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+# OFF-POLICY ACTOR-CRITIC WITH SHARED EXPERIENCE REPLAY
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We investigate the combination of actor-critic reinforcement learning algorithms with uniform large-scale experience replay and propose solutions for two challenges: (a) efficient actor-critic learning with experience replay (b) stability of off-policy learning where agents learn from other agents behaviour. We employ those insights to accelerate hyper-parameter sweeps in which all participating agents run concurrently and share their experience via a common replay module.
+
+To this end we analyze the bias-variance tradeoffs in V-trace, a form of importance sampling for actor-critic methods. Based on our analysis, we then argue for mixing experience sampled from replay with on-policy experience, and propose a new trust region scheme that scales effectively to data distributions where V-trace becomes unstable.
+
+We provide extensive empirical validation of the proposed solution. We further show the benefits of this setup by demonstrating state-of-the-art data efficiency on Atari among agents trained up until 200M environment frames.
+
+# 1 INTRODUCTION
+
+Value-based and actor-critic policy gradient methods are the two leading techniques of constructing general and scalable reinforcement learning agents (Sutton et al., 2018). Both have been combined with non-linear function approximation (Tesauro, 1995; Williams, 1992), and have achieved remarkable successes on multiple challenging domains; yet, these algorithms still require large amounts of data to determine good policies for any new environment. To improve data efficiency, experience replay agents store experience in a memory buffer (replay) (Lin, 1992), and reuse it multiple times to perform reinforcement learning updates (Riedmiller, 2005). Experience replay allows to generalize prioritized sweeping (Moore & Atkeson, 1993) to the non-tabular setting (Schaul et al., 2015), and can also be used to simplify exploration by including expert (e.g., human) trajectories (Hester et al., 2017). Overall, experience replay can be very effective at reducing the number of interactions with the environment otherwise required by deep reinforcement learning algorithms (Schaul et al., 2015). Replay is often combined with the value-based Q-learning (Mnih et al., 2015), as it is an off-policy algorithm by construction, and can perform well even if the sampling distribution from replay is not aligned with the latest agent’s policy. Combining experience replay with actor-critic algorithms can be harder due to their on-policy nature. Hence, most established actor-critic algorithms with replay such as (Wang et al., 2017; Gruslys et al., 2018; Haarnoja et al., 2018) employ and maintain Q-functions to learn from the replayed off-policy experience.
+
+In this paper, we demonstrate that off-policy actor-critic learning with experience replay can be achieved without surrogate Q-function approximators using V-trace by employing the following approaches: a) off-policy replay experience needs to be mixed with a proportion of on-policy experience. We show experimentally (Figure 2) and theoretically that the V-trace policy gradient is otherwise not guaranteed to converge to a locally optimal solution. b) a trust region scheme (Conn et al., 2000; Schulman et al., 2015; 2017) can mitigate bias and enable efficient learning in a strongly off-policy regime, where distinct agents share experience through a commonly shared replay module. Sharing experience permits the agents to benefit from parallel exploration (Kretchmar, 2002) (Figures 1 and 3).
+
+Our paper is structured as follows: In Section 2 we revisit pure importance sampling for actor-critic agents (Degris et al., 2012) and V-trace, which is notable for allowing to trade off bias and variance in its estimates. We recall that variance reduction is necessary (Figure 4 left) but is biased in V-trace. We derive proposition 2 stating that off-policy V-trace is not guaranteed to converge to a locally optimal solution – not even in an idealized scenario when provided with the optimal value function. Through theoretical analysis (Section 3) and experimental validation (Figure 2) we determine that mixing on-policy experience into experience replay alleviates the problem. Furthermore we propose a trust region scheme (Conn et al., 2000; Schulman et al., 2015; 2017) in Section 4 that enables efficient learning even in a strongly off-policy regime, where distinct agents share the experience replay module and learn from each others experience. We define the trust region in policy space and prove that the resulting estimator is correct (i.e. estimates an improved return).
+
+As a result, we present state-of-the-art data efficiency in Section 5 in terms of median human normalized performance across 57 Atari games (Bellemare et al., 2013), as well as improved learning efficiency on DMLab30 (Beattie et al., 2016) (Table 1).
+
+![](images/b3007e7cff52f28d5dcd2580f4b9f1ecf0d2295fbe77f8bc4a0c396fd1a51ce9.jpg)  
+Figure 1: Sharing experience between agents leads to more efficient hyper-parameter sweeps on 57 Atari games. Prior art results are presented as horizontal lines (with scores cited from Gruslys et al. (2018), Hessel et al. (2017) and Mnih et al. (2013)). Note that the only previous agent “R2D2” that achieved a score beyond $4 0 0 \%$ required more than 3,000 million environment steps (see Kapturowski et al. (2019), page 14, Figure 9). We present the pointwise best agent from hyper-parameter sweeps with and without experience replay (shared and not shared). Each sweep contains 9 agents with different learning rate and entropy cost combinations. Replay experiment were repeated twice and ran for 50M steps. To report scores at 200M we ran the baseline and one shared experience replay agent for 200M steps.
+
+Table 1: Comparison of state-of-the-art agents on 57 Atari games trained up until 200M environment steps (per game) and DMLab-30 trained until 10B steps (multi-task; all games combined). The first two rows are quoted from Xu et al. (2018) and Hessel et al. (2019), the third is our implementation of a pixel control agent from Hessel et al. (2019) and the last two rows are our proposed LASER (LArge Scale Experience Replay) agent. All agents use hyper-parameter sweeps expect for the marked.   
+
+<table><tr><td colspan="2">Atari Median</td><td></td><td>DMLab-30 MedianDMLab-30 Mean-Capped</td></tr><tr><td>IMPALA Meta-Gradient (no sweep)</td><td>287.6% at 200M</td><td></td><td>-</td></tr><tr><td>PopArt-IMPALA</td><td></td><td></td><td>73.5%</td></tr><tr><td>PopArt-IMPALA+PixelControl</td><td>=</td><td>85.5%</td><td>77.6%</td></tr><tr><td>LASER: Experience Replay (no sweep)</td><td>431% at 200M</td><td></td><td></td></tr><tr><td>LASER: Experience Replay</td><td>(233% at 50M)</td><td>95.4%</td><td>79.6%</td></tr><tr><td>LASER: Shared Experience Replay</td><td>(370% at 50M), 448% at200M</td><td>97.2%</td><td>81.7%</td></tr></table>
+
+# 2 THE ISSUE WITH IMPORTANCE SAMPLING: BIAS AND VARIANCE IN V-TRACE
+
+V-trace importance sampling is a popular off-policy correction for actor-critic agents (Espeholt et al., 2018). In this section we revisit how V-trace controls the (potentially infinite) variance that arises from naive importance sampling. We note that this comes at the cost of a biased estimate (see Proposition 1) and creates a failure mode (see Proposition 2) which makes the policy gradient biased. We discuss our solutions for said issues in Section 4.
+
+![](images/1a4a5bd49675dcb34c6ec97a071ce9916c6a5e1cd0c03fd7ca411195fadc711e.jpg)  
+Figure 2: Left: Learning entirely off-policy from experience replay fails, while combining on-policy data with experience replay leads to improved data efficiency: We present sweeps on DMLab-30 with experience replays of 10M capacity. A ratio of $8 7 . 5 \%$ implies that there are 7 replayed transitions in the batch for each online transition. Furthermore we consider an agent identical to “LASER $8 7 . 5 \%$ replay” which however draws all samples from replay. Its batch thus does not contain any online data and we observe a significant performance decrease (see Proposition 2 and 3). The shading represents the point-wise best and worst replica among 3 repetitions. The solid line is the mean. Right: The effect of capacity in experience replay with $\mathrm { \bar { 8 7 . 5 \% } }$ replay data per batch on sweeps on DMLab-30. Data-efficiency improves with larger capacity.
+
+![](images/54fbfdc3d2df252cf6e6f51c57fe74643b9658d64778eee642744ad1fdacddb7.jpg)  
+Figure 3: Left: Naively sharing experience between distinct agents in a hyper-parameter sweep fails (green) and is worse than the no-replay baseline (blue). The proposed trust region estimator mitigates the issue (red). Right: Combining population based training with trust region estimation improves performance further. All replay experiments use a capacity of 10 million observations and ${ \bar { 8 } } 7 . 5 \%$ replay data per batch.
+
+# 2.1 REINFORCEMENT LEARNING
+
+We follow the notation of Sutton et al. (2018) where an agent interacts with its environment, to collect rewards. On each discrete time-step $t$ , the agent selects an action $a _ { t }$ ; it receives in return a reward $r _ { t }$ and an observation $o _ { t + 1 }$ , encoding a partial view of the environment’s state $s _ { t + 1 }$ . In the fully observable case, the RL problem is formalized as a Markov Decision Process (Bellman, 1957): a tuple $( \boldsymbol { S } , \boldsymbol { A } , \boldsymbol { p } , \gamma )$ , where $s , A$ denotes finite sets of states and actions, $p$ models rewards and state transitions (so that $r _ { t } , s _ { t + 1 } \sim p ( s _ { t } , a _ { t } ) )$ , and $\gamma$ is a fixed discount factor. A policy is a mapping $\pi ( a | s )$ from states to action probabilities. The agent seeks an optimal policy $\pi ^ { * }$ that maximizes the value, defined as the expectation of the cumulative discounted returns $\begin{array} { r } { \dot { \boldsymbol { G } } _ { t } = \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } \boldsymbol { r } _ { t + k } } \end{array}$ .
+
+Off-policy learning is the problem of finding, or evaluating, a policy $\pi$ from data generated by a different policy $\mu$ . This arises in several settings. Experience replay (Lin, 1992) mixes data from multiple iterations of policy improvement. In large-scale RL, decoupling acting from learning (Nair et al., 2015; Horgan et al., 2018; Espeholt et al., 2018) causes the experience to lag behind the latest agent policy. Finally, it is often useful to learn multiple general value functions (Sutton et al., 2011; Mankowitz et al., 2018; Lample & Chaplot, 2016; Mirowski et al., 2017; Jaderberg et al., 2017b) or options (Sutton et al., 1999; Bacon et al., 2017) from a single stream of experience.
+
+# 2.2 NAIVE IMPORTANCE SAMPLING
+
+On-policy n-step bootstraps give more accurate value estimates in expectation with larger $n$ (Sutton et al., 2018). They are used in many reinforcement learning agents (Mnih et al., 2016; Schulman et al., 2017; Hessel et al., 2017). Unfortunately $n$ must be chosen suitably as the estimates variance increases with $n$ too.
+
+It is desirable to obtain benefits akin to n-step returns in the off-policy case. To this end multi-step importance sampling (Kahn, 1955) can be used. This however adds another source of (potentially infinite (Sutton et al., 2018)) variance to the estimate.
+
+Importance sampling can estimate the expected return $V ^ { \pi }$ from trajectories sampled from $\mu \neq \pi$ as long as $\mu$ is non-zero whereever $\pi$ is. We employ a previously estimated value function $V$ as a bootstrap to estimate expected returns. Following Degris et al. (2012), a multi-step formulation of the expected return is
+
+$$
+V ^ { \pi } ( s _ { t } ) = \mathbf { E } _ { \mu } \left[ V ( s _ { t } ) + \sum _ { k = 0 } ^ { K - 1 } \gamma ^ { k } \Big ( \prod _ { i = 0 } ^ { k } \frac { \pi _ { t + i } } { \mu _ { t + i } } \Big ) \delta _ { t + k } V \right]
+$$
+
+where $\mathbf { E } _ { \mu }$ denotes the expectation under policy $\mu$ up to an episode termination, $\delta _ { t } V ~ = ~ r _ { t } +$ $\gamma V ( s _ { t + 1 } ) - V ( s _ { t } )$ is the temporal difference error in consecutive states $s _ { t + 1 }$ , $s _ { t }$ , and $\pi _ { t } = \pi _ { t } ( a _ { t } | s _ { t } )$ . Importance sampling estimates can have high variance. Tree Backup (Precup et al., 2000), and $\mathbf { Q } ( \lambda )$ (Sutton et al., 2014) address this, but reduce the number of steps before bootstrapping even when this is undesirable (as in the on-policy case). RETRACE (Munos et al., 2016) makes use of full returns in the on-policy case, but it introduces a zero-mean random variable at each step, adding variance to empirical estimates in both on- and off-policy cases.
+
+# 2.3 BIAS-VARIANCE ANALYSIS & FAILURE MODE OF $\mathrm { v . }$ -TRACE IMPORTANCE SAMPLING
+
+V-trace (Espeholt et al., 2018) reduces the variance of importance sampling by trading off variance for a biased estimate of the return – resulting in a failure mode (see Proposition 2). It uses clipped importance sampling ratios to approximate $V ^ { \pi }$ by $\begin{array} { r } { V ^ { \tilde { \pi } } ( s _ { t } ) = V ( s _ { t } ) + \sum _ { k = 0 } ^ { K - 1 } \gamma ^ { k } \Big ( \prod _ { i = 0 } ^ { k - 1 } c _ { i } \Big ) \rho _ { t } \delta _ { t + k } V } \end{array}$ where $V$ is a learned state value estimate used to bootstrap, and $\rho _ { t } = \operatorname* { m i n } \left[ \pi _ { t } / \dot { \mu } _ { t } , \bar { \rho } \right]$ , $c _ { t } \dot { = } \operatorname* { m i n } \left[ \pi _ { t } / \mu _ { t } , \bar { c } \right]$ are the clipped importance ratios. Note that, differently from RETRACE, V-trace fully recovers the Monte Carlo return when on policy. It similarly reweights the policy gradient as:
+
+$$
+\nabla V ^ { \tilde { \pi } } ( s _ { t } ) \stackrel { \mathrm { d e f } } { = } \mathbf { E } _ { \mu } \left[ \rho _ { t } \nabla ( \log \pi _ { t } ) ( r _ { t } + \gamma V ^ { \tilde { \pi } } ( s _ { t + 1 } ) ) \right]
+$$
+
+Note that $\nabla V ^ { \tilde { \pi } } ( s _ { t } )$ recovers the naively importance sampled policy gradient for $\bar { \rho }  \infty$ . In the literature, it is common to subtract a baseline from the action-value estimate $r _ { t } + \gamma V ^ { \tilde { \pi } } ( s _ { t + 1 } )$ to reduce variance (Williams, 1992), omitted here for simplicity. The constants $\bar { \rho } \ge \bar { c } \ge 1$ (typically chosen $\bar { \rho } = \bar { c } = 1$ ) define the level of clipping, and improve stability by ensuring a bounded variance. For any given $\bar { \rho }$ , the bias introduced by V-trace in the value and policy gradient estimates increases with the difference between $\pi$ and $\mu$ . We analyze this in the following propositions.
+
+Proposition 1. The V-trace value estimate $V ^ { \tilde { \pi } }$ is biased: It does not match the expected return of $\pi$ but the return of a related implied policy $\tilde { \pi }$ defined by equation 3 that depends on the behaviour
+
+policy µ:
+
+$$
+\tilde { \pi } _ { \mu } ( a | x ) = \frac { \operatorname* { m i n } \left[ \bar { \rho } \mu ( a | x ) , \pi ( a | x ) \right] } { \sum _ { b \in A } \operatorname* { m i n } \left[ \bar { \rho } \mu ( b | x ) , \pi ( b | x ) \right] }
+$$
+
+Proof. See Espeholt et al. (2018).
+
+Note that the biased policy $\tilde { \pi } _ { \mu }$ can be very different from $\pi$ . Hence the V-trace value estimate $V ^ { \tilde { \pi } }$ may be very different from $V ^ { \pi }$ as well. As an illustrative example, consider two policies over a set of two actions, e.g. “left” and “right” represented as a tuple of probabilities. Let us investigate $\mu = ( \phi , 1 - \phi )$ and $\bar { \pi } = ( 1 - \phi , \phi )$ defined for any suitably small $\phi \leq 1$ . Observe that $\pi$ and $\mu$ share no trajectories (state-action sequences) in the limit as $\phi  0$ and they get more focused on one action. A practical example of this could be two policies, one almost always taking a left turn and one always taking the right. Given sufficient data of either policy it is possible to estimate the value of the other e.g. with naive importance sampling. However observe that V-trace with $\bar { \rho } = 1$ will always estimate a biased value - even given infinite data. Observe that $\sin \left[ \mu ( a | x ) , \pi ( a | x ) \right] = \sin \left[ \phi , 1 - \phi \right]$ for both actions. Thus $\tilde { \pi } _ { \mu }$ is uniform rather than resembling $\pi$ the policy. The $\mathrm { V } .$ -trace estimate $V ^ { \tilde { \pi } }$ would thus compute the average value of "left" and "right" – poorly representing the true $V ^ { \pi }$ .
+
+Proposition 2. The V-trace policy gradient is biased: given the the optimal value function $V ^ { * }$ the $V .$ -trace policy gradient does not converge to a locally optimal $\pi ^ { * }$ for all off-policy behaviour distributions $\mu$ .
+
+Proof. See Appendix C.
+
+# 3 MIXING ON- AND OFF-POLICY EXPERIENCE
+
+In Proposition 2 we presented a failure mode in $\mathrm { v } .$ -trace where the variance reduction biases the value estimate and policy gradient. V-trace computes biased $\mathbf { Q }$ -estimates $Q ^ { \omega } \neq Q$ resulting in a wrong local policy gradient: $\nabla { \bf E } _ { \pi ( a | s ) } \left[ Q ^ { \omega } ( s , a ) \right] \neq \nabla { \bf E } _ { \pi ( a | s ) } \left[ Q ( s , a ) \right] .$ . In equation 10 we show that $Q ^ { \omega } ( s , a ) = Q ( s , a ) \omega ( s , a )$ where $\begin{array} { r } { \omega ( s , a ) = \mathrm { m i n } \left[ 1 , \bar { \rho } \frac { \mu ( a | s ) } { \pi ( a | s ) } \right] \leq 1 } \end{array}$ .
+
+The question of how biased the resulting policy will be depends on whether the distortion changes the argmax of the Q-function. Little distortions that do not change the argmax will result in the same local fixpoint of the policy improvement. The policy will continue to select the optimal action and it will not be biased at this state. The policy will however be biased if the Q-function is distorted too much. For example consider a $\omega ( s , a )$ that swaps the argmax for the 2nd largest value, the regret will then be the difference between the maximum and the 2nd largest value. Intuitively speaking the more distorted the $Q ^ { \omega }$ , the larger will be the regret compared to the optimal policy.
+
+More precisely, the regret of learning a policy that maximizes the distorted $Q ^ { \omega }$ at state $s$ is:
+
+$$
+R ( s ) = Q ( s , a ^ { * } ) - Q ( s , a _ { \mathrm { a c t u a l } } ) = \operatorname* { m a x } _ { b } Q ( s , b ) - Q ( s , a _ { \mathrm { a c t u a l } } )
+$$
+
+where $a ^ { * } ~ = ~ \operatorname { a r g m a x } _ { b } ( Q , b )$ is the optimal action according to the real $Q$ and $\begin{array} { r l } { a _ { \mathrm { a c t u a l } } } & { { } = } \end{array}$ a $\mathrm { r g m a x } [ Q ^ { \omega } ( s , a ) ] = \mathrm { a r g m a x } [ Q ( s , a ) \omega ( s , a ) ]$ , is the optimal action according to the distorted $Q ^ { \omega }$ . For generality, we denote $A ^ { * }$ as the set of best actions - covering the case with multiple with identical optimal Q-values.
+
+Proposition 3 provides a mitigation: Clearly the V-trace policy gradient will converge to the same solution as the true on-policy gradient if the argmax of the Q-function is preserved at all states in a tabular setting. We show that this can be achieved by mixing a sufficient proportion $\alpha$ of on-policy experience into the computation.
+
+We show in equation 13 in the Appendix that choosing $\alpha$ such that
+
+$$
+\frac { \alpha } { 1 - \alpha } > _ { b \notin A ^ { * } } \left[ \frac { Q ^ { \omega } ( s , b ) - Q ^ { \omega } ( s , a ^ { * } ) } { Q ( s , a ^ { * } ) - Q ( s , b ) } \right] \frac { d ^ { \mu } ( s ) } { d ^ { \pi } ( s ) } \mathrm { f o r } Q ^ { \omega } ( s , a ) = Q ( s , a ) \omega ( s , a )
+$$
+
+will result in a policy that correctly chooses the best action at state $s$ . Note that $\frac { \alpha } { 1 - \alpha } \to \infty$ as $\alpha  1$
+
+Intuitively: the larger the action value gap of the real Q-function $Q ( s , a ^ { * } ) - Q ( s , b )$ the lower the right hand side and the less on-policy data is required. If $\begin{array} { r } { \operatorname* { m a x } _ { b } [ ( Q ( s , b ) \omega ( s , b ) - Q ( s , a ^ { * } ) \omega ( s , a ^ { * } ) ] } \end{array}$ is negative, then $\alpha$ may be as small as zero and we enabling even pure off-policy learning. Finally note that the right hand side decreases due to $d ^ { \mu } ( s ) / d ^ { \pi } ( s )$ if $\pi$ visits the state $s$ more often than $\mu$ .
+
+All of those conditions can be computed and checked if an accurate Q-function and state distribution is accessible. How to use imperfect Q-function estimates to adaptively choose such an $\alpha$ remain a question for future research.
+
+We provide experimental evidence for these results with function approximators in the 3-dimensional simulated environment DMLab-30 with various $\alpha \ge 1 / 8$ in Section 5.3 and Figure 2. We observe that $\alpha = 1 / 8$ is sufficient to facilitate stable learning. Furthermore it results in better data-efficiency than pure on-policy learning as it utilizes off-policy replay experience.
+
+Proposition 3. Mixing on-policy data into the V-trace policy gradient with the ratio α reduces the bias by providing a regularization to the implied state-action values. In the general function approximation case it changes the off-policy $V .$ -trace policy gradient from $\begin{array} { r l } { \sum _ { s } d ^ { \mu } ( \bar { s } ) \mathbf { E } _ { \pi } \left[ ( \dot { Q } ( s , a ) \nabla \log \pi ( a | s ) \right] } \end{array}$ to $\begin{array} { r } { \sum _ { s } \mathbf { E } _ { \pi } \left[ Q ^ { \alpha } ( s , a ) \nabla \log \pi ( a | s ) \right] } \end{array}$ where $Q ^ { \alpha } = Q d ^ { \pi } ( s ) \alpha + Q ^ { \overline { { { \omega } } } } \bar { d } ^ { \mu } ( s ) ( 1 - \alpha )$ is a regularized stateaction estimate and $d ^ { \pi }$ , $d ^ { \mu }$ are the state distributions for $\pi$ and $\mu$ . Note that there exists $\alpha \leq 1$ such that $Q ^ { \alpha }$ has the same argmax (i.e. best action) as $Q$ .
+
+# Proof. See Appendix C.
+
+Mixing online data with replay data has also been argued for by Zhang & Sutton (2017), as a heuristic way of reducing the sensitivity of reinforcement learning algorithms to the size of the replay memory. Proposition 3 grounds this in the theoretical properties of V-trace.
+
+# 4 TRUST REGION SCHEME FOR OFF-POLICY V-TRACE
+
+To mitigate the bias and variance problem of V-trace and importance sampling we propose a trust region scheme that adaptively selects only suitable behaviour distributions when estimating the state-value of $\pi$ . To this end we introduce a behaviour relevance function that classifies behaviour as relevant. We then define a trust-region estimator that computes expectations (such as expected returns, or the policy gradient) only on relevant transitions. In proposition 4 and 5 we show that this trust region estimator indeed computes new state-value estimates that improve over the current value function. While our analysis and proof is general we propose a suitable behaviour relevance function in section 4.3 that employs the Kullback Leibler divergence between target policy $\pi$ and implied policy $\tilde { \pi } _ { \mu } \colon \mathrm { K L } \left( \pi ( \cdot | s ) \bar { | } | \tilde { \pi } _ { \mu } ( \cdot | s ) \right)$ . We provide experimental validation in Figure 3.
+
+# 4.1 BEHAVIOUR RELEVANCE FUNCTIONS
+
+In off-policy learning we often consider a family of behaviour policies either indexed by training iteration $t$ : $M _ { T } = \{ \mu _ { t } | t < T \}$ for experience replay, or by a different agent $k$ : $M _ { K } = \{ \bar { \mu _ { k } } | k \in K \bar  \}$ when training multiple agents. In the classic experience replay case we then sample a time $t$ and locate the transition $\tau$ that was generated earlier via $\mu _ { t }$ . This extends naturally to the multiple agent case where we sample an agent index $k$ and then obtain a transition for such agent or tuples of $( k , t )$ . Without loss of generality we simplify this notation and index sampled behaviour policies by a random variable $z \sim Z$ that represents the selection process. While online reinforcement learning algorithms process transitions $\tau \sim \pi$ , off-policy algorithms process $\tau \sim \mu _ { z }$ for $z \sim Z$ . In this notation, given equation (1) and a bootstrap $V$ , the expectation of importance sampled off-policy returns at state $s _ { t }$ is described by:
+
+$$
+V _ { \mathrm { m i x } } ^ { \pi } ( s _ { t } ) = \mathbf { E } _ { z } \Big [ \mathbf { E } _ { \mu _ { z } | z } \big [ G ^ { \pi , \mu _ { z } } ( s _ { t } ) \big ] \Big ]
+$$
+
+where $\begin{array} { r } { G ^ { \pi , \mu } ( s _ { t } ) = V ( s _ { t } ) + \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } \Big ( \prod _ { i = 0 } ^ { k } \frac { \pi _ { t + i } } { \mu _ { t + i } } \Big ) \delta _ { t + k } V } \end{array}$ is a single importance sampled return.   
+Note that the on-policy return $\begin{array} { r } { G ^ { \pi , \pi } ( s _ { t } ) = V ( s _ { t } ) + \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } r _ { t + k } } \end{array}$ .
+
+Above $\mathbf { E } _ { \mu _ { z } | z }$ represents the expectation of sampling from a given $\mu _ { z }$ . The conditioning on $z$ is a notational reminder that this expectation does not sample $z$ or $\mu _ { z }$ but experience from $\mu _ { z }$ . For any
+
+sampled $z$ we obtain a $\mu _ { z }$ and observe that the inner expectation wrt. experience of $\mu _ { z }$ in equation (4) recovers the expected on-policy return in expectation:
+
+$$
+\begin{array} { r l } { \mathbf { E } _ { \mu _ { z } | z } \left[ G ^ { \pi , \mu _ { z } } ( s _ { t } ) \right] = \mathbf { E } _ { \mu _ { z } | z } \left[ V ( s _ { t } ) + \displaystyle \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } \Big ( \displaystyle \prod _ { i = 0 } ^ { k } \frac { \pi _ { t + i } } { \mu _ { z , t + i } } \Big ) \delta _ { t + k } V \right] } & { } \\ { = \mathbf { E } _ { \pi } \left[ V ( s _ { t } ) + \displaystyle \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } \Big ( \displaystyle \prod _ { i = 0 } ^ { k } \frac { \mu _ { z , t + i } } { \mu _ { z , t + i } } \Big ) \delta _ { t + k } V \right] } & { } \\ { = \mathbf { E } _ { \pi } \left[ V ( s _ { t } ) + \displaystyle \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } r _ { t + k } \right] = \mathbf { E } _ { \pi } \left[ G ^ { \pi , \pi } ( s _ { t } ) \right] = V ^ { \pi } ( s _ { t } ) } \end{array}
+$$
+
+Thus $V _ { \mathrm { m i x } } ^ { \pi } ( s _ { t } ) = \mathbf { E } _ { z } \left[ \mathbf { E } _ { \pi } \left[ G ^ { \pi , \pi } ( s _ { t } ) \right] \right] = \mathbf { E } _ { \pi } \left[ G ^ { \pi , \pi } ( s _ { t } ) \right] = V ^ { \pi } ( s _ { t } )$ . This holds provided that $\mu _ { z }$ is non-zero wherever $\pi$ is. This fairly standard assumption leads us straight to the core of the problem: it may be that some behaviours $\mu _ { z }$ are ill-suited for estimating the inner expectation. However, standard importance sampling applied to very off-policy experience divides by small $\mu$ resulting in high or even infinite variance. Similarly, $\mathrm { v } .$ -trace attempts to compute an estimate of the return following $\pi$ resulting in limited variance at the cost of a biased estimate in turn.
