | (200,5) | 0.000.100.150.200.250.30 | 5.2(1.1)18.2(1.9)29.0(2.3)40.5(2.5)50.5(2.5)62.5(2.4) | 1763(8)1692(12) | 6.2(1.2)16.8(1.9) | 1762(8)1699(12) | 0.2(0.2) | 1800(0) | 51.2(2.5)82.2(1.9) | 1586(11)1406(10) |
| 3.0(0.9) | 1788(3) |
| 1633(15)1559(17)1489(18)1407(18) | 25.2(2.2) | 1642(15) | 8.5(1.4) | 1762（7) | 91.8(1.4) | 1323（9) |
| 42.0(2.5)49.8(2.5)62.5(2.4) | 1548(17)1492(18)1413(18) | 17.2(1.9) | 1724(10) | 97.8(0.7) | 1257(7) |
| 33.0(2.4)53.5(2.5) | 1641(14) | 99.0(0.5)99.5(0.4) | 1218(6)1181(6) |
| 62.5(2.4) | 1522(16) |
| (20,50) | 0.000.100.150.200.250.30 | 5.8(1.2)19.0(2.0)28.5(2.3)39.0(2.4)50.7(2.5)65.2(2.4) | 1933(9)1839(16) | 5.0(1.1) | 1936(9) | 0.2(0.2) | 1978(1) | 51.5(2.5) | 1621(17)1360(15) |
| 19.0(2.0) | 1837(16) | 3.5(0.9) | 1961(5) | 81.0(2.0) |
| 1763(19)1680(21) | 28.0(2.2) | 1771(18) | 11.5(1.6) | 1911(10) | 90.8(1.4) | 1256(12)1171(8)1124(6)1092(5) |
| 1680(21) | 41.8(2.5) | 1685(20) | 24.0(2.1) | 1830(15) | 97.5(0.8) |
| 1592(22)1479(22) | 52.5(2.5) | 1568(22) | 44.2(2.5) | 1688(19)1539(20) | 99.0(0.5)99.2(0.4) |
| 63.7(2.4) | 1481(22) | 63.5(2.4) |
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