+
+The key idea of our proposed solution is to compute the return estimate for $\pi$ at each state only from a subset of suitable behaviours $\mu _ { z }$ :
+
+$$
+M _ { \beta , \pi } ( s ) = \{ \mu _ { z } | z \in Z { \mathrm { ~ a n d ~ } } \beta ( \pi , \mu , s ) < b \}
+$$
+
+as determined by a behaviour relevance function $\beta ( \pi , \mu , s ) : ( M _ { Z } , M _ { Z } , S )  \mathbb { R }$ and a threshold $b$ . The behaviour relevance function decides if experience from a behaviour is suitable to compute an expected return for $\pi$ . It can be chosen to control properties of $V _ { \mathrm { m i x } } ^ { \pi }$ by restricting the expectation on subsets of $Z$ . In particular it can be used to control the variance of an importance sampled estimator: Observe that the inner expectation $\begin{array} { r } { \mathsf { E } _ { \mu _ { z } } \left[ G ^ { \pi , \mu } ( s _ { t } ) \big | z \right] } \end{array}$ in equation (4) already matches the expected return $V ^ { \pi }$ . Thus we can condition the expectation on arbitrary subsets of $Z$ without changing the expected value of $V _ { \mathrm { m i x } } ^ { \pi }$ . This allows us to reject high variance $G ^ { \pi , \mu }$ without introducing a bias in $V _ { \mathrm { m i x } } ^ { \pi }$ . The same technique can be applied to $\mathrm { V } .$ -trace where we can reject return estimates with high bias.
+
+# 4.2 DERIVATION OF TRUST REGION ESTIMATORS
+
+Using a behaviour relevance function $\beta ( s )$ we can define a trust region estimator for regular importance sampling (IS) and $\mathrm { V } .$ -trace and show their correctness.
+
+We define the trust region estimator as the conditional expectation
+
+$$
+V _ { \mathrm { t r u s t e d } } ^ { \pi } ( s _ { t } ) = \mathbf { E } _ { z } \Big [ \mathbf { E } _ { \mu _ { z } \vert z } \big [ G ^ { \pi , \mu _ { z } , \beta } ( s _ { t } ) \big ] \Big \vert \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) \Big ]
+$$
+
+with $\lambda$ -returns $G$ , chosen as $G _ { \mathrm { I S } }$ for importance sampling and $G _ { \mathrm { V t r a c e } }$ for $\mathrm { V } .$ -trace:
+
+$$
+G _ { \mathrm { I S } } ^ { \pi , \mu _ { z } } ( s _ { t } ) = V ( s _ { t } ) + \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } \Big ( \prod _ { i = 0 } ^ { k } \lambda _ { \pi , \mu _ { z } } ( s _ { t + i } ) \frac { \pi _ { t + i } } { \mu _ { z , t + i } } \Big ) \delta _ { t + k } V
+$$
+
+$$
+G _ { \mathrm { V t r a c e } } ^ { \pi , \mu _ { z } } ( s _ { t } ) = V ( s _ { t } ) + \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } \Big ( \prod _ { i = 0 } ^ { k - 1 } \lambda _ { \pi , \mu _ { z } } ( s _ { t + i } ) c _ { z , t + i } \Big ) \lambda _ { \pi , \mu _ { z } } ( s _ { t + k } ) \rho _ { z , t + k } \delta _ { t + k } V
+$$
+
+where $\lambda _ { \pi , \mu } ( s _ { t } )$ is designed to constraint Monte-Carlo bootstraps to relevant behaviour: $\lambda _ { \pi , \mu } ( s _ { t } ) =$ $\mathbb { 1 } _ { \beta ( \pi , \mu , s _ { t } ) < b }$ and s bo $\begin{array} { r } { \rho _ { z , t + k } = \operatorname* { m i n } \left[ \frac { \pi _ { t + i } } { \mu _ { z , t + i } } , \bar { \rho } \right] } \end{array}$ and a mu $c _ { z , t + k }$ are behaviour dependent clipped importancep return estimators with adaptive length. Note $G _ { \mathrm { I S } } ^ { \pi , \mu _ { z } }$ $G _ { \mathrm { V t r a c e } } ^ { \bar { \pi } , \mu _ { z } }$   
+IS that only estimators with length $\geq 1$ e  are used in $V _ { \mathrm { t r u s t e d } } ^ { \pi }$ . Due to Minkowski’s inequality the trust region estimator thus shows at least the same contraction as a 1-step bootstrap, but can be faster due to its adaptive nature:   
+Proposition 4. Let that they all have th $G _ { \mathrm { I S } } ^ { \pi , \mu _ { z } }$ be a set  fix point mportance sampling estimaand contract with at least rs as defined in equation 7. Note. Then the contraction properties $V ^ { \pi }$ $\gamma$   
+carry over to $V _ { \mathrm { t r u s t e d } } ^ { \pi }$ . $I n$ particular $| V _ { \mathrm { t r u s t e d } } ^ { \pi } - V ^ { \pi } | _ { \infty } \leq \gamma | V - V ^ { \pi } | _ { \infty }$ .
+
+Proof. See Appendix C.
+
+Proposition 5. Let $G _ { \mathrm { V t r a c e } } ^ { \pi , \mu _ { z } }$ be a set of $V .$ -trace estimators (see equation 8) with corresponding fixed (see equation 3) to which they contract at a speed of an algorithm and behaviour specific $\eta _ { z }$ . Then $V _ { \mathrm { t r u s t e d } } ^ { \pi }$ moves towards $V ^ { \beta } = \mathbf { E } _ { z | \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } \left[ \bar { V ^ { z } } \right]$ shrinking the distance as follows $\begin{array} { r } { \big | V _ { \mathrm { t r u s t e d } } ^ { \pi } - V ^ { \beta } \big | _ { \infty } < \operatorname* { m a x } _ { \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } | \eta _ { z } ( V - V ^ { z } ) | _ { \infty } \leq \eta _ { \operatorname* { m a x } } \operatorname* { m a x } _ { \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } | ( V - V ^ { z } ) | _ { \infty } } \end{array}$ with $\eta _ { \mathrm { m a x } } = \operatorname* { m a x } _ { \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } \eta _ { z }$ .
+
+# Proof. See Appendix C.
+
+Note how the choice of π $\beta$ and t s $M _ { \beta , \pi }$ enables us discard ill-suited $G _ { \mathrm { V t r a c e } } ^ { \pi , \mu _ { z } }$ from the estimation of . Recall that $\mathrm { v } .$ -trace fixed points $V _ { z }$ are biased. Thus $\beta$ allows us to selectively create the V-trace target $V ^ { \beta } = \mathbf { E } _ { z | \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } \left[ V ^ { z } \right]$ and control its bias and the shrinkage $\begin{array} { r } { \operatorname* { m a x } _ { \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } | \eta _ { z } ( V ( s ) - V ^ { z } ( s ) ) | _ { \infty } } \end{array}$ (see Proposition 5). Similarly it can control cases where we can not use the exact importance sampled estimator. The same approach based on nested expectations can be applied to the expectation of the policy gradient estimate and allows to control the bias and greediness (see Proposition 2) there as well.
+
+# 4.3 IMPLEMENTATION DETAILS
+
+In Proposition 5 we have seen that the quality of the trust region V-trace return estimator depends on $\beta$ . A suitable choice of $\beta$ can move the return estimate $V ^ { \beta }$ closer to $V ^ { \pi }$ and improve the shrinkage by reducing $\begin{array} { r } { \operatorname* { m a x } _ { \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } \left| \eta _ { z } ( V ( s ) - V ^ { z } ( s ) ) \right| _ { \infty } } \end{array}$ . Hence, we employ a behaviour relevance function $\beta _ { \mathrm { K L } }$ that rejects high bias transitions by estimating the Kulback-Leibler divergence between the target policy $\pi$ and the implied policy $\tilde { \pi } _ { \mu _ { z } }$ for a sampled behaviour $\mu _ { z }$ . Recall from Proposition 1 that $\tilde { \pi } _ { \mu _ { z } }$ determines the fixed point of the $\mathrm { V } .$ -trace estimator for behaviour $\mu _ { z }$ and thus determines the bias in $V ^ { z }$ .
+
+$$
+\beta _ { \mathrm { K L } } ( \pi , \mu , s ) = \mathrm { K L } \left( \pi ( \cdot | s ) | | \tilde { \pi } _ { \mu } ( \cdot | s ) \right)
+$$
+
+Note that the behaviour probabilities $\mu _ { z }$ can be evaluated and saved to the replay when the agent executes the behaviour, similarly the target policy $\pi$ is represented by the agents neural network. Using both and equation 3, $\tilde { \pi } _ { \mu }$ can be computed. For large or infinite action spaces a Monte Carlo estimate of the KL divergence can be computed.
+
+It is possible to define separate behaviour relevance functions for the policy and value estimate. For simplicity we reject transitions entirely for all estimates and do not consider rejected transitions for the policy gradient and value gradient updates or auxiliary tasks. As described above we stop the Monte-Carlo bootstraps once they reach undesirable state-behaviour pairs. Note that this censoring procedure is computed from state dependent $\beta ( \pi , \mu , s )$ and ensures that the choice of bootstrapping does not depend on the sampled actions. Note that rejection by an action-based criteria such as small $\pi ( { a } | { s } ) / \mu ( { a } | { s } )$ would introduce an additional bias which we avoid by choosing $\beta _ { \mathrm { K L } }$ .
+
+# 5 EXPERIMENTS
+
+We present experiments to support the following claims:
+
+• Section 5.2: Uniform experience replay obtains comparable results as prioritized experience replay, while being simpler to implement and tune.   
+• Section 5.3: Using fresh experience before inserting it in experience replay is better than learning purely off-policy from experience replay – in line with Proposition 3.   
+• Section 5.4: Sharing experience without trust region performs poorly as suggested by Proposition 2. Off-Policy Trust-Region V-trace solves this issue.   
+• Section 5.5: Sharing experience can take advantage of parallel exploration and obtains state-of-the-art performance on Atari games, while also saving memory through sharing a single experience replay.
+
+# 5.1 EXPERIMENTAL SETUP & METHODOLOGY
+
+We use the V-trace distributed reinforcement learning agent (Espeholt et al., 2018) as our baseline. In our experiments we consider two experimental platforms: Atari and DeepMind Lab. On Atari we consider the common single task training regime, where a different agent is trained, from scratch, on each of the tasks. Following Xu et al. (2018) we use a discount of 0.995. Motivated by recent work by Kaiser et al. (2019), we use the IMPALA deep network and increased the number of channels $4 \times$ . We use $9 6 \%$ replay data per batch. Differently from Espeholt et al. (2018), we do not use gradient clipping by norm (Pascanu et al., 2012). Updates are computed on mini-batches of 32 (regular) and 128 (replay) trajectories, each corresponding to 19 steps in the environment. In the context of DeepMind Lab, we consider the multi-task suite DMLab-30 (Espeholt et al., 2018), as the visuals and the dynamics are more consistent across tasks. Furthermore the multi-task regime is particularly suitable for the investigation of strongly off-policy data distributions arising from sharing the replay across agents, as concurrently learning agents can easily be stuck in different policy plateaus, generating substantially different data (Schaul et al., 2019). As in Espeholt et al. (2018), in the multi-task setting each agent trains simultaneously on a uniform mixture of all tasks rather than individually on each game. The score of an agent is thus the median across all 30 tasks. Following Hessel et al. (2019), we augment our agent with multi-task Pop-Art normalization and PixelControl. We use a PreCo LSTM (Amos et al., 2018) instead of the vanilla one (Hochreiter & Schmidhuber, 1997). Updates are computed on mini-batches of multiple trajectories chosen as above, each corresponding to 79 steps in the environment. In early experiments we found that computing the entropy cost only on the online data provided slightly better results, hence we have done so throughout our experiments.
+
+In all our experiments, experience sampled from memory is mixed with online data within each minibatch – following Proposition 3. Episodes are removed in a first in first out order, so that replay always holds the most recent experience. Unless explicitly stated otherwise we consider hyper-parameter sweeps, some of which share experience via replay. In this setting multiple agents start from-scratch, run concurrently at identical speed, and add their new experience into a common replay buffer. All agents will then draw uniform samples from the replay buffer. On DMLab-30 we consider both regular hyper-parameter sweeps and sweeps with population based training (PBT) (Jaderberg et al., 2017a). On DMLab-30 sweeps contain 10 agents with hyper-parameters sampled similar as Espeholt et al. (2018) but fixed RMSProp $\epsilon = 0 . 1$ . On Atari sweeps contain 9 agents with different constant learning rate and entropy cost combinations $\{ 3 \cdot 1 0 ^ { - 4 } , 6 \cdot \bar { 1 } 0 ^ { - 4 } , 1 . 2 \cdot 1 0 ^ { - 3 } \} \times \{ 5 \cdot 1 0 ^ { - 3 } , 1 \cdot 1 0 ^ { - 2 } , 2 \cdot 1 0 ^ { - 2 } \}$ (distributed by factors $\{ 1 / 2 , 1 , 2 \}$ around the initial parameters reported in Espeholt et al. (2018)). Although our focus is on efficient hyper-parameter sweeps given crude initial parameters, we also present a single-agent LASER experiment using the same tuned schedule as Espeholt et al. (2018), a $8 7 . 5 \%$ replay ratio and a 15M replay. We store the entire episodes in the replay buffer and replay each episode from the beginning, using the most recent network parameters to recompute the LSTM states along the way: this is particularly critical when sharing experience between different agents, which may have arbitrarily different state representations.
+
+# 5.2 UNIFORM AND PRIORITIZED EXPERIENCE REPLAY
+
+Prioritized experience replay has the potential to provide more efficient learning compared to uniform experience replay (Schaul et al., 2015; Horgan et al., 2018). However, it also introduces a number of new hyper-parameters and design choices: the most critical are the priority metric, how strongly to bias the sampling distribution, and how to correct for the resulting bias. Uniform replay is instead almost parameter-free, requires little tuning and can be easily shared between multiple agents. Experiments provided in Figure 4 in the appendix showed little benefit of actor critic prioritized replay on DMLab-30. Furthermore priorities are typically computed from the agent specific metrics such as the TD-error, which are ill-defined when replay is shared among multiple agents. Hence we used uniform replay for our further investigations.
+
+# 5.3 MIXING ON- AND OFF-POLICY EXPERIENCE AND REPLAY CAPACITY
+
+Figure 2 (left) shows that performance degrades significantly when online data is not present in the batch. This experimentally validates Propositions 2 and 3 that highlight difficulties of learning purely off-policy. Furthermore Figure 2 (right) shows that best results are obtained with experience replay of 10M capacity and $8 7 . 5 \%$ ratio. A ratio of $8 7 . 5 \% = 7 / 8$ corresponds to 7 replay samples for each online sample. We have considered ratios of $1 / 2 , 3 / 4$ , and $7 / 8$ and observed stable training for all of them. Observe that among those values, larger ratios are more data-efficient as they take advantage of more replayed experience per training step.
+
+5.4 SHARED EXPERIENCE REPLAY WITH OFF-POLICY TRUST REGION V-TRACE
+
+In line with proposition 2 we observe in Figure 3 (left) that hyper-parameter sweeps without trustregion are even surpassed by the baseline without experience replay. State-of-the-art results are obtained in Figure 3 (right) when experience is shared with trust-region in a PBT sweep.
+
+Observe that this indicates parallel exploration benefits and saves memory at the same time: in our sweep of 10 replay agents the difference between $1 0 \times 1 0 \mathrm { M }$ (separate replays) and 10M (shared replay) is 10-fold. This effect would be even more pronounced with larger sweeps.
+
+As discussed in section 2.3, the bias in V-trace occurs due to the clipping of importance ratios. A potential solution of reducing the bias would be to increase the $\bar { \rho }$ threshold to clip less aggressively and accept increased variance. Figure 4 in the appendix shows that this is not a solution.
+
+# 5.5 EVALUATION ON ATARI
+
+We apply our proposed agent to Atari which has been a long established suite to evaluate reinforcement algorithms (Bellemare et al., 2013). Since we focus on sample-efficient learning we present our results in comparison to prior work at 200M steps (Figure 1). Shared experience replay obtains even better performance than not shared experience replay. This confirms the efficient use of parallel exploration (Kretchmar, 2002). The fastest prior agent to reach $4 0 0 \%$ is presented by Kapturowski et al. (2019) requiring more than 3,000M steps. LASER with shared replay achieves $4 2 3 \%$ in 60M per agent. Given 200M steps it achieves $4 4 8 \%$ . We also present a single (no sweep) LASER agent that achieves $4 3 1 \%$ in 200M steps.
+
+# 6 CONCLUSION
+
+We have presented LASER – an off-policy actor-critic agent which employs a large and shared experience replay to achieve data-efficiency. By sharing experience between concurrently running experiments in a hyper-parameter sweep it is able to take advantage of parallel exploration. As a result it achieves state-of-the-art data efficiency on 57 Atari games given 200M environment steps. Furthermore it achieves competitive results on both DMLab-30 and Atari under regular, not shared experience replay conditions.
+
+To facilitate this algorithm we have proposed two approaches: a) mixing replayed experience and on-policy data and b) a trust region scheme. We have shown theoretically and demonstrated through a series of experiments that they enable learning in strongly off-policy settings, which present a challenge for conventional importance sampling schemes.
+
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+# APPENDIX A ADDITIONAL EXPERIMENTS
+
+# A.1 REDUCED CLIPPING IN V-TRACE DOES NOT ENABLE SHARED EXPERIENCE REPLAY
+
+Increasing the clipping constant $\bar { \rho }$ in $\mathrm { v } .$ -trace reduces bias in favour of increased variance. We investigate if reducing bias in this manner enables sharing experience replay between multiple agents in a hyper-parameter sweep. Figure 4 (left) shows that this is not a solution, thus motivating our trust region scheme.
+
+![](images/63def57eb91c4f28226b0d99f0027c88e061dbe26b218e20bbad36d34778d456.jpg)  
+Figure 4: Left: Increasing the V-trace clipping constant $\bar { \rho }$ does not enable shared experience replay. In fact sharing experience replay in this particular way is worse than pure online learning. This motivates the use of our proposed trust region scheme. On a side note, increased clipping thresholds resulting in worse performance verifies the importance of variance reduction through clipping. Right: Median human normalized performance across 30 tasks for the best agent in a sweep, averaged across 2 replicas. All replay experiments use $5 0 \%$ replay ratio and a capacity of 3 million observations. We investigate if uncorrected LSTM states can be used in combination with different replay modes. We consider uniform sampling and prioritization via the critic’s loss, and include both full $\begin{array} { r } { \beta = 1 } \end{array}$ ) and partial $\beta = 0 . 5$ ) importance corrections
+
+A.2 PRIORITIZED AND UNIFORM EXPERIENCE REPLAY, LSTM STATES
+
+With prioritized experience replay each transition $\tau$ is sampled with probability $P ( \tau ) \propto p _ { \tau } ^ { \alpha }$ , for a suitable unnormalized priority score $p _ { \tau }$ and a global tunable parameter $\alpha$ . It is common (Schaul et al., 2015; Horgan et al., 2018; Hessel et al., 2017) to then weight updates computed from that sample by $1 / P ( \tau ) ^ { \beta }$ for $0 < \beta \leq 1$ , where $\beta = 1$ fully corrects for the bias introduced in the state distribution. In one step temporal difference methods, typical priorities are based on the immediate TD-error, and are typically recomputed after a transition is sampled from replay. This means low priorities might stay low and get stale – even if the transition suddenly becomes relevant. To alleviate this issue, the sampling distribution is mixed with a uniform, as controlled by a third hyper parameter $\epsilon$ .
+
+The performance of agents with prioritized experience replay can be quite sensitive to the hyperparameters $\alpha$ , $\beta$ , and $\epsilon$ .
+
+A critical practical consideration is how to implement random access for recurrent memory agents such as agents using an LSTM. Prioritized agents sample a presumably interesting transition from the past. This transition may be at any position within the episode. To infer the correct recurrent memory-state at this environment-state all earlier environment-states within that episode would need to be replayed. A prioritized agent with a random access pattern would thus require costly LSTM refreshes for each sampled transition. If LSTM states are not recomputed representational missmatch (Kapturowski et al., 2019) occurs.
+
+Sharing experience between multiple agents amplifies the issue of LSTM state representation missmatch. Here each agent has its own network parameters and the state representations between agents may be arbitrarily different.
+
+As a mitigation Kapturowski et al. (2019) use a burn-in window or to initialize with a constant starting state. We note that those solutions can only partially mitigate the fundamental issue and that counter examples such as arbitrarily long T-Mazes (Tolman, 1948; Olton, 1979) can be constructed easily.
+
+We thus advocate for uniform sampling. In our implementation we uniformly sample an episode. Then we replay each episode from the beginning, using the most recent network parameters to recompute the LSTM states along the way: this is particularly critical when sharing experience between different agents, which may have arbitrarily different state representations.
+
+This solution is exact and cost-efficient as it only requires one additional forward pass for each learning step (forward $^ +$ backward pass).
+
+An even more cost efficient approach would be to not refresh LSTM states at all. Naturally this comes at the cost of representational missmatch. However it would allow for an affordable implementation of prioritized experience replay. We investigate this in Figure 4 (right) and observe that it is not viable. We compare a baseline V-trace agent with no experience replay, one with uniform experience replay, and two different prioritized replay agents. We do not refresh LSTM states for any of the agents.
+
+The uniform replay agent is more data efficient then the baseline, and also saturates at a higher level of performance. The best prioritized replay agent uses full importance sampling corrections ( $\begin{array} { r } { \beta = 1 , } \end{array}$ ). However it performs no higher than with uniform replay. We therefore we used uniform replay with full state correction for all our investigations in the paper.
+
+# A.3 EVALUATION PROTOCOL
+
+For evaluation, we average episode returns within buckets of 1M (Atari) and 10M (DMLab) environment steps for each agent instance, and normalize scores on each game by using the scores of a human expert and a random agent (van Hasselt et al., 2016). In the multi-task setting, we then define the performance of each agent as the median normalized score of all levels that the agent trains on. Given the use of population based training, we need to perform the comparisons between algorithms at the level of sweeps. We do so by selecting the best performing agent instance within each sweep at any time. Note that for the multi-task setting, our approach of first averaging across many episodes, then taking the median across games, on DMLab further downsampling to 100M env steps, and only finally selecting the maximum within the sweep, results in substantially lower variance than if we were to compute the maximum before the median and smoothing.
+
+All DMLab-30 sweeps are repeated $3 \times$ with the exception of $\rho = 2$ and $\rho = 4$ in Figure 4. We then plot a shaded area between the point-wise best and worst replica and a solid line for the mean. Atari sweeps having 57 games are summarized and plotted by the median of the human-normalized scores.
+
+# APPENDIX B ALGORITHM PSEUDOCODE
+
+We present algorithm pseudocode for LASER with trust region (Algorithm 1). For clarity we present a version without LSTM and focus on the single agent case. The multi-agent case is a simple extension where all agents save to the same replay database and also sample from the same replay. Also each agent starts with different network parameters and hyper-parameters. The LSTM state recomputation can be achieved with Replayer Threads (nearly identical to Actor Threads) that sample entire epsiodes from replay, step through them while reevaluating the LSTM state and slice the experience into trajectories of length $T$ . Similar to regular LSTM Actor Threads from Espeholt et al. (2018) the Replayer Threads send each trajectory together with an LSTM state to the learning thread via a queue. The Learner Thread initializes the LSTM with the transmitted state when the LSTM is unrolled over the trajectory.
+
+# Algorithm 1 Single Agent LASER with Trust Region
+
+<table><tr><td>Initialize parameter vectors 0. Initialize π1 = 0. Actor Thread: while training is ongoing do</td></tr><tr><td>Sample trajectory unroll u = { t}tε{1...T} of length Tby acting in the environment using the latest πk Where Tt = (St,at,rt, μt = Tk(St|-)). Enqueue u into Lerner Queue, wait if full. Add u into Replay Database.</td></tr><tr><td>Remove oldest trajectory if database has reached desired capacity limit. end while</td></tr><tr><td>Learner Thread: Given: Batch size B,online fraction α. for training iteration k do</td></tr><tr><td>Form training batch U = {ub}b∈{1,.,B} of B trajectories of length T, by dequeuing Bα trajectories from Lerner Queue and sampling B(1 -α) trajectories from Replay Database.</td></tr><tr><td>Evaluate the target policy πk on the sampled transitions in U: i.e. πk(Sb,tl·). Compute behaviour relevance mask M with Mb,t = KL(πk(Sb,tl-)llμb,t) &lt; b where μb,t, Sb,t are obtained from Ub,t ·</td></tr><tr><td>Compute trust-region V-trace return Vt,b using 8 where Xπ,μ(Sb,t) = Mb,t. Let[Lv(0)lt,b=1(Vt,b-Vθ(st,b))². LetAt,b = Vt,b- Vθ(st,b) and [Lp(0)]t,b = pt,blog[πe(St,b|at,b)]At,b,Where ρ is the clipped v-trace importance sampling ratio.</td></tr></table>
+
+# APPENDIX C PROPOSITIONS
+
+We have stated five propositions in our paper for which we provide proofs below.
+
+Proposition 1. The V-trace value estimate $V ^ { \tilde { \pi } }$ is biased: It does not match the expected return of $\pi$ but the return of a related implied policy $\tilde { \pi }$ defined by equation 9 that depends on the behaviour policy $\mu$ :
+
+$$
+\tilde { \pi } _ { \mu } ( a | x ) = \frac { \operatorname* { m i n } \left[ \bar { \rho } \mu ( a | x ) , \pi ( a | x ) \right] } { \sum _ { b \in A } \operatorname* { m i n } \left[ \bar { \rho } \mu ( b | x ) , \pi ( b | x ) \right] }
+$$
+
+Proof. See Espeholt et al. (2018).
+
+Proposition 2. The V-trace policy gradient is biased: given the the optimal value function $V ^ { * }$ the $V .$ -trace policy gradient does not converge to a locally optimal $\pi ^ { * }$ for all off-policy behaviour distributions $\mu$ .
+
+Proof. Proof by contradiction:
+
+Consider a tabular counter example with a single (locally) optimal policy at $s _ { t }$ given by $\pi ^ { * } ( s _ { t } ) =$ argmaxπ $\textstyle \left[ \sum _ { a \in A } \pi ( a | s _ { t } ) Q ^ { * } ( a , s _ { t } ) \right]$ that always selects the action argmax $\mathbf { \Omega } _ { a } Q ^ { * } ( a , s _ { t } )$ .
+
+Even in this ideal tabular setting V-trace policy gradient estimates a different $\tilde { \pi } ^ { * }$ rather than the optimal $\pi ^ { * }$ as follows
+
+$$
+\begin{array} { r l } { \nabla V ^ { * , \pi } ( s _ { t } ) = \mathbf { E } _ { \mu } \left[ \rho _ { t } ( r _ { t } + \gamma V ^ { * } ( s _ { t + 1 } ) \nabla \log \pi ( a _ { t } | s _ { t } ) \right] } & { } \\ & { = \mathbf { E } _ { \mu } \left[ \rho _ { t } Q ^ { * } ( s _ { t } , a _ { t } ) \nabla \log \pi ( a _ { t } | s _ { t } ) \right] } \\ & { = \mathbf { E } _ { \mu } \left[ \operatorname* { m i n } \left[ \frac { \pi ( a _ { t } | s _ { t } ) } { \mu ( a _ { t } | s _ { t } ) } , \bar { \rho } \right] Q ^ { * } ( s _ { t } , a _ { t } ) \nabla \log \pi ( a _ { t } | s _ { t } ) \right] } \\ & { = \mathbf { E } _ { \mu } \left[ \frac { \pi ( a _ { t } | s _ { t } ) } { \mu ( a _ { t } | s _ { t } ) } \operatorname* { m i n } \left[ 1 , \bar { \rho } \frac { \mu ( a _ { t } | s _ { t } ) } { \pi ( a _ { t } | s _ { t } ) } \right] Q ^ { * } ( s _ { t } , a _ { t } ) \nabla \log \pi ( a _ { t } | s _ { t } ) \right] } \\ & { = \mathbf { E } _ { \pi } \left[ \operatorname* { m i n } \left[ 1 , \bar { \rho } \frac { \mu ( a _ { t } | s _ { t } ) } { \pi ( a _ { t } | s _ { t } ) } \right] Q ^ { * } ( s _ { t } , a _ { t } ) \nabla \log \pi ( a _ { t } | s _ { t } ) \right] } \\ & { = \mathbf { E } _ { \pi } \left[ \operatorname* { m i n } \left[ 1 , \bar { \rho } \frac { \mu ( a _ { t } | s _ { t } ) } { \pi ( a _ { t } | s _ { t } ) } \right] Q ^ { * } ( s _ { t } , a _ { t } ) \nabla \log \pi ( a _ { t } | s _ { t } ) \right] } \\ & { = \mathbf { E } _ { \pi } \left[ \operatorname* { o r } ( s _ { t } , a _ { t } ) Q ^ { * } ( s _ { t } , a _ { t } ) \nabla \log \pi ( a _ { t } | s _ { t } ) \right] } \\ & { = \mathbf { E } _ { \pi } \left[ Q ^ { * , \pi } ( s _ { t } , a _ { t } ) \nabla \log \pi ( a _ { t } | s _ { t } ) \right] } \end{array}
+$$
+
+Observe how the optimal Q-function $Q ^ { * }$ is scaled by $\begin{array} { r } { \omega ( s _ { t } , a _ { t } ) = \operatorname* { m i n } \left[ 1 , \bar { \rho } \frac { \mu ( a _ { t } | s _ { t } ) } { \pi ( a _ { t } | s _ { t } ) } \right] \leq 1 } \end{array}$ resulting in implied state-action values $Q ^ { \ast , \omega }$ . This penalizes actions where $\mu ( a _ { t } | \bar { s } _ { t } ) \bar { \rho } < \pi ( a _ { t } \bar { | } s _ { t } )$ and makes V-trace greedy w.r.t. to the remaining ones. Thus $\mu$ can be chosen adversarially to corrupt the optimal state action value. Note that $\bar { \rho }$ is a constant typically chosen to be 1.
+
+To prove the lemma consider a counter example such as an MDP with two actions and $Q ^ { * } = ( 2 , 5 )$ and $\mu = ( 0 . 9 , 0 . 1 )$ and initial $\pi = ( 0 . 5 , 0 . 5 )$ . Here the second action with expected return 5 is clearly favourable. Abusing notation $\mu / \pi = ( 1 . 8 , 0 . 2 )$ . Thus $Q ^ { \tilde { \pi } , \omega } = ( 2 * 1 , 5 * \bar { 0 . 2 } ) = ( 2 , 1 )$ . Therefore $\tilde { \pi } ^ { * } = ( 1 , 0 )$ wrongly selects the first action. □
+
+Proposition 3. Mixing on-policy data into the V-trace policy gradient with the ratio $\alpha$ reduces the bias by providing a regularization to the implied state-action values. In the general function approximation case it changes the off-policy $V .$ -trace policy gradient from $\begin{array} { r l } { \sum _ { s } d ^ { \mu } ( s ) \mathbf { E } _ { \pi } \left[ ( Q ( s , a ) \nabla \log \pi ( a | s ) \right] } \end{array}$ to $\begin{array} { r l } { \sum _ { s } \mathbf { E } _ { \pi } \left[ Q ^ { \alpha } ( s , a ) \nabla \log ^ { - } \pi ( a | s ) \right] } \end{array}$ where $Q ^ { \alpha } = Q d ^ { \pi } ( s ) \alpha + Q ^ { \overline { { { \omega } } } } \bar { d } ^ { \mu } ( s ) ( 1 - \bar { \alpha } )$ is a regularized stateaction estimate and $d ^ { \pi }$ , $d ^ { \mu }$ are the state distributions for $\pi$ and $\mu$ . Note that there exists $\alpha \leq 1$ such that $Q ^ { \alpha }$ has the same argmax (i.e. best action) as $Q$ .
+
+Proof. Note that the on-policy policy gradient is given by
+
+$$
+\nabla J _ { \mathrm { o n } } ( \pi ) = \sum _ { s } d ^ { \pi } ( s ) \mathbf { E } _ { \pi } \left[ Q ( s , a ) \nabla \log \pi ( a | s ) \right]
+$$
+
+Similarly the off-policy $\mathrm { V } .$ -trace gradient is given by
+
+$$
+\nabla J _ { \mathrm { o f f } } ( \pi ) = \sum _ { s } d ^ { \mu } ( s ) \mathbf { E } _ { \pi } \left[ \omega ( s , a ) Q ( s , a ) \nabla \log \pi ( a | s ) \right]
+$$
+
+with the V-trace distortion factor $\begin{array} { r } { \omega ( s _ { t } , a _ { t } ) = \operatorname* { m i n } \left[ 1 , \bar { \rho } \frac { \mu ( a _ { t } | s _ { t } ) } { \pi ( a _ { t } | s _ { t } ) } \right] \leq 1 } \end{array}$ that can de-emphasize action values and $Q ^ { \omega } ( s , a ) = \omega ( s , a ) Q ( s , a )$ .
+
+The $\alpha$ -interpolation of both gradients can be transformed as follows:
+
+$$
+\begin{array} { r l } { \tau [ ( \alpha , \alpha + 1 ) - \alpha ] \dot { \alpha } \dot { \theta } _ { 0 } [ \dot { \varepsilon } ] = } & { \sum _ { i = 1 } ^ { N } \dot { \varepsilon } [ \Phi _ { i } [ G _ { i } ^ { ( i ) } , G _ { i } ^ { ( i ) } , \varepsilon ] [ \Phi _ { i } [ \dot { \varepsilon } ] + \varepsilon _ { i } ^ { \prime } ( G _ { i } ^ { ( i ) } ) } \\ & { \mathrm { ( 1 ) } \quad \omega \dot { \varepsilon } ] \sum _ { i = 1 } ^ { N } \dot { \varepsilon } [ \Phi _ { i } [ G _ { i } ^ { ( i ) } , \varepsilon ] [ \Phi _ { i } [ \dot { \varepsilon } ] \varepsilon _ { i } \omega ^ { \prime } ] [ \Phi _ { i } [ \dot { \varepsilon } ] \varepsilon _ { i } \omega ^ { \prime } ] [ \dot { \varepsilon } ] } \\ & { - \sum _ { i = 1 } ^ { N } \dot { \varepsilon } [ \Phi _ { i } [ G _ { i } ^ { ( i ) } , \omega ] \nabla [ \Phi _ { i } [ \dot { \varepsilon } ] \varepsilon _ { i } \omega ^ { \prime } ] [ \Phi _ { i } [ \dot { \varepsilon } ] ] } \\ & { + \sum _ { i = 1 } ^ { N } \partial _ { i } ^ { ( j ) } [ \Phi _ { i } [ \varepsilon _ { i } \omega ^ { \prime } ] [ \Phi _ { i } [ \dot { \varepsilon } _ { i } ] ] [ \varepsilon _ { i } \omega ^ { \prime } ] [ \Phi _ { i } [ \dot { \varepsilon } ] ] } \\ & { - \sum _ { i = 1 } ^ { N } \dot { \varepsilon } [ \Phi _ { i } [ G _ { i } ^ { ( i ) } , \varepsilon ] [ \Phi _ { i } [ \varepsilon _ { i } \omega ^ { \prime } ] [ \Phi _ { i } [ \dot { \varepsilon } _ { i } ] ] ] } \\ & { + \sum _ { i = 1 } ^ { N } \dot { \varepsilon } [ \Phi _ { i } [ G _ { i } ^ { ( i ) } , \varepsilon ] [ \Phi _ { i } [ \varepsilon _ { i } \omega ^ { \prime } ] [ \Phi _ { i } [ \dot { \varepsilon } _ { i } ] ] ] } \\ &  - \sum _ { i = 1 } ^ { N } \dot { \varepsilon } [ \Phi _ { i } [ \varepsilon _ { i } ] [ \Phi _ { i } [ \varepsilon _ { i } ^ { \prime } ] - \varepsilon _ { i } [ \Phi _ { i } [ \varepsilon _ { i } ^ { \prime } ] ] [ \Phi _ { i } [ \varepsilon _ { i } ^ { \prime } ] ] [ \Phi _ { i } [ \varepsilon _ { i }  \end{array}
+$$
+
+Interpretation of Proposition 3 As discussed in section 3 the $\mathrm { v } .$ -trace policy gradient will have the correct local fixpoint at state $s$ if the argmax of the state-value function is preserved despite the distortion: i.e. if $\begin{array} { r } { \operatorname { a r g m a x } _ { a } [ Q ( s , a ) ] = \operatorname { a r g m a x } _ { a } [ Q ^ { \omega } ( s , a ) ] } \end{array}$ . Respectively when mixing in an $\alpha \in [ 0 , 1 )$ share of online data the fixpoint will be preserved if
+
+$$
+\begin{array} { r } { \operatorname { a r g m a x } _ { a } [ Q ( s , a ) ] = \operatorname { a r g m a x } _ { a } [ Q ^ { \alpha } ( s , a ) ] } \end{array}
+$$
+
+Let $a ^ { * } = \operatorname { a r g m a x } _ { b } ( Q , b )$ be any best action and $A ^ { * }$ be set of best actions. Then equation 12 is equivalent to:
+
+$$
+Q ^ { \alpha } ( s , a ^ { * } ) > Q ^ { \alpha } ( s , b ) \forall b \notin A ^ { * }
+$$
+
+Using the definition of $Q ^ { \alpha }$ this can be rewritten as:
+
+$$
+Q ( s , a ^ { * } ) d ^ { \pi } ( s ) \alpha + Q ^ { \omega } ( s , a ^ { * } ) d ^ { \mu } ( s ) ( 1 - \alpha ) > Q ( s , b ) d ^ { \pi } ( s ) \alpha + Q ^ { \omega } ( s , b ) d ^ { \mu } ( s ) ( 1 - \alpha ) \forall b \notin A ^ { * }
+$$
+
+Which can be rearranged to:
+
+$$
+[ Q ( s , a ^ { * } ) d ^ { \pi } ( s ) - Q ( s , b ) d ^ { \pi } ( s ) ] \alpha > [ Q ^ { \omega } ( s , b ) d ^ { \mu } ( s ) - Q ^ { \omega } ( s , a ^ { * } ) d ^ { \mu } ( s ) ] ( 1 - \alpha ) \forall b \notin A ^ { * }
+$$
+
+By definition $Q ( s , a ^ { * } ) d ^ { \pi } ( s ) - Q ( s , b ) d ^ { \pi } ( s ) > 0 \forall b \notin A ^ { * }$ , hence:
+
+$$
+\frac { \alpha } { 1 - \alpha } > \frac { Q ^ { \omega } ( s , b ) - Q ^ { \omega } ( s , a ^ { * } ) } { Q ( s , a ^ { * } ) - Q ( s , b ) } \frac { d ^ { \mu } ( s ) } { d ^ { \pi } ( s ) } \forall b \notin { \cal A } ^ { * }
+$$
+
+It follows that the policy gradient will have the same local fixpoint if
+
+$$
+\frac { \alpha } { 1 - \alpha } > \operatorname* { m a x } _ { b \notin A ^ { * } } \left[ \frac { Q ^ { \omega } ( s , b ) - Q ^ { \omega } ( s , a ^ { * } ) } { Q ( s , a ^ { * } ) - Q ( s , b ) } \right] \frac { d ^ { \mu } ( s ) } { d ^ { \pi } ( s ) }
+$$
+
+Note that $\textstyle { \frac { \alpha } { 1 - \alpha } } \to \infty$ as $\alpha  1$ . Mixing-in more online data thus increases the left hand side. Also note that the right hand side decreases due to $d ^ { \mu } ( s ) / d ^ { \pi } ( s )$ if $\pi$ visits the state $s$ more often than $\mu$ . Furthermore the larger the action value gap in the real Q-function $Q ( s , a ^ { * } ) - Q ( s , b )$ the lower the right hand side. Finally the denominator will be negative if $\begin{array} { r } { \operatorname* { m a x } _ { b \notin { A ^ { * } } } [ Q ^ { \omega } ( s , b ) ] < Q ^ { \omega } ( s , a ^ { * } ) } \end{array}$ thus enabling correct learning even in the pure off-policy case with $\alpha = 0$ .
+
+Note that all of those conditions can be computed and checked if an accurate Q-function and state distribution is accessible. How to use imperfect Q-function estimates to adaptively choose such an $\alpha$ remain a question for future research.
+
+Proposition 4. Let that they all have th $G _ { \mathrm { I S } } ^ { \pi , \mu _ { z } }$ be a set  fix point mportance sampling estimaand contract with at least rs as defined in equation 7. Note. Then the contraction properties carry over to $V _ { \mathrm { t r u s t e d } } ^ { \pi }$ . $I n$ particular $V ^ { \pi }$ $| V _ { \mathrm { t r u s t e d } } ^ { \pi } - V ^ { \pi } | _ { \infty } \leq \gamma | V - V ^ { \pi } | _ { \infty }$ $\gamma$ .
+
+Proof. Let us consider the set of importance sampling estimators as defined in 7 and note that they all contract to the same fixed point $V ^ { \pi }$ with at least $\begin{array} { r } { \left| \mathbf { E } _ { \mu _ { z } | z } \left[ G _ { \mathrm { I S } } ^ { \pi , \mu _ { z } } ( s ) \right] - V ^ { \pi } ( s ) \right| _ { \infty } \le } \end{array}$ $\gamma \left| V ( s ) - V ^ { \pi } ( s ) \right| _ { \infty }$ for any state $s$ .
+
+By Minkowski’s inequality the contraction properties of importance sampled Monte-Carlo bootstraps carry over to $V _ { \mathrm { t r u s t e d } } ^ { \pi }$ which is a $p ( z | \mu _ { z } \in \mathbf { \bar { M } } _ { \beta , \pi } ( s _ { t } ) )$ weighted average:
+
+$$
+\begin{array} { r l } & { \left| V _ { \mathrm { t r u s t e d } } ^ { \pi } ( s ) - V ^ { \pi } ( s ) \right| _ { \infty } = \left| \mathbf E _ { z } \Big [ \mathbf E _ { \mu _ { z } | z } \left[ G _ { \mathrm { I S } } ^ { \pi , \mu _ { z } } ( s ) \right] \Big | \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) \Big ] - V ^ { \pi } ( s ) \right| _ { \infty } } \\ & { \qquad = \Big | \mathbf E _ { z } \Big [ \mathbf E _ { \mu _ { z } | z } \left[ G _ { \mathrm { I S } } ^ { \pi , \mu _ { z } } ( s ) \right] - V ^ { \pi } ( s ) \Big | \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) \Big ] \Big | _ { \infty } } \\ & { \qquad \le \mathbf E _ { z } \Big [ \left| \mathbf E _ { \mu _ { z } | z } \left[ G _ { \mathrm { I S } } ^ { \pi , \mu _ { z } } ( s ) \right] - V ^ { \pi } ( s _ { t } ) \right| _ { \infty } \Big | \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) \Big ] } \\ & { \qquad < \mathbf E _ { z } \Big [ \gamma | V ( s ) - V ^ { \pi } ( s ) | _ { \infty } \Big | \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) \Big ] } \\ & { \qquad = \gamma \left| V ( s ) - V ^ { \pi } ( s ) \right| _ { \infty } } \end{array}
+$$
+
+Proposition 5. Let $G _ { \mathrm { V t r a c e } } ^ { \pi , \mu _ { z } }$ be a set of $V .$ -trace estimators (see equation 8) with corresponding fixed points (see equation $\cdot$ ) to which they contract at a speed of an algorithm and behaviour specific $\eta _ { z }$ . Then $V _ { \mathrm { t r u s t e d } } ^ { \pi }$ moves towards $V ^ { \beta ^ { * } } = \mathbf { { E } } _ { z \mid \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } \left[ \tilde { V ^ { z } } \right]$ shrinking the distance as follows $\begin{array} { r } { \big | V _ { \mathrm { t r u s t e d } } ^ { \pi } - V ^ { \beta } \big | _ { \infty } < \operatorname* { m a x } _ { \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } | \eta _ { z } ( V - V ^ { z } ) | _ { \infty } \leq \eta _ { \operatorname* { m a x } } \operatorname* { m a x } _ { \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } | ( V - V ^ { z } ) | _ { \infty } } \end{array}$ with $\eta _ { \mathrm { m a x } } = \operatorname* { m a x } _ { \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } \eta _ { z }$ .
+
+Proof. Recall the contraction properties of a $\mathbf { V }$ -trace importance sampled Monte-Carlo bootstraps $G _ { \mathrm { V t r a c e } } ^ { \pi , \bar { \mu } _ { z } }$ being
+
+$$
+\left| \mathbf { E } _ { \mu _ { z } | z } \left[ G _ { \mathrm { V t r a c e } } ^ { \pi , \mu _ { z } } ( s ) \right] - V ^ { z } ( s ) \right| _ { \infty } < \eta _ { z } \left| V ( s ) - V ^ { z } ( s ) \right| _ { \infty }
+$$
+
+for an algorithm and behaviour specific $\eta _ { z } ~ < ~ 1$ for a $z$ dependent fixed point $V ^ { z }$ and for any bootstrap $V$ . We then show that $V _ { \mathrm { t r u s t e d } } ^ { \pi }$ moves towards the weighted average of fixed points $V ^ { \beta } = \mathbf { E } _ { z | \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } \left[ V ^ { z } \right]$ , since
+
+$$
+\big | V _ { \mathrm { t r u s t e d } } ^ { \pi } ( s ) - V ^ { \beta } ( s ) \big | _ { \infty } < \eta _ { \mathrm { m a x } } \operatorname* { m a x } _ { \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } \big | V ( s ) - V ^ { z } ( s ) \big | _ { \infty }
+$$
+
+holds for any bootstrap function $V$ as we show below.
+
+$$
+\begin{array} { r l } & { | V _ { \mathrm { t r u s t e d } } ^ { \pi } ( s ) - V ^ { \beta } ( s ) | _ { \infty } = | \mathbf { E } _ { z } \Big [ \mathbf { E } _ { \mu _ { z } | z } \big [ G _ { \mathrm { V t r a c e } } ^ { \pi , \mu _ { z } } ( s ) \big ] - V ^ { z } ( s ) \Big | \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) \Big ] | _ { \infty } } \\ & { \qquad \leq \mathbf { E } _ { z } [ | \mathbf { E } _ { \mu _ { z } | z } \big [ G _ { \mathrm { V t r a c e } } ^ { \pi , \mu _ { z } } ( s ) \big ] - V ^ { z } ( s ) \big | _ { \infty } \Big | \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) ] } \\ & { \qquad < \mathbf { E } _ { z } \Big [ \big | \eta _ { z } \big ( V ( s ) - V ^ { z } ( s ) \big ) \big | _ { \infty } \Big | \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) \Big ] } \\ & { \qquad \leq \underset { \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } { \operatorname* { m a x } } \big | \eta _ { z } \big ( V ( s ) - V ^ { z } ( s ) \big ) \big | _ { \infty } } \\ & { \qquad \leq \eta _ { \operatorname* { m a x } } \underset { \mu _ { z } \in M _ { \beta , \pi } ( s _ { t } ) } { \operatorname* { m a x } } \big | V ( s ) - V ^ { z } ( s ) \big | _ { \infty } } \end{array}
+$$
+
+# APPENDIX D DETAILED ATARI RESULTS
+
+We display the Atari per-level performance of various agents at 50M and 200M environment steps in Table 2. The scores correspond to the agents presented in Figure 1. The LASER scores are computed by averaging the last 100 episode returns before 50M or respectively 200M environment frames have been experienced. Following the procedure defined by Mnih et al. (2015) we initialize the environment with a random number of no-op actions (up to 37 in our case). Again following Mnih et al. (2015) episodes are terminated after 30 minutes of gameplay. Note that Xu et al. (2018) have not published per-level scores. Rainbow scores are obtained from Hessel et al. (2017).
+
+Table 2: Per level performance of various agents at 50M and 200M environment steps (see Figure 1).   
+
+<table><tr><td rowspan="2">Game</td><td rowspan="2">LASER Shared (sweep at 50M)</td><td rowspan="2">LASERShared (sweep at 200M)</td><td rowspan="2">LASER (no sweep at 200M)</td><td rowspan="2">Rainbow (no sweep at 200M)</td></tr><tr><td></td></tr><tr><td>alien</td><td>18635.3</td><td>18277.3</td><td>35565.9</td><td>9491.7</td></tr><tr><td>amidar</td><td>1838.3</td><td>2695</td><td>1829.2</td><td>5131.2</td></tr><tr><td>assault</td><td>26027.1</td><td>40603.2</td><td>21560.4</td><td>14198.5</td></tr><tr><td>asterix</td><td>496735.0</td><td>240770</td><td>240090</td><td>428200</td></tr><tr><td>asteroids</td><td>232651</td><td>257420.1</td><td>213025</td><td>2712.8</td></tr><tr><td>atlantis</td><td>889934.0</td><td>866584</td><td>841200</td><td>826660</td></tr><tr><td>bank_heist</td><td>1333.1</td><td>1712.8</td><td>569.4</td><td>1358</td></tr><tr><td>battle_zone</td><td>66900</td><td>131880.0</td><td>64953.3</td><td>62010</td></tr><tr><td>beam_rider</td><td>80830.5</td><td>125795.2</td><td>90881.6</td><td>16850.2</td></tr><tr><td>berzerk</td><td>46651.6</td><td>64513.1</td><td>25579.5</td><td>2545.6</td></tr><tr><td>bowling</td><td>42.4</td><td>47.4</td><td>48.3</td><td>30</td></tr><tr><td>boxing</td><td>99.8</td><td>99.4</td><td>100.0</td><td>99.6</td></tr><tr><td>breakout</td><td>852.5</td><td>850.3</td><td>747.9</td><td>417.5</td></tr><tr><td>centipede</td><td>208008</td><td>409702.8</td><td>292792</td><td>8167.3</td></tr><tr><td>chopper_command</td><td>24814</td><td>727333</td><td>761699.0</td><td>16654</td></tr><tr><td>crazy_climber</td><td>160494</td><td>88818</td><td>167820</td><td>168788.5</td></tr><tr><td>defender</td><td>355447</td><td>369397.0</td><td>336953</td><td>55105</td></tr><tr><td>demon_attack</td><td>133557</td><td>138000.6</td><td>133530</td><td>111185</td></tr><tr><td>double_dunk</td><td>0.1</td><td>23.5</td><td>14</td><td>-0.3</td></tr><tr><td>enduro</td><td>0</td><td>0</td><td>0</td><td>2125.9</td></tr><tr><td>fishing_derby</td><td>45.4</td><td>62.6</td><td>45.2</td><td>31.3</td></tr><tr><td>freeway</td><td>34.0</td><td>34.0</td><td>0</td><td>34.0</td></tr><tr><td>frostbite</td><td>5297.4</td><td>2230.8</td><td>5083.5</td><td>9590.5</td></tr><tr><td>gopher</td><td>86222.2</td><td>39721.2</td><td>114820.7</td><td>70354.6</td></tr><tr><td>gravitar</td><td>1360.5</td><td>2812.0</td><td>1106.2</td><td>1419.3</td></tr><tr><td>hero</td><td>30159.2</td><td>36510.6</td><td>31628.7</td><td>55887.4</td></tr><tr><td>ice_hockey</td><td>20.2</td><td>38.7</td><td>17.4</td><td>1.1</td></tr><tr><td>jamesbond</td><td>21663</td><td>60402.5</td><td>37999.8</td><td>19809</td></tr><tr><td>kangaroo</td><td>13932</td><td>14187</td><td>14308</td><td>14637.5</td></tr><tr><td>krull</td><td>9559.3</td><td>5743.6</td><td>9387.5</td><td>8741.5</td></tr><tr><td>kung_fu_master</td><td>65032</td><td>81792</td><td>607443.0</td><td>52181</td></tr><tr><td>montezuma_revenge</td><td>1</td><td>1</td><td>0.3</td><td>384.0</td></tr><tr><td>ms_pacman</td><td>6089.3</td><td>6890.7</td><td>6565.5</td><td>5380.4</td></tr><tr><td>name_this_game</td><td>25998.9</td><td>27910.7</td><td>26219.5</td><td>13136</td></tr><tr><td>phoenix</td><td>458355</td><td>628711.6</td><td>519304</td><td>108529</td></tr><tr><td>pitfall</td><td>-0.2</td><td>-0.2</td><td>-0.6</td><td>0.0</td></tr><tr><td>pong</td><td>21.0</td><td>21.0</td><td>21.0</td><td>20.9</td></tr><tr><td>private_eye</td><td>100</td><td>100</td><td>96.3</td><td>4234.0</td></tr><tr><td>qbert</td><td>20283.8</td><td>24600.8</td><td>21449.6</td><td>33817.5</td></tr><tr><td>riverraid</td><td>24138.1</td><td>35491.5</td><td>40362.7</td><td>22920.8</td></tr><tr><td>road_runner</td><td>52942</td><td>63762.0</td><td>45289</td><td>62041</td></tr><tr><td>robotank</td><td>63.6</td><td>67.8</td><td>62.1</td><td>61.4</td></tr><tr><td>seaquest</td><td>1802.2</td><td>557213.3</td><td>2890.3</td><td>15898.9</td></tr><tr><td>skiing</td><td>-8904.8</td><td>-8980.1</td><td>-29968.4</td><td>-12957.8</td></tr><tr><td>solaris</td><td>2222.4</td><td>3017.6</td><td>2273.5</td><td>3560.3</td></tr><tr><td>space_invaders</td><td>36071.4</td><td>53124.3</td><td>51037.4</td><td>18789</td></tr><tr><td>star_gunner</td><td>331327</td><td>602540.0</td><td>321528</td><td>127029</td></tr><tr><td>surround</td><td>9.8</td><td>9.8</td><td>8.4</td><td>9.7</td></tr><tr><td>tennis</td><td>0</td><td>0</td><td>12.2</td><td>0</td></tr><tr><td>time_pilot</td><td>77899</td><td>113603.0</td><td>105316</td><td>12926</td></tr><tr><td>tutankham</td><td>251.8</td><td>268.5</td><td>278.9</td><td>241</td></tr><tr><td>up_n_down</td><td>341988</td><td>368586.5</td><td>345727</td><td>125755</td></tr><tr><td>venture</td><td>0</td><td>0</td><td>0</td><td>5.5</td></tr><tr><td>video_pinball</td><td>513121</td><td>397451</td><td>511835</td><td>533936.5</td></tr><tr><td>wizard_of_wor</td><td>22280</td><td>45335.0</td><td>29059.3</td><td>17862.5</td></tr><tr><td>yars_revenge</td><td>145055</td><td>144370</td><td>166292.3</td><td>102557</td></tr><tr><td>zaxxon</td><td>50486</td><td>106862.0</td><td>41118</td><td>22209.5</td></tr></table>

parse/train/SJzR2iRcK7/SJzR2iRcK7.md

@@ -0,0 +1,338 @@
+# MULTI-CLASS CLASSIFICATION WITHOUT MULTICLASS LABELS
+
+Yen-Chang $\mathbf { H s u ^ { 1 } }$ , Zhaoyang $\mathbf { L } \mathbf { v } ^ { 1 }$ , Joel Schlosser2, Phillip Odom2, and Zsolt Kira12
+
+1Georgia Institute of Technology 2Georgia Tech Research Institute 1{yenchang.hsu,zhaoyang.lv,zkira}@gatech.edu 2{joel.schlosser,phillip.odom}@gtri.gatech.edu
+
+# ABSTRACT
+
+This work presents a new strategy for multi-class classification that requires no class-specific labels, but instead leverages pairwise similarity between examples, which is a weaker form of annotation. The proposed method, meta classification learning, optimizes a binary classifier for pairwise similarity prediction and through this process learns a multi-class classifier as a submodule. We formulate this approach, present a probabilistic graphical model for it, and derive a surprisingly simple loss function that can be used to learn neural network-based models. We then demonstrate that this same framework generalizes to the supervised, unsupervised cross-task, and semi-supervised settings. Our method is evaluated against state of the art in all three learning paradigms and shows a superior or comparable accuracy, providing evidence that learning multi-class classification without multi-class labels is a viable learning option.
+
+# 1 INTRODUCTION
+
+One of the most common settings for machine learning is classification, which involves learning a function $f$ to map the input data $x$ to a class label $y \in \{ 1 , 2 , . . , C \}$ . The most successful method for learning such a function is deep neural networks, owing its popularity to its capability to approximate a complex nonlinear mapping between high-dimensional data (e.g. images) and the classes. Despite the success of deep learning, a neural network demands a large amount of class-specific labels for learning a discriminative model, i.e. $P ( y | x )$ . This type of labeling can be expensive to collect, requires a-priori knowledge of all classes, and limits the form of supervision required. For example, the classes may be ambiguous or non-expert human annotators may be able to more easily provide information about whether two instances are of the same class or not, rather than identifying the specific class. A final problem is that different methods are necessary depending on what type of data is available, ranging from supervised learning (known classes) to cross-task unsupervised learning (unknown classes in the target domain) and semi-supervised learning (mix of labeled and unlabeled with known classes). Unsupervised learning with unknown classes is especially difficult to support.
+
+To relax these limitations, we propose to reduce the problem of classification to a meta problem that underlies a set of learning problems. Instead of solving the target task directly (learning a multi-class discriminative model such as a neural network), we instead learn a model that does not require explicit class label $y$ but rather a weaker form of information. In the context of classification, the meta problem that we use is a binary decision problem. Note that such a conversion to a different task (e.g. binary) is called a problem reduction method (Allwein et al., 2000) which has had a long history in the literature, especially in ensemble methods and binarization techniques (Galar et al., 2011). The most well-known strategies are "one-vs-all" (Anand et al., 1995; Rifkin & Klautau, 2004) and "one-vs-one" (Knerr et al., 1990; Hastie & Tibshirani, 1998; Wu et al., 2004). Although they have varied ensembling strategies, all of them share the same task encapsulating scheme, as illustrated in Figure 1a; specifically the binary classifiers are the sub-modules of a multi-class classifier (i.e. the multi-class classifier consists of multiple binary classifiers). These schemes still require that the class label $y$ be available to create the inputs for each binary classifier, and therefore these strategies do not relax the labeling requirements mentioned earlier.
+
+![](images/21e2e1bc7e8d126096adc659175bdd711fd860919532c89a4e452cd8e66aa6fb.jpg)  
+Figure 1: Problem reduction schemes for multi-class classification. This work proposes scheme (b), which introduces a binary classifier that captures $s _ { i j }$ . Note that $s _ { i j }$ represents the probability that $x _ { i }$ and $x _ { j }$ belong to the same class.
+
+In this work, we propose a novel strategy to address the above limitations. Our method reverses the task encapsulation order so that a multi-class classifier becomes a sub-module of a binary classifier, as illustrated in Figure 1b. The connection between the two classifiers is elucidated in Section 3. There are two highlights in Figure 1b. First, class labels $y _ { i }$ are not required in the learning stage. Instead, our method uses pairs of data $( x _ { i } , x _ { j } )$ as input and pairwise similarity $s _ { i j }$ for the supervision. Second, there is only one binary classifier in the scheme and it is present only during the training stage. In other words, the ephemeral binary task assists the learning of a multi-class classifier without being involved in the inference. When using a neural network with softmax outputs for the multi-class classifier, the proposed scheme can learn a discriminative model with only pairwise information.
+
+We specifically make the following contributions: 1) We analyze the problem setting and show that the loss we can use for this encapsulation can be easily derived, and we present an intuitive probabilistic graphical model interpretation for doing so, 2) We evaluate its performance compared to vanilla supervised learning of neural networks which uses multi-class labels, and visualize the loss landscape to better understand the underlying optimization difficulty, and 3) We demonstrate support for learning classifiers in more challenging problem domains, e.g. in unsupervised cross-task transfer and semi-supervised learning. We show how our meta classification framework can support all three learning paradigms, and evaluate it against several state-of-the-art methods. The experimental results show that the same meta classification approach is superior or comparable to state of the art across the three problem domains (supervised learning, unsupervised cross-task learning, and semi-supervised learning), demonstrating flexibility to support even unknown types and numbers of classes.
+
+# 2 RELATED WORK
+
+Supervised learning and problem reduction: Allwein et al. (2000) presents a unifying framework for multi-class classification by reducing it to multiple binary problems. The concepts for achieving such reduction, one-vs-all and one-vs-one, have been widely adopted and analyzed (Galar et al., 2011). The two strategies have been used to create several popular algorithms, such as variants of support vector machine (Weston & Watkins, 1998), AdaBoost (Freund & Schapire, 1997; Schapire & Singer, 1999), and decision trees (Fürnkranz, 2003). Despite the long history of reduction, our proposed scheme (Figure 1b) has not been explored. Furthermore, the scheme can be deployed easily by replacing the learning objective, which is fully compatible with deep neural networks for classification, a desirable property for broad applicability.
+
+Unsupervised cross-task transfer learning: This learning scheme is proposed by Hsu et al. (2018). The method transfers the pairwise similarity as the meta knowledge to an unlabeled dataset of different classes. It then uses a constrained clustering algorithm with predicted pairwise constraints (binarized pairwise similarity) to discover the unseen classes. This learning scheme shares the same supervision (pairwise similarity) as ours, and therefore relates our method to constrained clustering algorithms. One class of such approaches uses the constraints to learn a distance metric, and then applies a generic clustering algorithm such as K-means or hierarchical clustering to obtain the cluster assignments. This includes DML (Xing et al., 2003), ITML (Davis et al., 2007), SKMS (Anand et al., 2014), SKKm (Anand et al., 2014; Amid et al., 2016), and SKLR (Amid et al., 2016). The second class of methods incorporates the constraints into the cluster assignment objective. Some constrained spectral clustering algorithms, e.g. CSP (Wang et al., 2014) and COSC (Rangapuram & Hein, 2012) use this strategy. There are also approaches combine both distance metric learning and a clustering objective jointly, such as MPCKMeans (Bilenko et al., 2004), CECM (Antoine et al., 2012), and Kullback–Leibler divergence based contrastive loss (KCL) (Hsu & Kira, 2016; Hsu et al., 2018). Our new learning objective for Figure 1b can replace the above objectives in the cross-task transfer learning scheme.
+
+![](images/bef284988ef65f5658dca7760ad85e579de870af6d3ea3572b11cccca6ef2cd5.jpg)  
+Figure 2: Graphical representation for the meta classification task; $X _ { i }$ represents the node of input data, $Y _ { i }$ represents the class label, $S _ { i j }$ is pairwise similarity between instances $i$ and $j$ , and $\theta$ represents the neural network parameters.
+
+Semi-supervised learning: Our meta classification strategy can easily plug into a semi-supervised learning scheme. Our comparison focuses on state-of-the-art methods which solely involve adding a consistency regularization (Laine & Aila, 2017; Sajjadi et al., 2016; Miyato et al., 2018; Tarvainen & Valpola, 2017) or Pseudo-Labeling (Lee, 2013) for training a neural network. Another line of strategy combines weak supervision, such as similar pairs, and unlabeled data to learn a binary classifier (Bao et al., 2018). We present a new method by augmenting Figure 1b with the Pseudo-Labeling strategy.
+
+# 3 META CLASSIFICATION LEARNING
+
+A natural way to analyze problems with observed and unobserved information is through a probabilistic graphical model. In figure 2, we show the graphical model for our problem, where classspecific labels $Y$ are latent while pairwise similarities $S$ are observed. Specifically, we denote $\mathbf { X } = \{ X _ { 1 } , . . , X _ { n } \}$ , $\mathbf { Y } = \{ Y _ { 1 } , . . , Y _ { n } \}$ , and $\mathbf { S } = \{ S _ { i j } \} _ { 1 \leq i , j \leq n }$ to represent the nodes for samples, class labels, and pairwise similarities, respectively. In the model, we have $Y _ { i } \in \{ 1 , 2 , . . , C \}$ and $S _ { i j } \in \{ 0 , 1 \}$ . Then we have $\mathsf { P } ( S _ { i j } = 1 | Y _ { i } , Y _ { j } ) = 1$ when $Y _ { i } = Y _ { j }$ and zero probability otherwise; similarly, $\bar { \mathsf { P } } ( S _ { i j } = 0 | Y _ { i } , Y _ { j } ) = \bar { 1 }$ when $Y _ { i } \neq Y _ { j }$ . The output of a discriminative classifier with parameters $\theta$ is $f ( x _ { i } ; \theta ) = \mathsf { P } ( Y _ { i } | x _ { i } ; \theta )$ , where $f ( \overline { { \boldsymbol { x } } } _ { i } ; \boldsymbol { \theta } )$ outputs a categorical distribution. Now we describe the likelihood that the model explains the observed labeling (either with class labeling or pairwise labeling).
+
+$$
+\mathcal { L } ( \boldsymbol { \theta } ; \mathbf { X } , \mathbf { Y } , \mathbf { S } ) = \mathbb { P } ( \mathbf { X } , \mathbf { Y } , \mathbf { S } ; \boldsymbol { \theta } ) = \mathbb { P } ( \mathbf { S } | \mathbf { Y } ) \mathbb { P } ( \mathbf { Y } | \mathbf { X } ; \boldsymbol { \theta } ) \mathbb { P } ( \mathbf { X } )
+$$
+
+When $\mathbf { S }$ is fully observed while $\mathbf { Y }$ is unknown, calculating the likelihood requires marginalizing $\mathbf { Y }$ by computing $\begin{array} { r l } { \sum _ { \mathbf { Y } } \mathbb { P } ( \mathbf { S } | \mathbf { Y } ) \mathbb { P } ( \mathbf { Y } | \mathbf { X } ; \boldsymbol { \theta } ) } & { { } } \end{array}$ , which is intractable. The pairwise term $\begin{array} { r } { \mathsf { P } ( \mathbf { S } | \mathbf { Y } ) = } \end{array}$ $\begin{array} { r } { \prod _ { i , j } \mathbb { P } ( S _ { i j } | Y _ { i } , Y _ { j } ) } \end{array}$ makes all $Y _ { i }$ dependent on each other and prohibits efficient factorization. Thus, we approximate the computation by imposing additional independences such that $S i j \perp \bf { S } \backslash \{ \it { S } _ { i j } \} \lvert \it { X } _ { i } , \it { X } _ { j }$ (see Appendix $\mathbf { D }$ for a discussion of these). Now we can compute the likelihood with the observed nodes $X _ { i } = x _ { i }$ and $S _ { i j } = s _ { i j }$ :
+
+$$
+\begin{array} { r l } & { { \mathcal { L } } ( \boldsymbol { \theta } ; \mathbf { X } , \mathbf { S } ) \approx \displaystyle \sum _ { \mathbf { Y } } \mathbb { P } ( \mathbf { S } | \mathbf { Y } ) \mathbb { P } ( \mathbf { Y } | \mathbf { X } ; \boldsymbol { \theta } ) } \\ & { \quad \quad \approx \displaystyle \prod _ { i , j } \Big ( \displaystyle \sum _ { Y _ { i } = Y _ { j } } \mathbb { 1 } \big [ s _ { i j } = \boldsymbol { 1 } ] \mathbb { P } ( Y _ { i } | x _ { i } ; \boldsymbol { \theta } ) \mathbb { P } ( Y _ { j } | x _ { j } ; \boldsymbol { \theta } ) + } \\ & { \quad \quad \quad \displaystyle \sum _ { Y _ { i } \neq Y _ { j } } \mathbb { 1 } \big [ s _ { i j } = 0 \big ] \mathbb { P } ( Y _ { i } | x _ { i } ; \boldsymbol { \theta } ) \mathbb { P } ( Y _ { j } | x _ { j } ; \boldsymbol { \theta } ) \Big ) . } \end{array}
+$$
+
+Equation (2) omits the $\mathtt { P } ( \mathbf { X } )$ since $\mathbf { X }$ are observed leaf nodes. It is straightforward to take a negative logarithm on equation 3 and derive a loss function:
+
+$$
+\begin{array} { l } { \displaystyle { L _ { m e t a } ( \theta ) = - \sum _ { i , j } \log \bigg ( \sum _ { Y _ { i } = Y _ { j } } \mathbb { I } [ s _ { i j } = 1 ] \mathbb { P } ( Y _ { i } | x _ { i } ; \theta ) \mathbb { P } ( Y _ { j } | x _ { j } ; \theta ) + } } \\ { \displaystyle { \sum _ { Y _ { i } \neq Y _ { j } } \mathbb { I } [ s _ { i j } = 0 ] \mathbb { P } ( Y _ { i } | x _ { i } ; \theta ) \mathbb { P } ( Y _ { j } | x _ { j } ; \theta ) \bigg ) } } \\ { = - \sum _ { i , j } s _ { i j } \log ( f ( x _ { i } ; \theta ) ^ { T } f ( x _ { j } ; \theta ) ) + ( 1 - s _ { i j } ) \log ( 1 - f ( x _ { i } ; \theta ) ^ { T } f ( x _ { j } ; \theta ) ) . } \end{array}
+$$
+
+Then we define the function $g$ by the probability of having the same class label, which is calculated by the inner product between two categorical distributions:
+
+$$
+g ( x _ { i } , x _ { j } , f ( \cdot , \theta ) ) = f ( x _ { i } ; \theta ) ^ { T } f ( x _ { j } ; \theta ) = \hat { s } _ { i j }
+$$
+
+Here we use $\hat { s } _ { i j }$ to denote the predicted similarity (as opposed to ground truth similarity $s _ { i j }$ ). By plugging equation 6 into equation 5, $L _ { m e t a }$ has the form of a binary cross-entropy loss:
+
+$$
+L _ { m e t a } = - \sum _ { i , j } s _ { i j } \log \hat { s } _ { i j } + ( 1 - s _ { i j } ) \log ( 1 - \hat { s } _ { i j } ) .
+$$
+
+In Figure 1b, the multi-class classifier corresponds to $f$ while the binary classifier corresponds to $g$ . In other words, it is surprisingly simple to wrap a multi-class classifier by a binary classifier as described above. Since there are no learnable parameters in $g$ , the weights optimized with the meta criterion $L _ { m e t a }$ are all in the neural network $f$ . To minimize $L _ { m e t a }$ , $f ( x _ { i } ; \theta )$ and $f ( x _ { j } ; \theta )$ must output a sharply peaked distribution with the peak happening only at the same output node when $s _ { i j } = 1$ In the case of $s _ { i j } = 0$ , the two distributions must have as little overlap as possible to minimize the loss. In the latter case, the two samples are pushed to be activated at the output nodes of different classes. Both properties of $f$ ’s output distribution are typical characteristics of a classifier learned with class labels and using multi-class cross-entropy. The properties also illustrate the intuition of why minimizing $L _ { m e t a }$ helps $f$ learn outputs similar to a multi-class classifier.
+
+Lastly, because of the likelihood nature of $L _ { m e t a }$ , we call the learning criterion a Meta Classification Likelihood (MCL) in the rest of the paper.
+
+# 4 LEARNING PARADIGMS
+
+The supervision used in MCL is the pairwise labeling $S$ . Due to its weaker form compared to class labels, we have the flexibility to collect it in a supervised, cross-task transfer, or semi-supervised manner. The collection method determines the learning paradigms. In the first two learning paradigms, other methods (see Related Work Section) have also used pair-wise constraints similarly; our novelty is the derivation of our new learning objective, MCL, which can replace the other objectives. In the semi-supervised learning scenario, the proposed Pseudo-MCL is a new method. Details are elaborated below.
+
+![](images/9d3e505b59279793334591fba2b346abc0cf8df1670faa6f0b53ee8d9562b4a1.jpg)  
+Figure 3: The training flows for each learning paradigm. $X _ { L }$ represents the labeled data with class label $Y _ { L }$ . $X _ { U L }$ is unlabeled data. $\hat { S }$ is the predicted pairwise similarity while $S$ is used as the learning target. The similarity prediction network (SPN) in (b) is learned on a labeled auxiliary dataset and transferred to the target dataset $X _ { U L }$ .
+
+# 4.1 SUPERVISED LEARNING
+
+Supervised pairwise labeling can be directly collected from humans, or converted from existing class labeling by having $S = \{ s _ { i j } \} _ { 1 \leq i , j \leq n }$ , where $s _ { i j } = 1$ if $x _ { i }$ and $x _ { j }$ belong to the same class, otherwise $s _ { i j } = 0$ . In our experiments, we use the latter setting to enable comparison to other supervised algorithms. Figure 3a illustrates the training process.
+
+# 4.2 UNSUPERVISED LEARNING
+
+Pairwise labeling can come from several natural cues, such as spatial and temporal proximity. For example, the patches in an image can be similar because of their spatial closeness, and the frames of video in a short time usually have similar content. Additionally, useful pairwise information can be found in the edges in social networks or in the network of academic citations. All of the above are potential applications of this work.
+
+Another strategy that is unsupervised in the target domain is to collect pairwise labels through transfer learning. Hsu et al. (2018) proposes a method in which a similarity prediction network (SPN) can be learned from a labeled auxiliary dataset. Then the SPN is applied on the unlabeled target dataset to predict $S$ (the probability of being in the same class). In the last step, the predicted $S$ is fed into a network (in that case optimized via Kullback–Leibler divergence based contrastive loss) to discover the categories in the unlabeled target dataset. Figure 3b illustrates above process. Note that the classes between the auxiliary dataset and target dataset may have an overlap (cross-domain transfer) or not (cross-task transfer) (Hsu et al., 2018). In both cases, the predicted pairwise similarity is noisy (especially in the latter case); therefore the transfer learning strategy creates a challenging scenario for learning classifiers. Its difficulty makes it a good benchmark to evaluate the robustness of our methods and is used in our experiments.
+
+# 4.3 SEMI-SUPERVISED LEARNING
+
+We propose a new strategy to obtain the $S$ for semi-supervised learning. Figure 3c illustrates the method under the typical semi-supervised learning setting, which takes a common dataset $D$ used for supervised learning and discards the labels for most of the dataset. The labeled and unlabeled portions in $D$ are $D _ { L } = \left( X _ { L } , Y _ { L } \right)$ and $D _ { U L } = X _ { U L }$ correspondingly. The main idea is to create a pseudosimilarity $S _ { L + U L }$ for the meta classifier (similar to Pseudo-Labeling (Lee, 2013)) by binarizing the predicted $\hat { S } _ { L + U L }$ at probability 0.5. We call the method Pseudo-MCL, and we note that here interestingly $g$ is not static as it iteratively improves as $f$ improves. Another way to create similarity is data augmentation, inspired by the $\Pi$ -model (Laine & Aila, 2017) or Stochastic Perturbations (Sajjadi et al., 2016). An image perturbed in different ways naturally belong to the same class, and thus provides free ground-truth similarity. The similarity from both methods can be easily combined to $S _ { L + U L }$ by having a logical-OR operation for the two binarized similarities. The learning objective is the sum of the multi-class cross-entropy and Pseudo-MCL, so the mapping between output nodes and classes are automatically decided by the supervised part of learning.
+
+# 5 EXPERIMENTS
+
+# 5.1 EXPERIMENTAL SETUP AND NETWORK OPTIMIZATION
+
+In all experiments, we use a standard gradient-based method for training a neural network by optimizing the learning criterion. For example, with stochastic gradient descent, we calculate MCL within a mini-batch of data. In that case, the $i$ and $j$ correspond to the index of data in a mini-batch $b$ . The outputs of $f ( \cdot ; \theta )$ are enumerated in $| b | ( | b | - 1 ) / 2$ pairs in a mini-batch before calculation of MCL. Our empirical finding is that this enumeration introduces a negligible overhead to the training time. We also note that for large datasets, this only samples from the full set of pairwise information.
+
+One limitation of learning a classifier without class labels is losing the mapping (the identifiability) between the output nodes and the semantic class. A simple method to obtain the mapping is by using a part of the training data with class labels and assigning the output nodes to the dominant class which activates the node (here we obtain the optimal assignment by the Hungarian algorithm (Kuhn, 1955), which is commonly used in evaluating the clustering accuracy (Yang et al., 2010)). Note, however, that for unsupervised problems we do not need to do this except to quantitatively evaluate our method; otherwise the outputs can be seen as arbitrary clusters.
+
+# 5.2 SUPERVISED LEARNING WITH WEAK LABELS
+
+This section empirically compares MCL to multi-class cross-entropy (CE) and the strong baseline using pairwise similarity (Kullback–Leibler divergence based contrastive loss (KCL) (Hsu & Kira, 2016; Hsu et al., 2018)), in a supervised learning setting. Specifically, we would like to demonstrate that we can achieve similar classification rates as cross-entropy (the standard objective for multi-class classification) using only pairwise similarity, and show that the previous pairwise criterion cannot do this likely due to a poor loss landscape. We compare the classification accuracy of these criteria with varied network depths and varied dataset difficulty. The visualization of loss landscape is provided in Appendix A. The formulation of KCL and how it relates to MCL is available in Appendix B.
+
+# 5.2.1 QUANTITATIVE ANALYSIS
+
+We compare the classification accuracy on three image datasets: MNIST (LeCun, 1998) is a 10-class handwritten digit dataset with 60000 images for training, and 10000 for testing; CIFAR10 and CIFAR100 (Krizhevsky, 2009) instances are colored $3 2 \times 3 2$ images of objects such as cat, dog, and ship. They both have 50000 images for training and 10000 for testing.
+
+Network Architectures: We use convolution neural networks with a varied number of layers: LeNet (LeCun et al., 1998) and VGG (Simonyan & Zisserman, 2014). We add VGG8, which only has one convolution layer before each pooling layer, as the supplement between LeNet and VGG11. The list of architectures also includes ResNet (He et al., 2016a) with pre-activation (He et al., 2016b)). The number of output nodes $K$ in the last fully connected layer is set to the true number of categories for this section. Since the learning objectives KCL and MCL both work on pairs of inputs, we have a pairwise enumeration layer (Hsu et al., 2018) between the network outputs and the loss function.
+
+Training Configurations: All networks in this section are trained from scratch with randomly initialized weights. By default, we use Adam (Kingma & Ba, 2014) to optimize the three criterion with mini-batch size 100 and initial learning rate 0.001. On MNIST the learning rate was dropped every 10 epochs by a factor of 0.1 with 30 epochs in total. On CIFAR10/100 we use the same setting except that the learning rate is dropped at 80 and 120 epochs with 140 epochs in total. For CIFAR100, the mini-batch size was 1000 and the learning rate dropped at epoch 100 and 150 with 180 epochs in total. In the experiments with ResNet, we use SGD instead of Adam since SGD converges to a higher accuracy when keeping other settings the same as above. The learning rate for SGD starts with 0.1 and decays with a factor of 0.1 at the number of epochs described above.
+
+Table 1: The classification error rate (lower is better) on three datasets with different objective functions and different neural network architectures. CE denotes that the network uses class-specific labels for training with a multi-class cross-entropy. MCL only uses the binarized similarity for learning with the meta-classification criterion. KCL is a strong baseline which also uses binarized similarity. The \* symbol indicates the worst cases of KCL. The performance in parenthesis means its network uses a better initialization (VGG16 and VGG8) or a learning schedule which is 10 times longer (VGG11). The two treatments are discussed in Section 5.2.1. We only use VGG8 for CIFAR100 since KCL performs the best with it on CIFAR10. Each value is the average of 3 runs.   
+
+<table><tr><td>Dataset</td><td>#class</td><td>Network</td><td>(Class label)</td><td colspan="2">(Pairwise label)</td></tr><tr><td>MNIST</td><td></td><td></td><td>CE</td><td>KCL</td><td>MCL</td></tr><tr><td></td><td>10</td><td>LeNet LeNet</td><td>0.6% 14.9%</td><td>0.5% 16.4%</td><td>0.6% 15.1%</td></tr><tr><td rowspan="8">CIFAR10</td><td rowspan="8">10</td><td>VGG8</td><td></td><td>10.2%</td><td></td></tr><tr><td></td><td>10.2%</td><td></td><td>10.2%</td></tr><tr><td>VGG11</td><td>8.9%</td><td>72.2(10.4)%</td><td>9.4%</td></tr><tr><td>VGG16</td><td>7.6%</td><td>*81.1(10.3)%</td><td>8.3%</td></tr><tr><td>ResNet18</td><td>6.7%</td><td>73.8%</td><td>6.6%</td></tr><tr><td>ResNet34</td><td>6.6%</td><td>79.3%</td><td>6.3%</td></tr><tr><td>ResNet50</td><td>6.6%</td><td>79.6%</td><td>5.9%</td></tr><tr><td>ResNet101</td><td>6.5%</td><td>79.9%</td><td>5.6%</td></tr><tr><td>CIFAR100</td><td>100</td><td>VGG8</td><td>35.4%</td><td>*45.3(40.2)%</td><td>36.1%</td></tr></table>
+
+Results and discussion: The results in Table 1 show that MCL achieves similar classification performance as CE with different network depths and three datasets. In contrast, KCL has degenerate performance when the networks are deeper or the dataset is more difficult. This might be due to a limitation of using KL-divergence, specifically that when two probability distributions are the same, the divergence will be zero no matter what the values are. This property may introduce bad local minima or small gradients for learning. To investigate such a perspective, we apply two strategies. First, we use a large learning rate (0.2) with SGD to avoid bad local minima and make the training schedule 10 times longer for exploring the parameter space. This setting helps KCL with VGG11, in that the error rate drops from $7 2 . 2 \%$ to $1 0 . 4 \%$ , but not with VGG16 (from $8 1 . 1 \%$ to $7 6 . 8 \%$ ). In the second strategy, we select the worst conditions (the values with \* notion) in Table 1 for KCL and pre-train the networks with only $4 \mathrm { k \Omega }$ labels with CE to initialize the networks. Then we use KCL with the full training set to finish the training. With a better initialization, KCL can reach a performance close to CE and MCL. The performance is shown with parenthesis in Table 1. The results of both strategies indicate that KCL has bad local minima or plateaus in its loss surface (see Section A in Appendix). Unlike KCL, MCL can converge to a performance close to CE with random initialization in all of our experiments. Furthermore, MCL outperforms CE with a deeper network (error rate $5 . 6 \%$ versus $6 . 5 \%$ with ResNet101). Such a result indicates that MCL is less prone to overfitting (in the Table 1, all ResNets achieve a training error less than $0 . 1 \%$ ).
+
+# 5.3 UNSUPERVISED CROSS-TASK TRANSFER LEARNING
+
+The second experiment follows the transfer learning scenario proposed by Hsu et al. (2018) and is summarized in Section 4.2. This scenario has two settings. The first is when the number of output nodes $K$ equal to the number of ground truth classes $C$ in a dataset. This setting is the same as a multi-class classification task, except no labels (both class labels or similarity labels) are provided in the target dataset. The second setting is having an unknown $C$ , which is closer to a clustering problem. One strategy to address the unknown $C$ is to set a large $K$ , and we rely on the clustering algorithm to use only a necessary number of clusters to describe the dataset while leaving the extra clusters empty.
+
+Table 2: Unsupervised cross-task transfer learning on Omniglot. The performance (higher is better) is averaged across 20 alphabets (datasets), in which each has 20 to 47 letters (classes). The ACC and NMI without brackets have the number of output nodes $K$ equal to the true number of classes in a dataset, while columns with " $\mathrm { { K = } } 1 0 0 )$ " represent the case where the number of classes is unknown and a fixed $K = 1 0 0$ is used.   
+
+<table><tr><td>Method</td><td>ACC</td><td>ACC (K=100)</td><td>NMI</td><td>NMI (K=100)</td></tr><tr><td>K-means (MacQueen et al., 1967)</td><td>21.7%</td><td>18.9%</td><td>0.353</td><td>0.464</td></tr><tr><td>LPNMF (Cai et al., 2009)</td><td>22.2%</td><td>16.3%</td><td>0.372</td><td>0.498</td></tr><tr><td>LSC (Chen &amp; Cai,2011)</td><td>23.6%</td><td>18.0%</td><td>0.376</td><td>0.500</td></tr><tr><td>ITML (Davis et al., 2007)</td><td>56.7%</td><td>47.2%</td><td>0.674</td><td>0.727</td></tr><tr><td>SKKm (Anand et al., 2014)</td><td>62.4%</td><td>46.9%</td><td>0.770</td><td>0.781</td></tr><tr><td>SKLR (Amid et al., 2016)</td><td>66.9%</td><td>46.8%</td><td>0.791</td><td>0.760</td></tr><tr><td>CSP (Wang et al., 2014)</td><td>62.5%</td><td>65.4%</td><td>0.812</td><td>0.812</td></tr><tr><td>MPCK-means (Bilenko et al., 2004)</td><td>81.9%</td><td>53.9%</td><td>0.871</td><td>0.816</td></tr><tr><td>KCL (Hsu et al., 2018)</td><td>82.4%</td><td>78.1%</td><td>0.889</td><td>0.874</td></tr><tr><td>MCL (ours)</td><td>83.3%</td><td>80.2%</td><td>0.897</td><td>0.893</td></tr></table>
+
+We use constrained clustering algorithms as the baselines since they can use the pairwise inputs from a similarity prediction network (SPN) (Hsu et al., 2018). In this section, the same set of binarized pairwise similarity prediction is provided to all algorithms for a fair comparison. The metric in this section is still the classification accuracy. The mapping between output nodes and classes is calculated by the Hungarian algorithm, in which each class only matches to one output node. The unmapped output nodes are all subject to the classification error. We also include the normalized mutual information (NMI) (Strehl & Ghosh, 2002) metric. We use two datasets in the evaluation.
+
+Omniglot (Lake et al., 2015): This dataset has 20 images for each of 1623 different handwritten characters. The characters are from 50 different alphabets and were separated into 30 background sets $( O m n i g l o t _ { b g } )$ ) and 20 evaluation sets $( O m n i g l o t _ { e v a l } )$ by the dataset author. The procedure uses the Omniglotbg set (964 characters in total) to learn the similarity function and applies it to the cross-task transfer learning on the 20 evaluation sets (this same input is used for all compared algorithms). In this test, the backbone network for classification has four convolution layers and has weights randomly initialized. Both MCL and KCL are optimized by Adam with mini-batch size 100.
+
+ImageNet (Deng et al., 2009): The 1000-class dataset is separated into 882-class and 118-class subsets as the random split in Vinyals et al. (2016). The procedure uses ImageN et882 for learning the similarity prediction function and randomly samples 30 classes ( $\mathrm { \sim } 3 9 \mathrm { k }$ images) from $I m a g e N e t _ { 1 1 8 }$ for the unlabeled target data. In this test, the backbone classification network is Resnet-18 and has weights initialized by classification on $I m a g e N e t _ { 8 8 2 }$ . Both learning objectives (KCL and MCL) are optimized by SGD with mini-batch size 100.
+
+Results and Discussion: We follow the evaluation procedure (including network architectures) used in Hsu et al. (2018), therefore the results can be directly compared. The results shown in Table 2 and 3 demonstrate a clear advantage for MCL over other methods. KCL also performs well, but MCL beats its performance with a larger gap when $C$ is unknown (ACC with $\mathrm { K } { = } 1 0 0 $ ). MCL also estimates the number of classes in a dataset better than KCL (Appendix Table 5). The advantage of MCL over KCL in this section is not due to the ease of optimization, since the network is shallow in the Omniglot experiment and the network is pre-trained in the ImageNet experiment. The advantage may due to the fact that MCL is free of hyper-parameters and so performs better than KCL which uses a heuristic threshold $\sigma = 2$ ) (Hsu & Kira, 2016) for its margin.
+
+Table 3: Unsupervised cross-task transfer learning on ImageNet. The values (higher is better) are the average of three random subsets in $I m a g e N e t _ { 1 1 8 }$ . Each subset has 30 classes. The "ACC" has $K = 3 0$ . All methods use the features (outputs of average pooling) from Resnet-18 pre-trained with ImageNet882 classification.   
+
+<table><tr><td>Method</td><td>ACC</td><td>ACC(K=100)</td><td>NMI</td><td>NMI(K=100)</td></tr><tr><td>K-means</td><td>71.9%</td><td>34.5%</td><td>0.713</td><td>0.671</td></tr><tr><td>LSC</td><td>73.3%</td><td>33.5%</td><td>0.733</td><td>0.655</td></tr><tr><td>LPNMF</td><td>43.0%</td><td>21.8%</td><td>0.526</td><td>0.500</td></tr><tr><td>KCL</td><td>73.8%</td><td>65.2%</td><td>0.750</td><td>0.715</td></tr><tr><td>MCL</td><td>74.4%</td><td>71.5%</td><td>0.762</td><td>0.765</td></tr></table>
+
+Table 4: Test error rates (lower is better) obtained by various semi-supervised learning approaches on CIFAR-10 with all but 4,000 labels removed. Supervised refers to using only 4,000 labeled samples from CIFAR-10 without any unlabeled data. All the methods use ResNet-18 and standard data augmentation.   
+
+<table><tr><td>Method</td><td>CIFAR10 4k labels</td></tr><tr><td>Supervised Pseudo-Label</td><td>25.4 ± 1.0% 19.8 ± 0.7%</td></tr><tr><td>II-model VAT</td><td>19.6 ± 0.4% 18.2 ± 0.4%</td></tr><tr><td>SPN-MCL</td><td>22.8 ± 0.5%</td></tr><tr><td>Pseudo-MCL</td><td>18.0 ± 0.4%</td></tr></table>
+
+# 5.4 SEMI-SUPERVISED LEARNING
+
+We evaluate the semi-supervised learning performance of the Pseudo-MCL on the standard benchmark dataset CIFAR-10. The Pseudo-MCL is compared to two state-of-the-art methods, which are VAT (Miyato et al., 2018) and Π-Model (Laine & Aila, 2017; Sajjadi et al., 2016). Our list of baselines additionally includes Pseudo-Labeling (Lee, 2013) and SPN-MCL since they share a similar strategy with Pseudo-MCL. The SPN-MCL uses the same strategy presented in the Section 4.2 for unsupervised learning, except that the SPN is trained with only the labeled portion (e.g. 4k labeled data) of CIFAR10 in this section. We also note that the SPN serves as a static function to provide the similarity for optimizing the regular MCL objective.
+
+Experiment Setting: To construct the $D _ { L }$ , four thousand labeled data are randomly sampled from the training set (50k images) of CIFAR10. This leaves $4 6 \mathrm { k }$ unlabeled data for $D _ { U L }$ . We use 5 random $D _ { L } / D _ { U L }$ splits to calculate the average performance. The images are augmented by the standard procedure which includes random cropping, random horizontal flipping, and normalization to zero mean with unit variance. The model for all method is the ResNet-18 (pre-activation version, He et al. (2016b)), which has no dropout as in a standard model. We use Adam to optimize the objective functions of all methods. The procedure begins with learning the supervised model with only the $4 \mathrm { k }$ labeled data; then all other methods have a fine-tuning with $D _ { L } + D _ { U L }$ based on the learned supervised model. The supervised model (with only 4k data) is trained with initial learning rate 0.001 and a decay with factor 0.1 at epochs 80 and 120 for a total of 140 epochs. All the semi-supervised methods are trained with initial learning rate 0.001 and have a decay with factor 0.1 at epoch 150 and 250 for a total of 300 epochs. We use a shared implementation among all methods so that the major difference between methods is the regularization term in the learning objective. Appendix C.1 provides the description for hyperparameter tuning.
+
+# Results and Discussion:
+
+Table 4 presents the comparison and shows that Pseudo-MCL is on-par with the state-of-the-art method VAT (Miyato et al., 2018). The performance difference between SPN-MCL and Pseudo-MCL clearly demonstrates the benefits of having the binary classifier and the multi-class classifier optimized together. Note that comparing our Table 4 and a recent review (Oliver et al., 2018), we have a lower baseline performance due to a lighter regularization (no dropout) and no extra data augmentation (such as adding Gaussian noise), but the relative ranking between methods is consistent. Therefore we confirm the effectiveness of Pseudo-MCL. Lastly, Pseudo-MCL is free of hyperparameter, which is a very appealing characteristic for learning with few data.
+
+# 6 CONCLUSION
+
+We presented a new strategy to learn a multi-class classification via a binary decision problem. We formulate the problem setting via a probabilistic graphical model and derive a simple likelihood objective that can be effectively optimized via neural networks. We show how this same framework can be used for three learning paradigms: supervised learning, unsupervised cross-task transfer learning, and semi-supervised learning. Results show comparable or improved results over state of the art, especially in the challenging unsupervised cross-task setting. This demonstrates the power of using pairwise similarity as weak labels to relax the requirement of class-specific labeling. We hope the presented perspective of meta classification inspires additional approaches to learning with fewer labeled data (e.g. domain adaptation and few-shot learning) as well as application to domains where weak labels are easier to obtain.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the National Science Foundation and National Robotics Initiative (grant # IIS-1426998) and DARPA’s Lifelong Learning Machines (L2M) program, under Cooperative Agreement HR0011-18-2-001.
+
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+Xiang Wang, Buyue Qian, and Ian Davidson. On constrained spectral clustering and its applications. Data Mining and Knowledge Discovery, pp. 1–30, 2014.
+
+Jason Weston and Chris Watkins. Multi-class support vector machines. Technical report, Citeseer, 1998.
+
+Ting-Fan Wu, Chih-Jen Lin, and Ruby C Weng. Probability estimates for multi-class classification by pairwise coupling. Journal of Machine Learning Research, 5(Aug):975–1005, 2004.
+
+Eric P Xing, Michael I Jordan, Stuart J Russell, and Andrew Y Ng. Distance metric learning with application to clustering with side-information. In Advances in neural information processing systems, pp. 521–528, 2003.
+
+Yi Yang, Dong Xu, Feiping Nie, Shuicheng Yan, and Yueting Zhuang. Image clustering using local discriminant models and global integration. IEEE Transactions on Image Processing, 2010.
+
+# APPENDICES
+
+# A LOSS LANDSCAPE VISUALIZATION
+
+![](images/da713a3cc64f309bbdcbb2faa381060b4b7e668bfc62c97182f97ae8c1c85097.jpg)  
+Figure 4: The loss landscape visualizations. Dark green represents a low loss value while yellow means high value. The bottom part of each diagram is the 2D contour of its 3D surface. The vertical axis of CE is logarithmic to better visualize its dynamic range (Li et al., 2017).
+
+We visualize the three loss functions: CE, MCL, and KCL. The loss surfaces are plotted with the function (Goodfellow et al., 2014; Im et al., 2016; Li et al., 2017):
+
+$$
+f ( \alpha , \beta ) = L ( \theta ^ { * } + \alpha \delta + \beta \eta ; D )
+$$
+
+where $\theta ^ { * }$ are the parameters of the model trained with loss function $L$ and labeled dataset $D = ( X , Y )$ . The $\delta$ and $\eta$ variables are two directions for a 2D projection of $\theta$ . The $\alpha$ and $\beta$ are the amount of shift along $\delta$ and $\eta$ from the origin $\theta ^ { * }$ . This method allows us to better understand the landscape of loss around the solution.
+
+To choose $\delta$ and $\eta$ , one straightforward method is to use random projections. However, it cannot be used to compare the geometry across different networks or loss functions, because of the scale invariance in network weights. One source of such invariance is batch normalization. In such cases, the size (i.e., norm) of a filter (assume a convolution layer) is irrelevant because the output of each layer is re-scaled during batch normalization. Li et al. (2017) propose Filter-wise Normalization to address the above concern. We adopt this strategy to normalize the two random projections and make the relative flatness between loss surfaces comparable. We call this a random projection method.
+
+Another way to choose $\delta$ and $\eta$ is to use solutions from different loss functions. Since we have three loss functions all able to solve the same multi-class classification problem, we can use one solution (e.g. $\theta _ { M C L } ^ { * }$ from MCL) for the $\theta ^ { * }$ and use the remaining two solutions (e.g. $\theta _ { C E } ^ { * }$ and $\theta _ { K C L } ^ { * }$ ) for the two projections (e.g. $\delta = \theta _ { C E } ^ { * } - \theta _ { M C L } ^ { * }$ and $\eta = \theta _ { K C L } ^ { * } - \theta _ { M C L } ^ { * } )$ . We call this a mutual projection method.
+
+Visualization Setting: This section uses CIFAR10 and VGG11. We choose VGG11 because it is the smallest network that KCL cannot be optimized well with a regular learning schedule. For each learning objectives, we use the best-learned models in that error rates are less than $1 0 . 4 \%$ (see Table 1). The parameters of three models $( \theta _ { C E } ^ { * } , \theta _ { M C L } ^ { * } , \theta _ { K C L } ^ { * } )$ are used to construct an interpolated one: $\theta = \theta ^ { * } + \alpha \delta + \beta \eta$ . A $9 1 \mathbf { x } 9 1$ grid is used to enumerate the combinations of $\alpha$ and $\beta$ , which are the scales for the two projected directions. The loss values associated with each $( \alpha , \beta )$ are plotted in the z-direction to form a surface for visualization. Similar to Li et al. (2017), the vertical axis of CE is logarithmic to better visualize its dynamic range. For more details please refer to Li et al. (2017).
+
+Results and Discussion: In the random projection (Figure 4a), the loss landscape with CE is similar to previous work (Li et al., 2017) which shows a nice convexity with a not-too-deep neural network (ResNet18). The solutions of MCL and KCL are both surrounded by a plateau of high loss, but MCL has a wider concave region. The same wide concavity can be seen in the mutual projection (Figure 4b). This is a possible explanation for why MCL still converges to a good local minimum with a randomly initialized network. Besides, the mutual projection shows that the geometry of MCL’s loss landscape is similar to CE’s surface, while KCL has a sharp low-loss region only around its solutions. This might be a reason why it requires a prolonged training schedule to find a good local minimum. Overall, MCL is qualitatively more similar to CE in the visualization of loss landscape.
+
+# B KCL VERSUS MCL
+
+From the view of optimization objective, the KLD-based Contrastive Loss (KCL) has a form close to our MCL although it is originally designed for clustering. In the KCL paper (Hsu & Kira, 2016; Hsu et al., 2018), it interprets the softmax output of a neural network as outputting a probability distribution over cluster assignments. Then a contrastive loss function is defined using KL-divergence to measure the distance between two distributions $\hat { \mathbf { y } } _ { i } = f ( x _ { i } ; \theta )$ and $\hat { \mathbf { y } } _ { j } ~ = ~ f ( x _ { j } ; \theta )$ . The cost between a similar pair $( x _ { i } , x _ { j } )$ , in which $s _ { i j } = 1$ , is given by:
+
+$$
+\begin{array} { r } { L _ { K C L } ^ { + } ( x _ { i } , x _ { j } ) = D _ { \mathrm { K L } } ( \hat { \bf y } _ { i } | | \hat { \bf y } _ { j } ) + D _ { \mathrm { K L } } ( \hat { \bf y } _ { j } | | \hat { \bf y } _ { i } ) . } \end{array}
+$$
+
+If $( x _ { i } , x _ { j } )$ is a dissimilar pair $( s _ { i j } = 0 )$ ), then $\hat { \mathbf { y } } _ { i }$ and ${ \hat { \mathbf { y } } } _ { j }$ are expected to be different distributions, which is described by a hinge-loss function with a hyper-parameter $\sigma$ for the margin.
+
+$$
+\begin{array} { r } { L _ { K C L } ^ { - } ( x _ { i } , x _ { j } ) = L _ { h } ( D _ { \mathrm { K L } } ( \hat { \bf y } _ { i } | | \hat { \bf y } _ { j } ) , \sigma ) + L _ { h } ( D _ { \mathrm { K L } } ( \hat { \bf y } _ { j } | | \hat { \bf y } _ { i } ) , \sigma ) , } \end{array}
+$$
+
+Then the total contrastive loss (KCL) has the form:
+
+$$
+L _ { K C L } = \sum _ { i , j } s _ { i j } L _ { K C L } ^ { + } ( x _ { i } , x _ { j } ) + ( 1 - s _ { i j } ) L _ { K C L } ^ { - } ( x _ { i } , x _ { j } ) .
+$$
+
+In comparing KCL and MCL, we find that they are similar in using pairwise similarity and have no requirement on the number of output nodes $K$ no matter what the true number of classes $C$ is. They can also be plugged into the training of neural networks in the same way, in that switching MCL to KCL can easily be done by replacing the learning criterion. Although they are similar in terms of usage, their formulation has a fundamental difference. KCL is inspired by metric learning, in that KL-divergence is the metric for evaluating the pairwise distance. Our MCL is inspired by the concept of meta classification learning and explained by a maximum likelihood estimation. The most significant difference is that MCL is free of hyperparameter. Therefore MCL does not require cross-validation for hyperparameter tuning. This property is crucial for unsupervised learning or when only a few instances of labeled data are available.
+
+Table 5: Estimates for the number of characters across the 20 datasets in $O m n i g l o t _ { e v a l }$ when $C$   
+is unknown. The bold number means the prediction has error smaller or equal to 3. The number   
+of dand inant clusters is defiis the size of cluster d by . For $\begin{array} { r } { N D C = \sum _ { i = 1 } ^ { K } \left[ C _ { i } > = E [ C _ { i } ] \right] } \end{array}$ , where e alpha $[ \cdot ]$ is an Iverson Bracket has 1000 images and $C _ { i }$ $i$ $E [ C _ { i } ]$   
+$K = 1 0 0$ . The $A D i f$ represents average difference (Hsu et al., 2018).
+
+<table><tr><td>Alphabet</td><td>#class</td><td>SKMS</td><td>KCL</td><td>MCL</td></tr><tr><td>Angelic</td><td>20</td><td>16</td><td>26</td><td>22</td></tr><tr><td>Atemayar Q.</td><td>26</td><td>17</td><td>34</td><td>26</td></tr><tr><td>Atlantean</td><td>26</td><td>21</td><td>41</td><td>25</td></tr><tr><td>Aurek_Besh</td><td>26</td><td>14</td><td>28</td><td>22</td></tr><tr><td>Avesta</td><td>26</td><td>8</td><td>32</td><td>23</td></tr><tr><td>Ge_ez</td><td>26</td><td>18</td><td>32</td><td>25</td></tr><tr><td>Glagolitic</td><td>45</td><td>18</td><td>45</td><td>36</td></tr><tr><td>Gurmukhi</td><td>45</td><td>12</td><td>43</td><td>31</td></tr><tr><td>Kannada</td><td>41</td><td>19</td><td>44</td><td>30</td></tr><tr><td>Keble</td><td>26</td><td>16</td><td>28</td><td>23</td></tr><tr><td>Malayalam</td><td>47</td><td>12</td><td>47</td><td>35</td></tr><tr><td>Manipuri</td><td>40</td><td>17</td><td>41</td><td>33</td></tr><tr><td>Mongolian</td><td>30</td><td>28</td><td>36</td><td>29</td></tr><tr><td>Old Church S.</td><td>45</td><td>23</td><td>45</td><td>38</td></tr><tr><td>Oriya</td><td>46</td><td>22</td><td>49</td><td>32</td></tr><tr><td>Sylheti</td><td>28</td><td>11</td><td>50</td><td>30</td></tr><tr><td>Syriac_Serto</td><td>23</td><td>19</td><td>38</td><td>24</td></tr><tr><td>Tengwar</td><td>25</td><td>12</td><td>41</td><td>26</td></tr><tr><td>Tibetan</td><td>42</td><td>15</td><td>42</td><td>34</td></tr><tr><td>ULOG</td><td>26</td><td>15</td><td>40</td><td>27</td></tr><tr><td>ADif</td><td></td><td>16.3</td><td>6.35</td><td>5.1</td></tr></table>
+
+# C EXPERIMENTAL SETTING
+
+# C.1 HYPERPARAMETER TUNING FOR SEMI-SUPERVISED LEARNING
+
+All the semi-supervised learning objectives $L _ { S S L }$ here can be represented as a weighted sum of a supervised term $L _ { s u p }$ and an unsupervised regularization term $L _ { r e g }$ :
+
+$$
+{ \cal L } _ { S S L } = \alpha { \cal L } _ { s u p } ( X _ { L } , Y _ { L } ) + \beta { \cal L } _ { r e g } ( X _ { L } \cup X _ { U L } )
+$$
+
+For a fair comparison, one should give the same budget for tuning the hyperparameters, such as $\alpha$ and $\beta$ . One strategy is applying an exhaustive grid search in the hyperparameter space. Such searching requires doing cross-validation and may not be applicable when the number of labeled data is small. We adopt another strategy that gives zero tuning budget for all. We decide the $\alpha$ and $\beta$ by natural statistics, which is the ratio between the amount of data be seen by the $L _ { s u p }$ and $L _ { r e g }$ . Specifically:
+
+$$
+\alpha = \frac { | D _ { L } | } { | D | + | D _ { L } | } , \beta = \frac { | D | } { | D | + | D _ { L } | }
+$$
+
+One method, VAT (Miyato et al., 2018), has extra hyperparameters (e.g. the $\epsilon$ ) in its design. In that case, we use the values decided in the original paper for this dataset.
+
+# D ASSUMPTIONS IN META CLASSIFICATION LIKELIHOOD
+
+# D.1 SIMPLIFIED LIKELIHOOD
+
+In section 3, the original likelihood (eq. 2) relies on an additional independence assumption to simplify its negative logarithm form to a binary cross-entropy. Such an simplification raises the question of whether equation (3) is over-simplified. For the supervised learning case (Section 4.1 with results in Section 5.2), where the constraints are ground truth, the global solution of our likelihood is also the solution for the original likelihood. This is because if an instance is misclassified, then it will break some pair-wise constraints in both likelihoods and no longer be optimal.
+
+Of course, in practice, there could be two issues. First, the optimization methods for more complex models (e.g. stochastic gradient descent) may find local minima. Although it is hard to show theory for this in the general case, where local optima may be found, in such cases our visualization of the loss landscape (see Appendix A) provides some evidence that our method has a landscape that reduces poor local minima compared to prior work (KCL, Hsu et al. (2018)). The second potential issue is when constraints may be noisy. In such cases, for example, if the noise is high and there is a dependency structure to be leveraged, jointly optimizing across many or all constraints with the original likelihood may provide additional performance (at the expense of tractability). In practice, noisy constraints actually occur in our cross-task transfer learning experiments where our similarity prediction has significant errors (e.g. in Table 3 ImageNet experiments the similar pair precision, similar pair recall, dissimilar pair precision, and dissimilar pair recall are 0.812, 0.655, 0.982, and 0.992 respectively). The strong performance in terms of classification accuracy for the cross-task transfer experiments (Tables 2 and 3) shows that our simplification is robust to noise.
+
+Overall, the fact that we have demonstrated our method on five image datasets and three application scenarios (Section 5.2 for supervised learning, 5.3 for unsupervised cross-task transfer learning, and 5.4 for semi-supervised learning) empirically support that the proposed likelihood can overcome these two issues. It would be interesting future work to develop methods that can incorporate constraints jointly, however.
+
+# D.2 SEPARABILITY ASSUMPTIONS
+
+Note that we assume separability of semantic categories in a dataset. This means that when the constraints are given (supervised learning), there is sufficient information (in the features) to separate or to group the samples. In the case of no given constraints (unsupervised or semi-supervised learning), there is also sufficient information to estimate the pairwise similarity. However, these are common assumptions that are inherent in discriminative models.

parse/train/SJzR2iRcK7/SJzR2iRcK7_content_list.json

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+    {
+        "type": "text",
+        "text": "MULTI-CLASS CLASSIFICATION WITHOUT MULTICLASS LABELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
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+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yen-Chang $\\mathbf { H s u ^ { 1 } }$ , Zhaoyang $\\mathbf { L } \\mathbf { v } ^ { 1 }$ , Joel Schlosser2, Phillip Odom2, and Zsolt Kira12 ",
+        "bbox": [
+            220,
+            169,
+            790,
+            185
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1Georgia Institute of Technology 2Georgia Tech Research Institute 1{yenchang.hsu,zhaoyang.lv,zkira}@gatech.edu 2{joel.schlosser,phillip.odom}@gtri.gatech.edu ",
+        "bbox": [
+            284,
+            185,
+            732,
+            243
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            280,
+            544,
+            295
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This work presents a new strategy for multi-class classification that requires no class-specific labels, but instead leverages pairwise similarity between examples, which is a weaker form of annotation. The proposed method, meta classification learning, optimizes a binary classifier for pairwise similarity prediction and through this process learns a multi-class classifier as a submodule. We formulate this approach, present a probabilistic graphical model for it, and derive a surprisingly simple loss function that can be used to learn neural network-based models. We then demonstrate that this same framework generalizes to the supervised, unsupervised cross-task, and semi-supervised settings. Our method is evaluated against state of the art in all three learning paradigms and shows a superior or comparable accuracy, providing evidence that learning multi-class classification without multi-class labels is a viable learning option. ",
+        "bbox": [
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+            313,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            510,
+            336,
+            526
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One of the most common settings for machine learning is classification, which involves learning a function $f$ to map the input data $x$ to a class label $y \\in \\{ 1 , 2 , . . , C \\}$ . The most successful method for learning such a function is deep neural networks, owing its popularity to its capability to approximate a complex nonlinear mapping between high-dimensional data (e.g. images) and the classes. Despite the success of deep learning, a neural network demands a large amount of class-specific labels for learning a discriminative model, i.e. $P ( y | x )$ . This type of labeling can be expensive to collect, requires a-priori knowledge of all classes, and limits the form of supervision required. For example, the classes may be ambiguous or non-expert human annotators may be able to more easily provide information about whether two instances are of the same class or not, rather than identifying the specific class. A final problem is that different methods are necessary depending on what type of data is available, ranging from supervised learning (known classes) to cross-task unsupervised learning (unknown classes in the target domain) and semi-supervised learning (mix of labeled and unlabeled with known classes). Unsupervised learning with unknown classes is especially difficult to support. ",
+        "bbox": [
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+            542,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To relax these limitations, we propose to reduce the problem of classification to a meta problem that underlies a set of learning problems. Instead of solving the target task directly (learning a multi-class discriminative model such as a neural network), we instead learn a model that does not require explicit class label $y$ but rather a weaker form of information. In the context of classification, the meta problem that we use is a binary decision problem. Note that such a conversion to a different task (e.g. binary) is called a problem reduction method (Allwein et al., 2000) which has had a long history in the literature, especially in ensemble methods and binarization techniques (Galar et al., 2011). The most well-known strategies are \"one-vs-all\" (Anand et al., 1995; Rifkin & Klautau, 2004) and \"one-vs-one\" (Knerr et al., 1990; Hastie & Tibshirani, 1998; Wu et al., 2004). Although they have varied ensembling strategies, all of them share the same task encapsulating scheme, as illustrated in Figure 1a; specifically the binary classifiers are the sub-modules of a multi-class classifier (i.e. the multi-class classifier consists of multiple binary classifiers). These schemes still require that the class label $y$ be available to create the inputs for each binary classifier, and therefore these strategies do not relax the labeling requirements mentioned earlier. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/21e2e1bc7e8d126096adc659175bdd711fd860919532c89a4e452cd8e66aa6fb.jpg",
+        "image_caption": [
+            "Figure 1: Problem reduction schemes for multi-class classification. This work proposes scheme (b), which introduces a binary classifier that captures $s _ { i j }$ . Note that $s _ { i j }$ represents the probability that $x _ { i }$ and $x _ { j }$ belong to the same class. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a novel strategy to address the above limitations. Our method reverses the task encapsulation order so that a multi-class classifier becomes a sub-module of a binary classifier, as illustrated in Figure 1b. The connection between the two classifiers is elucidated in Section 3. There are two highlights in Figure 1b. First, class labels $y _ { i }$ are not required in the learning stage. Instead, our method uses pairs of data $( x _ { i } , x _ { j } )$ as input and pairwise similarity $s _ { i j }$ for the supervision. Second, there is only one binary classifier in the scheme and it is present only during the training stage. In other words, the ephemeral binary task assists the learning of a multi-class classifier without being involved in the inference. When using a neural network with softmax outputs for the multi-class classifier, the proposed scheme can learn a discriminative model with only pairwise information. ",
+        "bbox": [
+            174,
+            406,
+            825,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We specifically make the following contributions: 1) We analyze the problem setting and show that the loss we can use for this encapsulation can be easily derived, and we present an intuitive probabilistic graphical model interpretation for doing so, 2) We evaluate its performance compared to vanilla supervised learning of neural networks which uses multi-class labels, and visualize the loss landscape to better understand the underlying optimization difficulty, and 3) We demonstrate support for learning classifiers in more challenging problem domains, e.g. in unsupervised cross-task transfer and semi-supervised learning. We show how our meta classification framework can support all three learning paradigms, and evaluate it against several state-of-the-art methods. The experimental results show that the same meta classification approach is superior or comparable to state of the art across the three problem domains (supervised learning, unsupervised cross-task learning, and semi-supervised learning), demonstrating flexibility to support even unknown types and numbers of classes. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            715,
+            344,
+            732
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Supervised learning and problem reduction: Allwein et al. (2000) presents a unifying framework for multi-class classification by reducing it to multiple binary problems. The concepts for achieving such reduction, one-vs-all and one-vs-one, have been widely adopted and analyzed (Galar et al., 2011). The two strategies have been used to create several popular algorithms, such as variants of support vector machine (Weston & Watkins, 1998), AdaBoost (Freund & Schapire, 1997; Schapire & Singer, 1999), and decision trees (Fürnkranz, 2003). Despite the long history of reduction, our proposed scheme (Figure 1b) has not been explored. Furthermore, the scheme can be deployed easily by replacing the learning objective, which is fully compatible with deep neural networks for classification, a desirable property for broad applicability. ",
+        "bbox": [
+            174,
+            750,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised cross-task transfer learning: This learning scheme is proposed by Hsu et al. (2018). The method transfers the pairwise similarity as the meta knowledge to an unlabeled dataset of different classes. It then uses a constrained clustering algorithm with predicted pairwise constraints (binarized pairwise similarity) to discover the unseen classes. This learning scheme shares the same supervision (pairwise similarity) as ours, and therefore relates our method to constrained clustering algorithms. One class of such approaches uses the constraints to learn a distance metric, and then applies a generic clustering algorithm such as K-means or hierarchical clustering to obtain the cluster assignments. This includes DML (Xing et al., 2003), ITML (Davis et al., 2007), SKMS (Anand et al., 2014), SKKm (Anand et al., 2014; Amid et al., 2016), and SKLR (Amid et al., 2016). The second class of methods incorporates the constraints into the cluster assignment objective. Some constrained spectral clustering algorithms, e.g. CSP (Wang et al., 2014) and COSC (Rangapuram & Hein, 2012) use this strategy. There are also approaches combine both distance metric learning and a clustering objective jointly, such as MPCKMeans (Bilenko et al., 2004), CECM (Antoine et al., 2012), and Kullback–Leibler divergence based contrastive loss (KCL) (Hsu & Kira, 2016; Hsu et al., 2018). Our new learning objective for Figure 1b can replace the above objectives in the cross-task transfer learning scheme. ",
+        "bbox": [
+            176,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/bef284988ef65f5658dca7760ad85e579de870af6d3ea3572b11cccca6ef2cd5.jpg",
+        "image_caption": [
+            "Figure 2: Graphical representation for the meta classification task; $X _ { i }$ represents the node of input data, $Y _ { i }$ represents the class label, $S _ { i j }$ is pairwise similarity between instances $i$ and $j$ , and $\\theta$ represents the neural network parameters. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            374,
+            99,
+            627,
+            188
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            277,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Semi-supervised learning: Our meta classification strategy can easily plug into a semi-supervised learning scheme. Our comparison focuses on state-of-the-art methods which solely involve adding a consistency regularization (Laine & Aila, 2017; Sajjadi et al., 2016; Miyato et al., 2018; Tarvainen & Valpola, 2017) or Pseudo-Labeling (Lee, 2013) for training a neural network. Another line of strategy combines weak supervision, such as similar pairs, and unlabeled data to learn a binary classifier (Bao et al., 2018). We present a new method by augmenting Figure 1b with the Pseudo-Labeling strategy. ",
+        "bbox": [
+            173,
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+            550
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 META CLASSIFICATION LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            580,
+            496,
+            597
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A natural way to analyze problems with observed and unobserved information is through a probabilistic graphical model. In figure 2, we show the graphical model for our problem, where classspecific labels $Y$ are latent while pairwise similarities $S$ are observed. Specifically, we denote $\\mathbf { X } = \\{ X _ { 1 } , . . , X _ { n } \\}$ , $\\mathbf { Y } = \\{ Y _ { 1 } , . . , Y _ { n } \\}$ , and $\\mathbf { S } = \\{ S _ { i j } \\} _ { 1 \\leq i , j \\leq n }$ to represent the nodes for samples, class labels, and pairwise similarities, respectively. In the model, we have $Y _ { i } \\in \\{ 1 , 2 , . . , C \\}$ and $S _ { i j } \\in \\{ 0 , 1 \\}$ . Then we have $\\mathsf { P } ( S _ { i j } = 1 | Y _ { i } , Y _ { j } ) = 1$ when $Y _ { i } = Y _ { j }$ and zero probability otherwise; similarly, $\\bar { \\mathsf { P } } ( S _ { i j } = 0 | Y _ { i } , Y _ { j } ) = \\bar { 1 }$ when $Y _ { i } \\neq Y _ { j }$ . The output of a discriminative classifier with parameters $\\theta$ is $f ( x _ { i } ; \\theta ) = \\mathsf { P } ( Y _ { i } | x _ { i } ; \\theta )$ , where $f ( \\overline { { \\boldsymbol { x } } } _ { i } ; \\boldsymbol { \\theta } )$ outputs a categorical distribution. Now we describe the likelihood that the model explains the observed labeling (either with class labeling or pairwise labeling). ",
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+        "type": "equation",
+        "img_path": "images/18b1294ab29e9d32632e164e867369d1bb7758a2471a5569213a1c5b6a537bcd.jpg",
+        "text": "$$\n\\mathcal { L } ( \\boldsymbol { \\theta } ; \\mathbf { X } , \\mathbf { Y } , \\mathbf { S } ) = \\mathbb { P } ( \\mathbf { X } , \\mathbf { Y } , \\mathbf { S } ; \\boldsymbol { \\theta } ) = \\mathbb { P } ( \\mathbf { S } | \\mathbf { Y } ) \\mathbb { P } ( \\mathbf { Y } | \\mathbf { X } ; \\boldsymbol { \\theta } ) \\mathbb { P } ( \\mathbf { X } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "When $\\mathbf { S }$ is fully observed while $\\mathbf { Y }$ is unknown, calculating the likelihood requires marginalizing $\\mathbf { Y }$ by computing $\\begin{array} { r l } { \\sum _ { \\mathbf { Y } } \\mathbb { P } ( \\mathbf { S } | \\mathbf { Y } ) \\mathbb { P } ( \\mathbf { Y } | \\mathbf { X } ; \\boldsymbol { \\theta } ) } & { { } } \\end{array}$ , which is intractable. The pairwise term $\\begin{array} { r } { \\mathsf { P } ( \\mathbf { S } | \\mathbf { Y } ) = } \\end{array}$ $\\begin{array} { r } { \\prod _ { i , j } \\mathbb { P } ( S _ { i j } | Y _ { i } , Y _ { j } ) } \\end{array}$ makes all $Y _ { i }$ dependent on each other and prohibits efficient factorization. Thus, we approximate the computation by imposing additional independences such that $S i j \\perp \\bf { S } \\backslash \\{ \\it { S } _ { i j } \\} \\lvert \\it { X } _ { i } , \\it { X } _ { j }$ (see Appendix $\\mathbf { D }$ for a discussion of these). Now we can compute the likelihood with the observed nodes $X _ { i } = x _ { i }$ and $S _ { i j } = s _ { i j }$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/847371e9bd6ac7c9e94a54e66af4b64c8c936daa9011522f6e928a07567b0b94.jpg",
+        "text": "$$\n\\begin{array} { r l } & { { \\mathcal { L } } ( \\boldsymbol { \\theta } ; \\mathbf { X } , \\mathbf { S } ) \\approx \\displaystyle \\sum _ { \\mathbf { Y } } \\mathbb { P } ( \\mathbf { S } | \\mathbf { Y } ) \\mathbb { P } ( \\mathbf { Y } | \\mathbf { X } ; \\boldsymbol { \\theta } ) } \\\\ & { \\quad \\quad \\approx \\displaystyle \\prod _ { i , j } \\Big ( \\displaystyle \\sum _ { Y _ { i } = Y _ { j } } \\mathbb { 1 } \\big [ s _ { i j } = \\boldsymbol { 1 } ] \\mathbb { P } ( Y _ { i } | x _ { i } ; \\boldsymbol { \\theta } ) \\mathbb { P } ( Y _ { j } | x _ { j } ; \\boldsymbol { \\theta } ) + } \\\\ & { \\quad \\quad \\quad \\displaystyle \\sum _ { Y _ { i } \\neq Y _ { j } } \\mathbb { 1 } \\big [ s _ { i j } = 0 \\big ] \\mathbb { P } ( Y _ { i } | x _ { i } ; \\boldsymbol { \\theta } ) \\mathbb { P } ( Y _ { j } | x _ { j } ; \\boldsymbol { \\theta } ) \\Big ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Equation (2) omits the $\\mathtt { P } ( \\mathbf { X } )$ since $\\mathbf { X }$ are observed leaf nodes. It is straightforward to take a negative logarithm on equation 3 and derive a loss function: ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/be148a9bec5214d60ff5cc5d943523a76e9278ad834f11d111a88b849005577b.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle { L _ { m e t a } ( \\theta ) = - \\sum _ { i , j } \\log \\bigg ( \\sum _ { Y _ { i } = Y _ { j } } \\mathbb { I } [ s _ { i j } = 1 ] \\mathbb { P } ( Y _ { i } | x _ { i } ; \\theta ) \\mathbb { P } ( Y _ { j } | x _ { j } ; \\theta ) + } } \\\\ { \\displaystyle { \\sum _ { Y _ { i } \\neq Y _ { j } } \\mathbb { I } [ s _ { i j } = 0 ] \\mathbb { P } ( Y _ { i } | x _ { i } ; \\theta ) \\mathbb { P } ( Y _ { j } | x _ { j } ; \\theta ) \\bigg ) } } \\\\ { = - \\sum _ { i , j } s _ { i j } \\log ( f ( x _ { i } ; \\theta ) ^ { T } f ( x _ { j } ; \\theta ) ) + ( 1 - s _ { i j } ) \\log ( 1 - f ( x _ { i } ; \\theta ) ^ { T } f ( x _ { j } ; \\theta ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Then we define the function $g$ by the probability of having the same class label, which is calculated by the inner product between two categorical distributions: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/811f2dd7f4a2910cd109e88993e2beddbd733047840dae23cc620f65ac695949.jpg",
+        "text": "$$\ng ( x _ { i } , x _ { j } , f ( \\cdot , \\theta ) ) = f ( x _ { i } ; \\theta ) ^ { T } f ( x _ { j } ; \\theta ) = \\hat { s } _ { i j }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Here we use $\\hat { s } _ { i j }$ to denote the predicted similarity (as opposed to ground truth similarity $s _ { i j }$ ). By plugging equation 6 into equation 5, $L _ { m e t a }$ has the form of a binary cross-entropy loss: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d9fbf022dde97cde0d5a8a0e11a801a3b9d4c93cd6fce96baa117bd93a9af062.jpg",
+        "text": "$$\nL _ { m e t a } = - \\sum _ { i , j } s _ { i j } \\log \\hat { s } _ { i j } + ( 1 - s _ { i j } ) \\log ( 1 - \\hat { s } _ { i j } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In Figure 1b, the multi-class classifier corresponds to $f$ while the binary classifier corresponds to $g$ . In other words, it is surprisingly simple to wrap a multi-class classifier by a binary classifier as described above. Since there are no learnable parameters in $g$ , the weights optimized with the meta criterion $L _ { m e t a }$ are all in the neural network $f$ . To minimize $L _ { m e t a }$ , $f ( x _ { i } ; \\theta )$ and $f ( x _ { j } ; \\theta )$ must output a sharply peaked distribution with the peak happening only at the same output node when $s _ { i j } = 1$ In the case of $s _ { i j } = 0$ , the two distributions must have as little overlap as possible to minimize the loss. In the latter case, the two samples are pushed to be activated at the output nodes of different classes. Both properties of $f$ ’s output distribution are typical characteristics of a classifier learned with class labels and using multi-class cross-entropy. The properties also illustrate the intuition of why minimizing $L _ { m e t a }$ helps $f$ learn outputs similar to a multi-class classifier. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Lastly, because of the likelihood nature of $L _ { m e t a }$ , we call the learning criterion a Meta Classification Likelihood (MCL) in the rest of the paper. ",
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+    {
+        "type": "text",
+        "text": "4 LEARNING PARADIGMS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "The supervision used in MCL is the pairwise labeling $S$ . Due to its weaker form compared to class labels, we have the flexibility to collect it in a supervised, cross-task transfer, or semi-supervised manner. The collection method determines the learning paradigms. In the first two learning paradigms, other methods (see Related Work Section) have also used pair-wise constraints similarly; our novelty is the derivation of our new learning objective, MCL, which can replace the other objectives. In the semi-supervised learning scenario, the proposed Pseudo-MCL is a new method. Details are elaborated below. ",
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+    {
+        "type": "image",
+        "img_path": "images/9d3e505b59279793334591fba2b346abc0cf8df1670faa6f0b53ee8d9562b4a1.jpg",
+        "image_caption": [
+            "Figure 3: The training flows for each learning paradigm. $X _ { L }$ represents the labeled data with class label $Y _ { L }$ . $X _ { U L }$ is unlabeled data. $\\hat { S }$ is the predicted pairwise similarity while $S$ is used as the learning target. The similarity prediction network (SPN) in (b) is learned on a labeled auxiliary dataset and transferred to the target dataset $X _ { U L }$ . "
+        ],
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+    {
+        "type": "text",
+        "text": "4.1 SUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Supervised pairwise labeling can be directly collected from humans, or converted from existing class labeling by having $S = \\{ s _ { i j } \\} _ { 1 \\leq i , j \\leq n }$ , where $s _ { i j } = 1$ if $x _ { i }$ and $x _ { j }$ belong to the same class, otherwise $s _ { i j } = 0$ . In our experiments, we use the latter setting to enable comparison to other supervised algorithms. Figure 3a illustrates the training process. ",
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+    {
+        "type": "text",
+        "text": "4.2 UNSUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Pairwise labeling can come from several natural cues, such as spatial and temporal proximity. For example, the patches in an image can be similar because of their spatial closeness, and the frames of video in a short time usually have similar content. Additionally, useful pairwise information can be found in the edges in social networks or in the network of academic citations. All of the above are potential applications of this work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Another strategy that is unsupervised in the target domain is to collect pairwise labels through transfer learning. Hsu et al. (2018) proposes a method in which a similarity prediction network (SPN) can be learned from a labeled auxiliary dataset. Then the SPN is applied on the unlabeled target dataset to predict $S$ (the probability of being in the same class). In the last step, the predicted $S$ is fed into a network (in that case optimized via Kullback–Leibler divergence based contrastive loss) to discover the categories in the unlabeled target dataset. Figure 3b illustrates above process. Note that the classes between the auxiliary dataset and target dataset may have an overlap (cross-domain transfer) or not (cross-task transfer) (Hsu et al., 2018). In both cases, the predicted pairwise similarity is noisy (especially in the latter case); therefore the transfer learning strategy creates a challenging scenario for learning classifiers. Its difficulty makes it a good benchmark to evaluate the robustness of our methods and is used in our experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.3 SEMI-SUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We propose a new strategy to obtain the $S$ for semi-supervised learning. Figure 3c illustrates the method under the typical semi-supervised learning setting, which takes a common dataset $D$ used for supervised learning and discards the labels for most of the dataset. The labeled and unlabeled portions in $D$ are $D _ { L } = \\left( X _ { L } , Y _ { L } \\right)$ and $D _ { U L } = X _ { U L }$ correspondingly. The main idea is to create a pseudosimilarity $S _ { L + U L }$ for the meta classifier (similar to Pseudo-Labeling (Lee, 2013)) by binarizing the predicted $\\hat { S } _ { L + U L }$ at probability 0.5. We call the method Pseudo-MCL, and we note that here interestingly $g$ is not static as it iteratively improves as $f$ improves. Another way to create similarity is data augmentation, inspired by the $\\Pi$ -model (Laine & Aila, 2017) or Stochastic Perturbations (Sajjadi et al., 2016). An image perturbed in different ways naturally belong to the same class, and thus provides free ground-truth similarity. The similarity from both methods can be easily combined to $S _ { L + U L }$ by having a logical-OR operation for the two binarized similarities. The learning objective is the sum of the multi-class cross-entropy and Pseudo-MCL, so the mapping between output nodes and classes are automatically decided by the supervised part of learning. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 EXPERIMENTAL SETUP AND NETWORK OPTIMIZATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In all experiments, we use a standard gradient-based method for training a neural network by optimizing the learning criterion. For example, with stochastic gradient descent, we calculate MCL within a mini-batch of data. In that case, the $i$ and $j$ correspond to the index of data in a mini-batch $b$ . The outputs of $f ( \\cdot ; \\theta )$ are enumerated in $| b | ( | b | - 1 ) / 2$ pairs in a mini-batch before calculation of MCL. Our empirical finding is that this enumeration introduces a negligible overhead to the training time. We also note that for large datasets, this only samples from the full set of pairwise information. ",
+        "bbox": [
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+        "type": "text",
+        "text": "One limitation of learning a classifier without class labels is losing the mapping (the identifiability) between the output nodes and the semantic class. A simple method to obtain the mapping is by using a part of the training data with class labels and assigning the output nodes to the dominant class which activates the node (here we obtain the optimal assignment by the Hungarian algorithm (Kuhn, 1955), which is commonly used in evaluating the clustering accuracy (Yang et al., 2010)). Note, however, that for unsupervised problems we do not need to do this except to quantitatively evaluate our method; otherwise the outputs can be seen as arbitrary clusters. ",
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+    {
+        "type": "text",
+        "text": "5.2 SUPERVISED LEARNING WITH WEAK LABELS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This section empirically compares MCL to multi-class cross-entropy (CE) and the strong baseline using pairwise similarity (Kullback–Leibler divergence based contrastive loss (KCL) (Hsu & Kira, 2016; Hsu et al., 2018)), in a supervised learning setting. Specifically, we would like to demonstrate that we can achieve similar classification rates as cross-entropy (the standard objective for multi-class classification) using only pairwise similarity, and show that the previous pairwise criterion cannot do this likely due to a poor loss landscape. We compare the classification accuracy of these criteria with varied network depths and varied dataset difficulty. The visualization of loss landscape is provided in Appendix A. The formulation of KCL and how it relates to MCL is available in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "5.2.1 QUANTITATIVE ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compare the classification accuracy on three image datasets: MNIST (LeCun, 1998) is a 10-class handwritten digit dataset with 60000 images for training, and 10000 for testing; CIFAR10 and CIFAR100 (Krizhevsky, 2009) instances are colored $3 2 \\times 3 2$ images of objects such as cat, dog, and ship. They both have 50000 images for training and 10000 for testing. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Network Architectures: We use convolution neural networks with a varied number of layers: LeNet (LeCun et al., 1998) and VGG (Simonyan & Zisserman, 2014). We add VGG8, which only has one convolution layer before each pooling layer, as the supplement between LeNet and VGG11. The list of architectures also includes ResNet (He et al., 2016a) with pre-activation (He et al., 2016b)). The number of output nodes $K$ in the last fully connected layer is set to the true number of categories for this section. Since the learning objectives KCL and MCL both work on pairs of inputs, we have a pairwise enumeration layer (Hsu et al., 2018) between the network outputs and the loss function. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Training Configurations: All networks in this section are trained from scratch with randomly initialized weights. By default, we use Adam (Kingma & Ba, 2014) to optimize the three criterion with mini-batch size 100 and initial learning rate 0.001. On MNIST the learning rate was dropped every 10 epochs by a factor of 0.1 with 30 epochs in total. On CIFAR10/100 we use the same setting except that the learning rate is dropped at 80 and 120 epochs with 140 epochs in total. For CIFAR100, the mini-batch size was 1000 and the learning rate dropped at epoch 100 and 150 with 180 epochs in total. In the experiments with ResNet, we use SGD instead of Adam since SGD converges to a higher accuracy when keeping other settings the same as above. The learning rate for SGD starts with 0.1 and decays with a factor of 0.1 at the number of epochs described above. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/b545077dc08f421a8f57089cdd13496e9024a42726c6bc0611a1d5e3a0f6242a.jpg",
+        "table_caption": [
+            "Table 1: The classification error rate (lower is better) on three datasets with different objective functions and different neural network architectures. CE denotes that the network uses class-specific labels for training with a multi-class cross-entropy. MCL only uses the binarized similarity for learning with the meta-classification criterion. KCL is a strong baseline which also uses binarized similarity. The \\* symbol indicates the worst cases of KCL. The performance in parenthesis means its network uses a better initialization (VGG16 and VGG8) or a learning schedule which is 10 times longer (VGG11). The two treatments are discussed in Section 5.2.1. We only use VGG8 for CIFAR100 since KCL performs the best with it on CIFAR10. Each value is the average of 3 runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>#class</td><td>Network</td><td>(Class label)</td><td colspan=\"2\">(Pairwise label)</td></tr><tr><td>MNIST</td><td></td><td></td><td>CE</td><td>KCL</td><td>MCL</td></tr><tr><td></td><td>10</td><td>LeNet LeNet</td><td>0.6% 14.9%</td><td>0.5% 16.4%</td><td>0.6% 15.1%</td></tr><tr><td rowspan=\"8\">CIFAR10</td><td rowspan=\"8\">10</td><td>VGG8</td><td></td><td>10.2%</td><td></td></tr><tr><td></td><td>10.2%</td><td></td><td>10.2%</td></tr><tr><td>VGG11</td><td>8.9%</td><td>72.2(10.4)%</td><td>9.4%</td></tr><tr><td>VGG16</td><td>7.6%</td><td>*81.1(10.3)%</td><td>8.3%</td></tr><tr><td>ResNet18</td><td>6.7%</td><td>73.8%</td><td>6.6%</td></tr><tr><td>ResNet34</td><td>6.6%</td><td>79.3%</td><td>6.3%</td></tr><tr><td>ResNet50</td><td>6.6%</td><td>79.6%</td><td>5.9%</td></tr><tr><td>ResNet101</td><td>6.5%</td><td>79.9%</td><td>5.6%</td></tr><tr><td>CIFAR100</td><td>100</td><td>VGG8</td><td>35.4%</td><td>*45.3(40.2)%</td><td>36.1%</td></tr></table>",
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+        "text": "Results and discussion: The results in Table 1 show that MCL achieves similar classification performance as CE with different network depths and three datasets. In contrast, KCL has degenerate performance when the networks are deeper or the dataset is more difficult. This might be due to a limitation of using KL-divergence, specifically that when two probability distributions are the same, the divergence will be zero no matter what the values are. This property may introduce bad local minima or small gradients for learning. To investigate such a perspective, we apply two strategies. First, we use a large learning rate (0.2) with SGD to avoid bad local minima and make the training schedule 10 times longer for exploring the parameter space. This setting helps KCL with VGG11, in that the error rate drops from $7 2 . 2 \\%$ to $1 0 . 4 \\%$ , but not with VGG16 (from $8 1 . 1 \\%$ to $7 6 . 8 \\%$ ). In the second strategy, we select the worst conditions (the values with \\* notion) in Table 1 for KCL and pre-train the networks with only $4 \\mathrm { k \\Omega }$ labels with CE to initialize the networks. Then we use KCL with the full training set to finish the training. With a better initialization, KCL can reach a performance close to CE and MCL. The performance is shown with parenthesis in Table 1. The results of both strategies indicate that KCL has bad local minima or plateaus in its loss surface (see Section A in Appendix). Unlike KCL, MCL can converge to a performance close to CE with random initialization in all of our experiments. Furthermore, MCL outperforms CE with a deeper network (error rate $5 . 6 \\%$ versus $6 . 5 \\%$ with ResNet101). Such a result indicates that MCL is less prone to overfitting (in the Table 1, all ResNets achieve a training error less than $0 . 1 \\%$ ). ",
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+        "type": "text",
+        "text": "5.3 UNSUPERVISED CROSS-TASK TRANSFER LEARNING ",
+        "text_level": 1,
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+        "text": "The second experiment follows the transfer learning scenario proposed by Hsu et al. (2018) and is summarized in Section 4.2. This scenario has two settings. The first is when the number of output nodes $K$ equal to the number of ground truth classes $C$ in a dataset. This setting is the same as a multi-class classification task, except no labels (both class labels or similarity labels) are provided in the target dataset. The second setting is having an unknown $C$ , which is closer to a clustering problem. One strategy to address the unknown $C$ is to set a large $K$ , and we rely on the clustering algorithm to use only a necessary number of clusters to describe the dataset while leaving the extra clusters empty. ",
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+        "table_caption": [
+            "Table 2: Unsupervised cross-task transfer learning on Omniglot. The performance (higher is better) is averaged across 20 alphabets (datasets), in which each has 20 to 47 letters (classes). The ACC and NMI without brackets have the number of output nodes $K$ equal to the true number of classes in a dataset, while columns with \" $\\mathrm { { K = } } 1 0 0 )$ \" represent the case where the number of classes is unknown and a fixed $K = 1 0 0$ is used. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>ACC</td><td>ACC (K=100)</td><td>NMI</td><td>NMI (K=100)</td></tr><tr><td>K-means (MacQueen et al., 1967)</td><td>21.7%</td><td>18.9%</td><td>0.353</td><td>0.464</td></tr><tr><td>LPNMF (Cai et al., 2009)</td><td>22.2%</td><td>16.3%</td><td>0.372</td><td>0.498</td></tr><tr><td>LSC (Chen &amp; Cai,2011)</td><td>23.6%</td><td>18.0%</td><td>0.376</td><td>0.500</td></tr><tr><td>ITML (Davis et al., 2007)</td><td>56.7%</td><td>47.2%</td><td>0.674</td><td>0.727</td></tr><tr><td>SKKm (Anand et al., 2014)</td><td>62.4%</td><td>46.9%</td><td>0.770</td><td>0.781</td></tr><tr><td>SKLR (Amid et al., 2016)</td><td>66.9%</td><td>46.8%</td><td>0.791</td><td>0.760</td></tr><tr><td>CSP (Wang et al., 2014)</td><td>62.5%</td><td>65.4%</td><td>0.812</td><td>0.812</td></tr><tr><td>MPCK-means (Bilenko et al., 2004)</td><td>81.9%</td><td>53.9%</td><td>0.871</td><td>0.816</td></tr><tr><td>KCL (Hsu et al., 2018)</td><td>82.4%</td><td>78.1%</td><td>0.889</td><td>0.874</td></tr><tr><td>MCL (ours)</td><td>83.3%</td><td>80.2%</td><td>0.897</td><td>0.893</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We use constrained clustering algorithms as the baselines since they can use the pairwise inputs from a similarity prediction network (SPN) (Hsu et al., 2018). In this section, the same set of binarized pairwise similarity prediction is provided to all algorithms for a fair comparison. The metric in this section is still the classification accuracy. The mapping between output nodes and classes is calculated by the Hungarian algorithm, in which each class only matches to one output node. The unmapped output nodes are all subject to the classification error. We also include the normalized mutual information (NMI) (Strehl & Ghosh, 2002) metric. We use two datasets in the evaluation. ",
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+    {
+        "type": "text",
+        "text": "Omniglot (Lake et al., 2015): This dataset has 20 images for each of 1623 different handwritten characters. The characters are from 50 different alphabets and were separated into 30 background sets $( O m n i g l o t _ { b g } )$ ) and 20 evaluation sets $( O m n i g l o t _ { e v a l } )$ by the dataset author. The procedure uses the Omniglotbg set (964 characters in total) to learn the similarity function and applies it to the cross-task transfer learning on the 20 evaluation sets (this same input is used for all compared algorithms). In this test, the backbone network for classification has four convolution layers and has weights randomly initialized. Both MCL and KCL are optimized by Adam with mini-batch size 100. ",
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+    {
+        "type": "text",
+        "text": "ImageNet (Deng et al., 2009): The 1000-class dataset is separated into 882-class and 118-class subsets as the random split in Vinyals et al. (2016). The procedure uses ImageN et882 for learning the similarity prediction function and randomly samples 30 classes ( $\\mathrm { \\sim } 3 9 \\mathrm { k }$ images) from $I m a g e N e t _ { 1 1 8 }$ for the unlabeled target data. In this test, the backbone classification network is Resnet-18 and has weights initialized by classification on $I m a g e N e t _ { 8 8 2 }$ . Both learning objectives (KCL and MCL) are optimized by SGD with mini-batch size 100. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Results and Discussion: We follow the evaluation procedure (including network architectures) used in Hsu et al. (2018), therefore the results can be directly compared. The results shown in Table 2 and 3 demonstrate a clear advantage for MCL over other methods. KCL also performs well, but MCL beats its performance with a larger gap when $C$ is unknown (ACC with $\\mathrm { K } { = } 1 0 0 $ ). MCL also estimates the number of classes in a dataset better than KCL (Appendix Table 5). The advantage of MCL over KCL in this section is not due to the ease of optimization, since the network is shallow in the Omniglot experiment and the network is pre-trained in the ImageNet experiment. The advantage may due to the fact that MCL is free of hyper-parameters and so performs better than KCL which uses a heuristic threshold $\\sigma = 2$ ) (Hsu & Kira, 2016) for its margin. ",
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+    {
+        "type": "table",
+        "img_path": "images/0ef5f0c8c5a50775023833c97d618960eea71c9ea94173391bc7d6896cc89848.jpg",
+        "table_caption": [
+            "Table 3: Unsupervised cross-task transfer learning on ImageNet. The values (higher is better) are the average of three random subsets in $I m a g e N e t _ { 1 1 8 }$ . Each subset has 30 classes. The \"ACC\" has $K = 3 0$ . All methods use the features (outputs of average pooling) from Resnet-18 pre-trained with ImageNet882 classification. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>ACC</td><td>ACC(K=100)</td><td>NMI</td><td>NMI(K=100)</td></tr><tr><td>K-means</td><td>71.9%</td><td>34.5%</td><td>0.713</td><td>0.671</td></tr><tr><td>LSC</td><td>73.3%</td><td>33.5%</td><td>0.733</td><td>0.655</td></tr><tr><td>LPNMF</td><td>43.0%</td><td>21.8%</td><td>0.526</td><td>0.500</td></tr><tr><td>KCL</td><td>73.8%</td><td>65.2%</td><td>0.750</td><td>0.715</td></tr><tr><td>MCL</td><td>74.4%</td><td>71.5%</td><td>0.762</td><td>0.765</td></tr></table>",
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/0fdf1c2b61797dddace9fda17c99e4578a8ca38c78b963b9f03195ae5c37fb2f.jpg",
+        "table_caption": [
+            "Table 4: Test error rates (lower is better) obtained by various semi-supervised learning approaches on CIFAR-10 with all but 4,000 labels removed. Supervised refers to using only 4,000 labeled samples from CIFAR-10 without any unlabeled data. All the methods use ResNet-18 and standard data augmentation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>CIFAR10 4k labels</td></tr><tr><td>Supervised Pseudo-Label</td><td>25.4 ± 1.0% 19.8 ± 0.7%</td></tr><tr><td>II-model VAT</td><td>19.6 ± 0.4% 18.2 ± 0.4%</td></tr><tr><td>SPN-MCL</td><td>22.8 ± 0.5%</td></tr><tr><td>Pseudo-MCL</td><td>18.0 ± 0.4%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.4 SEMI-SUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the semi-supervised learning performance of the Pseudo-MCL on the standard benchmark dataset CIFAR-10. The Pseudo-MCL is compared to two state-of-the-art methods, which are VAT (Miyato et al., 2018) and Π-Model (Laine & Aila, 2017; Sajjadi et al., 2016). Our list of baselines additionally includes Pseudo-Labeling (Lee, 2013) and SPN-MCL since they share a similar strategy with Pseudo-MCL. The SPN-MCL uses the same strategy presented in the Section 4.2 for unsupervised learning, except that the SPN is trained with only the labeled portion (e.g. 4k labeled data) of CIFAR10 in this section. We also note that the SPN serves as a static function to provide the similarity for optimizing the regular MCL objective. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Experiment Setting: To construct the $D _ { L }$ , four thousand labeled data are randomly sampled from the training set (50k images) of CIFAR10. This leaves $4 6 \\mathrm { k }$ unlabeled data for $D _ { U L }$ . We use 5 random $D _ { L } / D _ { U L }$ splits to calculate the average performance. The images are augmented by the standard procedure which includes random cropping, random horizontal flipping, and normalization to zero mean with unit variance. The model for all method is the ResNet-18 (pre-activation version, He et al. (2016b)), which has no dropout as in a standard model. We use Adam to optimize the objective functions of all methods. The procedure begins with learning the supervised model with only the $4 \\mathrm { k }$ labeled data; then all other methods have a fine-tuning with $D _ { L } + D _ { U L }$ based on the learned supervised model. The supervised model (with only 4k data) is trained with initial learning rate 0.001 and a decay with factor 0.1 at epochs 80 and 120 for a total of 140 epochs. All the semi-supervised methods are trained with initial learning rate 0.001 and have a decay with factor 0.1 at epoch 150 and 250 for a total of 300 epochs. We use a shared implementation among all methods so that the major difference between methods is the regularization term in the learning objective. Appendix C.1 provides the description for hyperparameter tuning. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Results and Discussion: ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 4 presents the comparison and shows that Pseudo-MCL is on-par with the state-of-the-art method VAT (Miyato et al., 2018). The performance difference between SPN-MCL and Pseudo-MCL clearly demonstrates the benefits of having the binary classifier and the multi-class classifier optimized together. Note that comparing our Table 4 and a recent review (Oliver et al., 2018), we have a lower baseline performance due to a lighter regularization (no dropout) and no extra data augmentation (such as adding Gaussian noise), but the relative ranking between methods is consistent. Therefore we confirm the effectiveness of Pseudo-MCL. Lastly, Pseudo-MCL is free of hyperparameter, which is a very appealing characteristic for learning with few data. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We presented a new strategy to learn a multi-class classification via a binary decision problem. We formulate the problem setting via a probabilistic graphical model and derive a simple likelihood objective that can be effectively optimized via neural networks. We show how this same framework can be used for three learning paradigms: supervised learning, unsupervised cross-task transfer learning, and semi-supervised learning. Results show comparable or improved results over state of the art, especially in the challenging unsupervised cross-task setting. This demonstrates the power of using pairwise similarity as weak labels to relax the requirement of class-specific labeling. We hope the presented perspective of meta classification inspires additional approaches to learning with fewer labeled data (e.g. domain adaptation and few-shot learning) as well as application to domains where weak labels are easier to obtain. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "This work was supported by the National Science Foundation and National Robotics Initiative (grant # IIS-1426998) and DARPA’s Lifelong Learning Machines (L2M) program, under Cooperative Agreement HR0011-18-2-001. ",
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+        "text": "APPENDICES ",
+        "text_level": 1,
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+            103,
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+    {
+        "type": "text",
+        "text": "A LOSS LANDSCAPE VISUALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            137,
+            503,
+            154
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/da713a3cc64f309bbdcbb2faa381060b4b7e668bfc62c97182f97ae8c1c85097.jpg",
+        "image_caption": [
+            "Figure 4: The loss landscape visualizations. Dark green represents a low loss value while yellow means high value. The bottom part of each diagram is the 2D contour of its 3D surface. The vertical axis of CE is logarithmic to better visualize its dynamic range (Li et al., 2017). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We visualize the three loss functions: CE, MCL, and KCL. The loss surfaces are plotted with the function (Goodfellow et al., 2014; Im et al., 2016; Li et al., 2017): ",
+        "bbox": [
+            173,
+            606,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9434bdf3e0f287146fe617b46b97dd060b1a876b6cd8dda83cb779ec4164a0fd.jpg",
+        "text": "$$\nf ( \\alpha , \\beta ) = L ( \\theta ^ { * } + \\alpha \\delta + \\beta \\eta ; D )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            656,
+            606,
+            674
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta ^ { * }$ are the parameters of the model trained with loss function $L$ and labeled dataset $D = ( X , Y )$ . The $\\delta$ and $\\eta$ variables are two directions for a 2D projection of $\\theta$ . The $\\alpha$ and $\\beta$ are the amount of shift along $\\delta$ and $\\eta$ from the origin $\\theta ^ { * }$ . This method allows us to better understand the landscape of loss around the solution. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To choose $\\delta$ and $\\eta$ , one straightforward method is to use random projections. However, it cannot be used to compare the geometry across different networks or loss functions, because of the scale invariance in network weights. One source of such invariance is batch normalization. In such cases, the size (i.e., norm) of a filter (assume a convolution layer) is irrelevant because the output of each layer is re-scaled during batch normalization. Li et al. (2017) propose Filter-wise Normalization to address the above concern. We adopt this strategy to normalize the two random projections and make the relative flatness between loss surfaces comparable. We call this a random projection method. ",
+        "bbox": [
+            173,
+            750,
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+            848
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Another way to choose $\\delta$ and $\\eta$ is to use solutions from different loss functions. Since we have three loss functions all able to solve the same multi-class classification problem, we can use one solution (e.g. $\\theta _ { M C L } ^ { * }$ from MCL) for the $\\theta ^ { * }$ and use the remaining two solutions (e.g. $\\theta _ { C E } ^ { * }$ and $\\theta _ { K C L } ^ { * }$ ) for the two projections (e.g. $\\delta = \\theta _ { C E } ^ { * } - \\theta _ { M C L } ^ { * }$ and $\\eta = \\theta _ { K C L } ^ { * } - \\theta _ { M C L } ^ { * } )$ . We call this a mutual projection method. ",
+        "bbox": [
+            174,
+            853,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Visualization Setting: This section uses CIFAR10 and VGG11. We choose VGG11 because it is the smallest network that KCL cannot be optimized well with a regular learning schedule. For each learning objectives, we use the best-learned models in that error rates are less than $1 0 . 4 \\%$ (see Table 1). The parameters of three models $( \\theta _ { C E } ^ { * } , \\theta _ { M C L } ^ { * } , \\theta _ { K C L } ^ { * } )$ are used to construct an interpolated one: $\\theta = \\theta ^ { * } + \\alpha \\delta + \\beta \\eta$ . A $9 1 \\mathbf { x } 9 1$ grid is used to enumerate the combinations of $\\alpha$ and $\\beta$ , which are the scales for the two projected directions. The loss values associated with each $( \\alpha , \\beta )$ are plotted in the z-direction to form a surface for visualization. Similar to Li et al. (2017), the vertical axis of CE is logarithmic to better visualize its dynamic range. For more details please refer to Li et al. (2017). ",
+        "bbox": [
+            173,
+            102,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Results and Discussion: In the random projection (Figure 4a), the loss landscape with CE is similar to previous work (Li et al., 2017) which shows a nice convexity with a not-too-deep neural network (ResNet18). The solutions of MCL and KCL are both surrounded by a plateau of high loss, but MCL has a wider concave region. The same wide concavity can be seen in the mutual projection (Figure 4b). This is a possible explanation for why MCL still converges to a good local minimum with a randomly initialized network. Besides, the mutual projection shows that the geometry of MCL’s loss landscape is similar to CE’s surface, while KCL has a sharp low-loss region only around its solutions. This might be a reason why it requires a prolonged training schedule to find a good local minimum. Overall, MCL is qualitatively more similar to CE in the visualization of loss landscape. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B KCL VERSUS MCL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            375,
+            370,
+            392
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "From the view of optimization objective, the KLD-based Contrastive Loss (KCL) has a form close to our MCL although it is originally designed for clustering. In the KCL paper (Hsu & Kira, 2016; Hsu et al., 2018), it interprets the softmax output of a neural network as outputting a probability distribution over cluster assignments. Then a contrastive loss function is defined using KL-divergence to measure the distance between two distributions $\\hat { \\mathbf { y } } _ { i } = f ( x _ { i } ; \\theta )$ and $\\hat { \\mathbf { y } } _ { j } ~ = ~ f ( x _ { j } ; \\theta )$ . The cost between a similar pair $( x _ { i } , x _ { j } )$ , in which $s _ { i j } = 1$ , is given by: ",
+        "bbox": [
+            173,
+            411,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/925221fa125381528d286f4409d01ea407876a3f159c4a727ae82342b303f42a.jpg",
+        "text": "$$\n\\begin{array} { r } { L _ { K C L } ^ { + } ( x _ { i } , x _ { j } ) = D _ { \\mathrm { K L } } ( \\hat { \\bf y } _ { i } | | \\hat { \\bf y } _ { j } ) + D _ { \\mathrm { K L } } ( \\hat { \\bf y } _ { j } | | \\hat { \\bf y } _ { i } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            526,
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+            546
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "If $( x _ { i } , x _ { j } )$ is a dissimilar pair $( s _ { i j } = 0 )$ ), then $\\hat { \\mathbf { y } } _ { i }$ and ${ \\hat { \\mathbf { y } } } _ { j }$ are expected to be different distributions, which is described by a hinge-loss function with a hyper-parameter $\\sigma$ for the margin. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2f6e34ac8ab65d64ac6bdbb1a2bc18ba2fc1d1bc56be8c205db207194de93dec.jpg",
+        "text": "$$\n\\begin{array} { r } { L _ { K C L } ^ { - } ( x _ { i } , x _ { j } ) = L _ { h } ( D _ { \\mathrm { K L } } ( \\hat { \\bf y } _ { i } | | \\hat { \\bf y } _ { j } ) , \\sigma ) + L _ { h } ( D _ { \\mathrm { K L } } ( \\hat { \\bf y } _ { j } | | \\hat { \\bf y } _ { i } ) , \\sigma ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            628,
+            705,
+            647
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Then the total contrastive loss (KCL) has the form: ",
+        "bbox": [
+            173,
+            685,
+            508,
+            700
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3f4b29ac8c4ea213bc2defe14aeb88885da1bd80fe4f3d46aef0fcfebf74b5f7.jpg",
+        "text": "$$\nL _ { K C L } = \\sum _ { i , j } s _ { i j } L _ { K C L } ^ { + } ( x _ { i } , x _ { j } ) + ( 1 - s _ { i j } ) L _ { K C L } ^ { - } ( x _ { i } , x _ { j } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            728,
+            692,
+            763
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In comparing KCL and MCL, we find that they are similar in using pairwise similarity and have no requirement on the number of output nodes $K$ no matter what the true number of classes $C$ is. They can also be plugged into the training of neural networks in the same way, in that switching MCL to KCL can easily be done by replacing the learning criterion. Although they are similar in terms of usage, their formulation has a fundamental difference. KCL is inspired by metric learning, in that KL-divergence is the metric for evaluating the pairwise distance. Our MCL is inspired by the concept of meta classification learning and explained by a maximum likelihood estimation. The most significant difference is that MCL is free of hyperparameter. Therefore MCL does not require cross-validation for hyperparameter tuning. This property is crucial for unsupervised learning or when only a few instances of labeled data are available. ",
+        "bbox": [
+            173,
+            784,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Estimates for the number of characters across the 20 datasets in $O m n i g l o t _ { e v a l }$ when $C$   \nis unknown. The bold number means the prediction has error smaller or equal to 3. The number   \nof dand inant clusters is defiis the size of cluster d by . For $\\begin{array} { r } { N D C = \\sum _ { i = 1 } ^ { K } \\left[ C _ { i } > = E [ C _ { i } ] \\right] } \\end{array}$ , where e alpha $[ \\cdot ]$ is an Iverson Bracket has 1000 images and $C _ { i }$ $i$ $E [ C _ { i } ]$   \n$K = 1 0 0$ . The $A D i f$ represents average difference (Hsu et al., 2018). ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/13134581eba67b403ce07dd48b7903a8d0e505a9a5ff7882e3ae27537604967f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Alphabet</td><td>#class</td><td>SKMS</td><td>KCL</td><td>MCL</td></tr><tr><td>Angelic</td><td>20</td><td>16</td><td>26</td><td>22</td></tr><tr><td>Atemayar Q.</td><td>26</td><td>17</td><td>34</td><td>26</td></tr><tr><td>Atlantean</td><td>26</td><td>21</td><td>41</td><td>25</td></tr><tr><td>Aurek_Besh</td><td>26</td><td>14</td><td>28</td><td>22</td></tr><tr><td>Avesta</td><td>26</td><td>8</td><td>32</td><td>23</td></tr><tr><td>Ge_ez</td><td>26</td><td>18</td><td>32</td><td>25</td></tr><tr><td>Glagolitic</td><td>45</td><td>18</td><td>45</td><td>36</td></tr><tr><td>Gurmukhi</td><td>45</td><td>12</td><td>43</td><td>31</td></tr><tr><td>Kannada</td><td>41</td><td>19</td><td>44</td><td>30</td></tr><tr><td>Keble</td><td>26</td><td>16</td><td>28</td><td>23</td></tr><tr><td>Malayalam</td><td>47</td><td>12</td><td>47</td><td>35</td></tr><tr><td>Manipuri</td><td>40</td><td>17</td><td>41</td><td>33</td></tr><tr><td>Mongolian</td><td>30</td><td>28</td><td>36</td><td>29</td></tr><tr><td>Old Church S.</td><td>45</td><td>23</td><td>45</td><td>38</td></tr><tr><td>Oriya</td><td>46</td><td>22</td><td>49</td><td>32</td></tr><tr><td>Sylheti</td><td>28</td><td>11</td><td>50</td><td>30</td></tr><tr><td>Syriac_Serto</td><td>23</td><td>19</td><td>38</td><td>24</td></tr><tr><td>Tengwar</td><td>25</td><td>12</td><td>41</td><td>26</td></tr><tr><td>Tibetan</td><td>42</td><td>15</td><td>42</td><td>34</td></tr><tr><td>ULOG</td><td>26</td><td>15</td><td>40</td><td>27</td></tr><tr><td>ADif</td><td></td><td>16.3</td><td>6.35</td><td>5.1</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C EXPERIMENTAL SETTING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            532,
+            419,
+            549
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C.1 HYPERPARAMETER TUNING FOR SEMI-SUPERVISED LEARNING",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "All the semi-supervised learning objectives $L _ { S S L }$ here can be represented as a weighted sum of a supervised term $L _ { s u p }$ and an unsupervised regularization term $L _ { r e g }$ : ",
+        "bbox": [
+            173,
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+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1e9360aa7d8d0355de954e8e2f98b41e594baffbc271a69395adedb089206e32.jpg",
+        "text": "$$\n{ \\cal L } _ { S S L } = \\alpha { \\cal L } _ { s u p } ( X _ { L } , Y _ { L } ) + \\beta { \\cal L } _ { r e g } ( X _ { L } \\cup X _ { U L } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            637,
+            658,
+            655
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For a fair comparison, one should give the same budget for tuning the hyperparameters, such as $\\alpha$ and $\\beta$ . One strategy is applying an exhaustive grid search in the hyperparameter space. Such searching requires doing cross-validation and may not be applicable when the number of labeled data is small. We adopt another strategy that gives zero tuning budget for all. We decide the $\\alpha$ and $\\beta$ by natural statistics, which is the ratio between the amount of data be seen by the $L _ { s u p }$ and $L _ { r e g }$ . Specifically: ",
+        "bbox": [
+            173,
+            662,
+            825,
+            733
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4270e8eb7ad1fb42dcc0adbd38957e36f7b718e1039cc47ddb0a40a9f68fd19c.jpg",
+        "text": "$$\n\\alpha = \\frac { | D _ { L } | } { | D | + | D _ { L } | } , \\beta = \\frac { | D | } { | D | + | D _ { L } | }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            750,
+            616,
+            784
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "One method, VAT (Miyato et al., 2018), has extra hyperparameters (e.g. the $\\epsilon$ ) in its design. In that case, we use the values decided in the original paper for this dataset. ",
+        "bbox": [
+            171,
+            791,
+            825,
+            820
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D ASSUMPTIONS IN META CLASSIFICATION LIKELIHOOD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            838,
+            669,
+            856
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D.1 SIMPLIFIED LIKELIHOOD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            869,
+            393,
+            883
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In section 3, the original likelihood (eq. 2) relies on an additional independence assumption to simplify its negative logarithm form to a binary cross-entropy. Such an simplification raises the question of whether equation (3) is over-simplified. For the supervised learning case (Section 4.1 with results in Section 5.2), where the constraints are ground truth, the global solution of our likelihood is also the solution for the original likelihood. This is because if an instance is misclassified, then it will break some pair-wise constraints in both likelihoods and no longer be optimal. ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            159
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Of course, in practice, there could be two issues. First, the optimization methods for more complex models (e.g. stochastic gradient descent) may find local minima. Although it is hard to show theory for this in the general case, where local optima may be found, in such cases our visualization of the loss landscape (see Appendix A) provides some evidence that our method has a landscape that reduces poor local minima compared to prior work (KCL, Hsu et al. (2018)). The second potential issue is when constraints may be noisy. In such cases, for example, if the noise is high and there is a dependency structure to be leveraged, jointly optimizing across many or all constraints with the original likelihood may provide additional performance (at the expense of tractability). In practice, noisy constraints actually occur in our cross-task transfer learning experiments where our similarity prediction has significant errors (e.g. in Table 3 ImageNet experiments the similar pair precision, similar pair recall, dissimilar pair precision, and dissimilar pair recall are 0.812, 0.655, 0.982, and 0.992 respectively). The strong performance in terms of classification accuracy for the cross-task transfer experiments (Tables 2 and 3) shows that our simplification is robust to noise. ",
+        "bbox": [
+            174,
+            166,
+            825,
+            347
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Overall, the fact that we have demonstrated our method on five image datasets and three application scenarios (Section 5.2 for supervised learning, 5.3 for unsupervised cross-task transfer learning, and 5.4 for semi-supervised learning) empirically support that the proposed likelihood can overcome these two issues. It would be interesting future work to develop methods that can incorporate constraints jointly, however. ",
+        "bbox": [
+            174,
+            354,
+            825,
+            424
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "D.2 SEPARABILITY ASSUMPTIONS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            440,
+            426,
+            455
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Note that we assume separability of semantic categories in a dataset. This means that when the constraints are given (supervised learning), there is sufficient information (in the features) to separate or to group the samples. In the case of no given constraints (unsupervised or semi-supervised learning), there is also sufficient information to estimate the pairwise similarity. However, these are common assumptions that are inherent in discriminative models. ",
+        "bbox": [
+            174,
+            467,
+            825,
+            536
+        ],
+        "page_idx": 15
+    }
+]

parse/train/WA39qkJvLi/WA39qkJvLi.md

@@ -0,0 +1,249 @@
+# Uncertainty-Driven Loss for Single Image Super-Resolution
+
+Qian $\mathbf { N i n g ^ { 1 } }$ , Weisheng $\mathbf { D o n g } ^ { 1 }$ ∗, Xin Li2, Jinjian $\mathbf { W } \mathbf { u } ^ { 1 }$ , Guangming Shi1 1School of Artificial Intelligence, Xidian University, Xi’an 710071, China 2Lane Dep. of CSEE, West Virginia University, Morgantown WV 26506, USA ningqian@stu.xidian.edu.cn, {wsdong,jinjian.wu}@mail.xidian.edu.cn xin.li@mail.wvu.edu, gmshi@xidian.edu.cn
+
+# Abstract
+
+In low-level vision such as single image super-resolution (SISR), traditional MSE or $\mathcal { L } _ { 1 }$ loss function treats every pixel equally with the assumption that the importance of all pixels is the same. However, it has been long recognized that texture and edge areas carry more important visual information than smooth areas in photographic images. How to achieve such spatial adaptation in a principled manner has been an open problem in both traditional model-based and modern learning-based approaches toward SISR. In this paper, we propose a new adaptive weighted loss for SISR to train deep networks focusing on challenging situations such as textured and edge pixels with high uncertainty. Specifically, we introduce variance estimation characterizing the uncertainty on a pixel-by-pixel basis into SISR solutions so the targeted pixels in a high-resolution image (mean) and their corresponding uncertainty (variance) can be learned simultaneously. Moreover, uncertainty estimation allows us to leverage conventional wisdom such as sparsity prior for regularizing SISR solutions. Ultimately, pixels with large certainty (e.g., texture and edge pixels) will be prioritized for SISR according to their importance to visual quality. For the first time, we demonstrate that such uncertainty-driven loss can achieve better results than $M S E$ or $\mathcal { L } _ { 1 }$ loss for a wide range of network architectures. Experimental results on three popular SISR networks show that our proposed uncertainty-driven loss has achieved better PSNR performance than traditional loss functions without any increased computation during testing. The code is available at https://see.xidian.edu.cn/faculty/wsdong/Projects/UDL-SR.htm
+
+# 1 Introduction
+
+Single image super-resolution (SISR) aims at reconstructing high-resolution (HR) images from their corresponding degraded low-resolution (LR) images. Since the publication of super-resolution with convolutional neural network (SRCNN) [1], there has been a flurry of works on deep learning-based approaches toward SISR - e.g., EDSR [2], DPDNN [3], RCAN [4], SAN [5], and MoG-DUN [6]. The unifying theme along this line of research appears to be that deeper, bigger, and more complex networks can achieve improved SISR performance by facilitating the reconstruction of high-frequency details such as textures and edges in photographic images. Such improvement has been achieved by novel network architectures (e.g., skip connections [2]), new attention mechanism (e.g., residue channel attention [4]), and closed-loop supervision [7]. Surprisingly, most of these existing methods have adopted $M S E$ or $\mathcal { L } _ { 1 }$ loss to optimize the parameters of networks.
+
+The commonly used practice, such as $M S E$ or $\mathcal { L } _ { 1 }$ loss, treats every pixel equally regardless of whether the pixel is in texture/edge regions or smooth areas. The optimality of such non-adaptive loss function has been questioned in the literature of SISR calling for the proposition of perceptual loss function (e.g., [9]). From a Bayesian perspective, the assumption underlying the $M S E$ or $\bar { \mathcal { L } } _ { 1 }$ loss is that each pixel obeys the independent and identically distribution with the same variance. Taking $\mathcal { L } _ { 1 }$ loss as an example, the likelihood of all pixels in an image can be formulated as
+
+![](images/222bd7ad66e205fc8caca654c095a992888dc8615acaff7cc921e0978b1f6352.jpg)  
+Figure 1: Illustration of the difference (d) between HR image (b) and SR image (c) reconstructed by EDSR network [2] on dataset Set14 [8]. The image reconstructed by EDSR network is shown in (c) and (d) shows the absolute difference between the HR image and SR image. Best viewed in color.
+
+$$
+p ( \pmb { x } \mid \pmb { y } , \pmb { W } ) = \prod _ { l = 1 } ^ { M } c \exp ( - \frac { \vert \vert \pmb { x } ^ { ( l ) } - \pmb { f } ^ { ( W ) } ( \pmb { y } ^ { ( l ) } ) \vert \vert _ { 1 } } { \sigma } ) ,
+$$
+
+where $_ { \textbf { \em x } }$ and $\textbf {  { y } }$ denote the pair of HR and LR image, $f ^ { ( W ) } ( \cdot )$ denotes an arbitrary SISR network parameterized by $W$ , and $c , \sigma$ denote spatially invariant constants. However, such assumption of stationarity or spatial invariance of image prior model is invalid for photographic images in the real world. For instance, if one compares the ground-truth (HR image) and the SR image reconstructed by EDSR [2] as shown in Fig. 1 (c), it can be observed that texture areas (e.g., hair of baboon) are not restored as good as smooth areas (e.g., nose of baboon). Fig. 1 (d) depicts the absolute difference between the HR image and reconstructed SR image, from which we can observe spatial variation of the difference map. Such observation implies that the uncertainty of texture and edge areas as characterized by the variance is much larger than that in smooth areas. How to address such uncertainty-driven loss for SISR sets up the stage for this paper.
+
+In this paper, we propose a new adaptive weighted loss (uncertainty-driven loss) for SISR by assigning texture and edge areas with higher weights during the training process. Unlike previous work of perceptual loss [9] focusing on characterizing content and style consistency, we target at explicitly estimating the variance field underlying the unknown HR image in the first step, which can be exploited as an auxiliary signal for guiding the SISR solution in the second step. A direct consequence of our two-step learning approach is that it delivers not only higher visual quality but also improved objective performance such as PSNR and SSIM. Moreover, uncertainty estimation perspective allows us to easily incorporate existing models such as Jefferey’s prior [10, 11] into the proposed SISR solution. It follows that the network training boils down to two sequential steps in which the variance map is estimated from the first step and serves as the attention signal for the second step. The main technical contributions are summarized as follows.
+
+• Uncertainty modeling and estimation. We propose to cast SISR into a Bayesian estimation framework under which SR image (mean) and uncertainty (variance) are derived simultaneously. Unlike previous works in which pixels with large uncertainty are attenuated for high-level vision tasks, we advocate to prioritize them for low-level vision tasks such as SISR.   
+Uncertainty-driven loss (UDL). The estimation of variance map facilitates the training of SISR network by dividing it into two steps. In the first step, an estimating sparsity uncertainty (ESU) loss function was derived from the classical Jeffrey’s prior to estimate the variance map. In the second step, the estimated variance map serves as the guidance signal leading to adaptive weighted loss named uncertainty-driven loss ${ \mathcal { L } } _ { \mathrm { U D L } }$ .   
+Universality of UDL. The proposed uncertainty loss can easily be employed in any existing SISR network to improve performance and do not increase any additional computation cost during testing.   
+• Experimental results on three different baseline networks show that our proposed uncertaintydriven loss has achieved better PSNR performance than traditional $M S E$ or $\mathcal { L } _ { 1 }$ loss.
+
+# 2 Related Work
+
+# 2.1 Uncertainty in Deep Learning
+
+Many works [12–14] have introduced uncertainty into the regression with input-dependent noises problems, and studied the nature and behavior of uncertainty for a long time. More recently, modeling uncertainty in deep learning have improved the performance and robustness of deep networks in many computer vision tasks [15–17] such as image classification [18], image segmentation [15, 16], and face recognition [17, 19]. The uncertainty in deep learning can be roughly divided into two categories [20]. Epistemic/model uncertainty describes how much the model is uncertain about its predictions. Another type is aleatoric/data uncertainty which refers to noise inherent in observation data. In [15], they presented a Bayesian deep learning framework combining aleatoric uncertainty with epistemic uncertainty for per-pixel semantic segmentation and depth regression tasks. Chang et al.[17] investigated the data uncertainty with estimated mean and variance in face recognition. Those uncertainty-based loss function proposed by those works [15–17] can be summarized as
+
+$$
+\mathcal { L } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \frac { | | \pmb { x } _ { i } - \pmb { f } ( \pmb { y } _ { i } ) | | _ { 2 } } { 2 \sigma _ { i } ^ { 2 } } + \frac { 1 } { 2 } \ln \sigma _ { i } ^ { 2 } ,
+$$
+
+where $f ( \pmb { y } _ { i } )$ and $\sigma _ { i } ^ { 2 }$ denote the learned mean and variance respectively. Using above loss function indeed improved their robustness to noisy data. In those tasks, the pixels with high uncertainty were regarded as unreliable pixels which would bear loss attenuation. On the contrary, in SISR tasks, the pixels with high uncertainty (e.g., complex texture or edge areas) should be prioritized since those regions visually more important than pixels in smooth areas. That can explain why applying above loss into SISR directly leads performance decline.
+
+# 2.2 Modeling Uncertainty for SISR
+
+To the best of our knowledge, only two works [21, 22] have studied the behavior of uncertainty for SISR in the open literature. [22] used batch-normalization uncertainty to analyze SISR uncertainty, improving the robustness of the network against adversarial attack. The most recent advance related to our work is Gradient Rescaling Attention Model (GRAM) [21], which analyses the effect of aleatoric/data uncertainty on SISR reconstruction. By decreasing the loss attenuation of large variance pixels, GRAM achieves better results than applying above uncertainty loss into SISR directly. However, GRAM [21] loss remains attenuated when the variance of pixels is high, which contradicts the intuition of prioritizing texture and edge pixels. Thus, GRAM [21] is still inferior to baseline methods since the proposed method fails to prioritize the pixels of large variance. Different from GRAM, we propose a novel uncertainty-driven loss (UDL) to enforce the network concentrating more on the pixels with large variance aiming at better reconstruction of texture and edge regions. By quantifying the uncertainty in SISR under deep Bayesian framework, our proposed method has achieved better results than baseline methods.
+
+# 3 Methodology
+
+Unlike traditional $M S E$ or $\mathcal { L } _ { 1 }$ loss treating every pixel equally, the proposed new adaptive weighted loss for SISR aims at prioritizing texture and edge pixels that are visually more important than pixels in smooth areas. Toward this objective, we first introduce an approach of estimating intermediate results of SR image (mean) and uncertainty (variance) simultaneously in SISR. Then, with Jeffrey’s prior term, a regularized approach of estimating sparse uncertainty is proposed for more accurate uncertainty estimation. An important new insight brought by this paper is that unlike high-level vision tasks where pixels with large uncertainty are assigned lower weights to attenuate their impact $I I 5 J ,$ one should prioritize these pixels in low-level vision tasks such as SISR. Such observation implies that the attenuation of weighting coefficients in loss function needs to be properly translated into the attention mechanism given the specific vision problem as the context.
+
+In previous study [15], it has been shown that explicitly representing aleatoric uncertainty can lead to performance and robustness improvement to noise data in high-level vision tasks such as image segmentation. Such improvement can be explained away by attenuating the weights of pixels with large uncertainty. However, attenuation has to go the opposite direction in low-level vision tasks such as SISR - i.e., larger weights should be assigned to the pixels with high uncertainty (e.g., texture and edge pixels) because they are visually more important than pixels in smooth regions. It should be noted that existing work such as gradient rescaling strategy in GRAM [21] fails to recognize such difference and does not prioritize pixels with high uncertainty. In this paper, we propose a new adaptive weighted loss named uncertainty-driven loss (UDL) for properly turning attenuation into attention for SISR.
+
+![](images/84f5f28ea0e80d3f72e497ee576be39823ce4c585fb5ad50377ffdc6a519dba9.jpg)  
+Figure 2: The overview of training SISR network with proposed ${ \mathcal { L } } _ { \mathrm { U D L } }$ loss. The whole training process can divided into two steps; the first step estimates the uncertainty $\pmb \theta$ precisely and the second step generates the final mean value $f ( \boldsymbol { y } )$ . In step1 shown in (a), the mean value $f ( \boldsymbol { y } )$ and variance $\pmb \theta$ are pretrained by $\mathcal { L } _ { \mathrm { E S U } }$ loss. During step2, as shown in (b), the mean value $f ( y )$ network is trained by ${ \mathcal { L } } _ { \mathrm { U D L } }$ loss, while the network of inferring variance $\pmb \theta$ is fixed. Note that the mean value $f ( \boldsymbol { y } )$ network of step2 starts training from the pretrained network of step1. The Nearest Upsampling denotes interpolation operator.
+
+# 3.1 Estimating Uncertainty (EU) in SISR.
+
+As discussed in [15], there are two classes of uncertainty in Bayesian modeling: aleatoric uncertainty capturing noise inherent in observation data and epistemic uncertainty accounting for uncertainty of model about its predictions. We opt to study the former (aleatoric uncertainty) and explore its application into SISR by designing new uncertainty-driven loss (UDL) functions in this paper. In order to better quantify aleatoric uncertainty in SISR, we use ${ \mathbf { } } _ { \mathbf { } } \mathbf { } _ { \mathbf { } } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf { } \Psi \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf { } \Psi \mathbf \Psi \Psi \mathbf { } \mathbf \Psi \Psi \Psi \mathbf { } \mathbf \Psi \Psi \Psi \mathbf { } \mathbf \Psi \Psi \Psi \mathbf { }$ to denote the low-resolution (LR) image and the corresponding high-resolution (HR) image respectively. Let $f ( \cdot )$ denotes an arbitrary SISR network and the aleatoric uncertainty can be denoted by an additive term $\theta _ { i }$ . This way, the overall observation model can be formulated as
+
+$$
+\pmb { x } _ { i } = f ( \pmb { y } _ { i } ) + \epsilon \pmb { \theta } _ { i } ,
+$$
+
+where $\epsilon$ represents the Laplace distribution with zero-mean and unit-variance. Existing deep-learning based SISR methods target at training a network to learn the SR image (mean) $f ( \pmb { y } _ { i } )$ only. To more accurately characterize aleatoric uncertainty for SISR, we propose to estimate not only the SR image (mean) $\dot { f } ( \pmb { y } _ { i } )$ but also the uncertainty (variance) $\theta _ { i }$ simultaneously.
+
+For a given LR image $\mathbf { \nabla } _ { \mathbf { \psi } _ { 3 } } \psi _ { i }$ and corresponding HR image $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ , a Laplace distribution 2 is assumed for characterizing the likelihood function by
+
+$$
+p ( \pmb { x } _ { i } , \pmb { \theta } _ { i } | \pmb { y } _ { i } ) = \frac { 1 } { 2 \pmb { \theta } _ { i } } \exp ( - \frac { | | \pmb { x } _ { i } - \pmb { f } ( \pmb { y } _ { i } ) | | _ { 1 } } { \pmb { \theta } _ { i } } ) ,
+$$
+
+where $f ( \pmb { y } _ { i } )$ and $\theta _ { i }$ denote the SR image (mean) and the uncertainty (variance) which are learned by deep neural networks (DNNs) respectively. Then, the log likelihood can be formulated as follows,
+
+$$
+\ln p ( { \pmb x } _ { i } , \pmb \theta _ { i } | { \pmb y } _ { i } ) = - \frac { | | { \pmb x } _ { i } - f ( { \pmb y } _ { i } ) | | _ { 1 } } { \pmb \theta _ { i } } - \ln \pmb \theta _ { i } - \ln 2
+$$
+
+![](images/84479f818a49b71b89a280268aba9b89fc85314b12af5b6bb3ff38e05c6f146e.jpg)  
+Figure 3: SISR visual quality comparisons of EDSR-S [2] with different loss function on ‘Img_005’ from Set5 [23] (bicubic-downsampling $\times 4 )$ ). Best viewed in color.
+
+For numerical stability, we train the networks to estimate log variance $\begin{array} { r } { s _ { i } = \ln \theta _ { i } } \end{array}$ as shown in Fig. 2 (a). At last, the maximum likelihood estimation of (5) can be reformulated as the minimization of following loss function for estimating uncertainty (EU) in SISR.
+
+$$
+\mathcal { L } _ { E U } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \exp ( - s _ { i } ) \vert \vert \pmb { x } _ { i } - f ( \pmb { y } _ { i } ) \vert \vert _ { 1 } + s _ { i }
+$$
+
+Jeffrey’s Prior for Estimating Sparse Uncertainty (ESU) in SISR. The loss function $\mathcal { L } _ { \mathrm { E U } }$ includes two terms; the first one is associated with fidelity term and the second one prevents the network from predicting infinite uncertainty for all pixels. Those two terms reach equilibrium but there is no prior that imposed on the uncertainty estimation. Therefore, based on the observation that the uncertainty is sparse in view of the whole image as shown in Fig. 2, we propose to impose Jeffrey’s prior [10] $\begin{array} { r } { p ( \dot { \boldsymbol { w } } ) \propto \frac { 1 } { w } } \end{array}$ on uncertainty $\theta _ { i }$ , which can be expressed as
+
+$$
+\nu ( x _ { i } , \theta _ { i } | y _ { i } ) = p ( x _ { i } | y _ { i } , \theta _ { i } ) p ( \theta _ { i } ) \propto \frac { 1 } { 2 \theta _ { i } } \exp ( - \frac { \left| | x _ { i } - f ( y _ { i } ) | \right| _ { 1 } } { \theta _ { i } } ) \frac { 1 } { \theta _ { i } } = \frac { 1 } { 2 \theta _ { i } ^ { 2 } } \exp ( - \frac { \left| | x _ { i } - f ( y _ { i } ) | \right| _ { 1 } } { \theta _ { i } } )
+$$
+
+Then the log likelihood and loss function can be separately formulated as follows,
+
+$$
+\ln p ( \pmb { x } _ { i } | \pmb { y } _ { i } ) = - \frac { | | \pmb { x } _ { i } - \pmb { f } ( \pmb { y } _ { i } ) | | _ { 1 } } { \pmb { \theta } _ { i } } - 2 \ln \pmb { \theta } _ { i } - \ln 2
+$$
+
+$$
+\mathcal { L } _ { E S U } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \exp ( - s _ { i } ) | | x _ { i } - f ( \pmb { y } _ { i } ) | | _ { 1 } + 2 s _ { i }
+$$
+
+The limitations of $\mathcal { L } _ { \bf E U }$ and ${ \mathcal { L } } _ { \mathbf { E S U } }$ loss. Applying $\mathcal { L } _ { \mathrm { E U } }$ and $\mathcal { L } _ { \mathrm { E S U } }$ loss leads to more accurate estimation of uncertainty (variance field), but counter-intuitively, they do not directly improve the performance of SISR. We have conducted experiments comparing those three different loss functions to verify the above claim. As shown in Tab. 1, the average PSNR and SSIM results of $\mathcal { L } _ { \mathrm { E S U } }$ and $\mathcal { L } _ { \mathrm { E U } }$ are notably lower than the original results. The reason behind this observation is that both ${ \mathcal { L } } _ { \mathrm { E U } }$ and $\mathcal { L } _ { \mathrm { E S U } }$ loss functions have incorporated the variance term $( \pmb \theta _ { i } )$ into the divisor of the absolution difference term. Consequently, a pixel with a large variance will be penalized after the division and has less impact on the overall loss function. Note that such attenuation of pixels with large uncertainty is preferred for high-level vision tasks, as demonstrated in previous works [15–17] on image classification [18], image segmentation [15, 16], and face recognition [17, 19].
+
+Low-level vision tasks such as SISR are much different. As shown in Fig. 1, pixels with large uncertainty carry visually important information such as textured and edges. They need to be prioritized (opposite to attenuation) and given larger instead of smaller weights. To verify such claim, we have presented a simple example comparing the visual results between $\mathcal { L } _ { \mathrm { E U } }$ and $\mathcal { L } _ { \mathrm { E S U } }$ as shown in Fig. 3. It can be seen that the uncertainty captured by $\mathcal { L } _ { \mathrm { E S U } }$ loss is better than $\mathcal { L } _ { \mathrm { E U } }$ loss. The improvement of $\mathcal { L } _ { \mathrm { E S U } }$ in Eq. (9) over $\mathcal { L } _ { \mathrm { E U } }$ in Eq. (6) is attributed to the prioritization of pixels with large uncertainty ( $\boldsymbol { s } _ { i }$ values). Fig. 3 (f) clearly demonstrate superiority of exploiting the sparsity constraint with the uncertainty estimation.
+
+Table 1: Average PSNR and SSIM results for BI degradation on five datasets for investigating three different loss. The best performance is shown in bold. We record the results in $1 . 2 \times 1 0 ^ { \overline { { 5 } } }$ iterations.   
+
+<table><tr><td rowspan="2">Base Model</td><td rowspan="2">Scale</td><td rowspan="2">Loss</td><td colspan="2">Set5[23]</td><td colspan="2">Set14 [8]</td><td colspan="2">BSD100[24]</td><td colspan="2">Urban100 [25]</td><td colspan="2">Manga109 [26]</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td rowspan="3">EDSR-S[2]</td><td rowspan="3">×4</td><td>Original</td><td>30.93</td><td>0.8740</td><td>27.80</td><td>0.7627</td><td>27.05</td><td>0.7190</td><td>24.71</td><td>0.7351</td><td>28.14</td><td>0.8693</td></tr><tr><td>LEU</td><td>30.19</td><td>0.8627</td><td>27.29</td><td>0.7538</td><td>26.78</td><td>0.7120</td><td>24.21</td><td>0.7179</td><td>26.78</td><td>0.8481</td></tr><tr><td>LESU</td><td>30.31</td><td>0.8637</td><td>27.39</td><td>0.7543</td><td>26.83</td><td>0.7124</td><td>24.27</td><td>0.7192</td><td>26.92</td><td>0.8496</td></tr></table>
+
+# 3.2 Uncertainty-Driven Loss (UDL) for SISR
+
+Improvement of $\mathcal { L } _ { \mathrm { E S U } }$ over $\mathcal { L } _ { \mathrm { E U } }$ inspired us to go one step further. To better prioritize pixels with large uncertainty, we propose a new adaptive weighted loss named uncertainty-driven loss (UDL) for SISR. Unlike $\mathcal { L } _ { \mathrm { E S U } }$ loss putting a larger weight to the second term than $\mathcal { L } _ { \mathrm { E U } }$ , we suggest that the first term can also be modified to directly associate the aleatoric/data uncertainty of $f ( \pmb { y } _ { i } )$ . That is, instead of using $e x p ( - s _ { i } )$ to attenuate the importance of pixels with large uncertainty, we need to use a monotonically increasing function to prioritize them. Linear scaling would be a natural option, which leads to the following loss function
+
+$$
+\mathcal { L } _ { U D L } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \hat { s _ { i } } | | \pmb { x } _ { i } - f ( \pmb { y } _ { i } ) | | _ { 1 } ,
+$$
+
+where ${ \hat { s } } _ { i } = s _ { i } - \operatorname* { m i n } ( s _ { i } )$ is a non-negative linear scaling function. To prevent uncertainty value from degenerating into zeros, the result of uncertainty estimation network in the first step will be passed to the second step as the attention signal $\displaystyle s = \ln \theta$ ), as shown in Fig. 2. By leveraging the log variance to represent the challenging and cumbersome pixels with higher uncertainty, we propose a new weighted loss named uncertainty-driven loss ${ \mathcal { L } } _ { \mathrm { U D L } }$ . In ${ \mathcal { L } } _ { \mathrm { U D L } }$ loss, texture and edge pixels with higher uncertainty tend to have larger weights than those in smooth regions. In summary, the uncertainty estimation $\pmb \theta$ serves as the bridge connecting two steps: it is the output of the first step; but passed on to the second step as the guidance required for calculating ${ \mathcal { L } } _ { \mathrm { U D L } }$ loss.
+
+# 3.3 Two-step Training of Dual Networks
+
+As shown in Fig. 2, the whole training process can be divided into two steps; the first step estimates the uncertainty $\pmb { \theta }$ precisely and the second step generates the final mean value $f ( \boldsymbol { y } )$ with the aid from the estimated uncertainty $\pmb \theta$ from step1. More specifically, the mean value $f ( y )$ and variance $\theta$ are pre-trained by $\mathcal { L } _ { \mathrm { E S U } }$ loss during step1 as shown in Fig. 2 (a). After the uncertainty has been estimated, the mean value $f ( \boldsymbol { y } )$ network is trained by ${ \mathcal { L } } _ { \mathrm { U D L } }$ loss with variance $\theta$ as shown in Fig. 2 (b), while the network of inferring variance $\pmb { \theta }$ is fixed.
+
+Note that the mean value $f ( \boldsymbol { y } )$ network of step2 starts training from the pre-trained network of step1. Such partial parameter sharing is a salient property of our proposed dual networks with parallel symmetric attention [27]. In theory, we can extend the two-step training into multiple-step training by alternating between the estimation of uncertainty (variance $\pmb \theta$ ) and mean value $f ( \boldsymbol { y } )$ . Conceptually, an improved estimation of unknown HR image can leads to an improved estimation of aleatoric uncertainty and vice versa. This line of reasoning will lead to the pursuit of a deep equilibrium model [28] for SISR; but it is beyond the scope of this paper.
+
+# 3.4 Discussions: Why UDL Outperforms GRAM?
+
+To the best of our knowledge, only one work GRAM [21] has studied data uncertainty in SISR, which is the most related to our work. We will discuss connections and differences between proposed UDL and GRAM [21] here. First, both GRAM [21] and our work has found out that applying the traditional uncertainty loss designed for high-level computer vision tasks into SISR task directly results in performance decline. For high-level computer vision tasks, the pixels with higher uncertainty indicates less confidence in final inference, which needs loss attenuation. However, for SISR tasks, the pixels with higher uncertainty (e.g., texture and edge pixels) should be prioritized with larger weights because they are visually more important than pixels in smooth regions. To solve this problem, GRAM [21] proposes to use uncertainty to generate an attention mask that decreases loss attenuation. However, GRAM [21] loss still is attenuated when the variance of pixels is high. Thus, GRAM [21] is still inferior to baseline method since it still does not prioritize pixels with high uncertainty.
+
+Different from GRAM, we propose an uncertainty-driven loss to assign the pixels with high variance more weight to prioritize them. Besides, modeling uncertainty under Bayesian framework allows us to leverage sparsity prior for a more precise estimation of uncertainty. Ultimately, our proposed method consists of those two technical contributions that achieve better results than baseline methods and outperform GRAM.
+
+# 4 Experiments
+
+# 4.1 Experimental Settings
+
+Datasets and Metrics. 800 high-quality (2K resolution) images from the DIV2K dataset [29] have been used for training. Following EDSR [2], five standard benchmark datasets: Set5 [23], Set14 [8], BSD100[24], Urban100 [25], Manga109 [26] are used for testing. Performance evaluation in terms of of PSNR and SSIM [30] metrics is conducted on the luminance (Y) channel only.
+
+Training Setting. We randomly select 16 RGB LR patches sized by $4 8 \times 4 8$ as the inputs. The image patches are randomly rotated by $9 0 ^ { \circ }$ , $1 8 0 ^ { \circ }$ , $2 7 0 ^ { \circ }$ and flipped horizontally. The ADAM algorithm [31] with $\beta _ { 1 } = 0 . 9 , \beta _ { 2 } = 0 . 9 9 9$ , $\epsilon \overset { \cdot } { = } 1 0 ^ { - 8 }$ is adopted to optimize the network. The initial learning rate is $1 0 ^ { - 4 }$ and decreases by half for every $2 \times 1 0 ^ { 5 }$ minibatch updates.
+
+Degradation models. To demonstrate the effectiveness of our proposed uncertainty-driven loss in varying degradation scenarios, we have designed the following experiments with two different degradation models. Let BI denotes bicubic downsampling. The second one is BD which uses Gaussian blur followed by nearest downsampling to generate LR images. Specifically, we apply $1 1 \times 1 1$ sized Gaussian kernel with a standard deviation 1.6 for blurring in our experiments.
+
+SISR Networks. We choose three different networks to verify the effectiveness of proposed ${ \mathcal { L } } _ { \mathrm { U D L } }$ loss. The first one is EDSR-S or called baseline network in [2]. EDSR-S [2] mainly consists of 16 Resblock with 64 channels, having $1 . 5 M$ parameters. The second one is DPDNN[3] where denoiser network is U-net under model-guided framework. The last one is a big network EDSR[2], consisting of 32 Resblock with 256 channels, having $4 3 M$ parameters. The analysis of training cost can be found in our supplementary material.
+
+# 4.2 Ablation Study
+
+Table 2: Average PSNR and SSIM results for BI degradation on five datasets for investigating three different loss. The best performance is shown in bold. We record the results in $4 \times 1 0 ^ { 5 }$ iterations.   
+
+<table><tr><td rowspan="2">Base Model</td><td rowspan="2">Scale</td><td rowspan="2">Loss</td><td colspan="2">Set5[23]</td><td colspan="2">Set14[8]</td><td colspan="2">BSD100 [24]</td><td colspan="2">Urban100[25]</td><td colspan="2">Manga109 [26]</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td rowspan="3">EDSR-S[2]</td><td rowspan="3">×4</td><td>Original</td><td>31.61</td><td>0.8862</td><td>28.22</td><td>0.7721</td><td>27.30</td><td>0.7271</td><td>25.25</td><td>0.7575</td><td>29.31</td><td>0.8907</td></tr><tr><td>LEU+LUDL</td><td>31.83</td><td>0.8895</td><td>28.33</td><td>0.7754</td><td>27.37</td><td>0.7297</td><td>25.49</td><td>0.7665</td><td>29.70</td><td>0.8959</td></tr><tr><td>LESU+LUDL</td><td>31.90</td><td>0.8897</td><td>28.37</td><td>0.7755</td><td>27.40</td><td>0.7301</td><td>25.54</td><td>0.7671</td><td>29.77</td><td>0.8967</td></tr></table>
+
+To further verify the effectiveness of sparse uncertainty estimation at step1, we have conducted an ablation study to compare the final PSNR/SSIM results of ${ \mathcal { L } } _ { \mathrm { U D L } }$ with $\mathcal { L } _ { \mathrm { E U } }$ or with $\mathcal { L } _ { \mathrm { E S U } }$ at step1. In our ablation study, we have used $\times 4$ bicubic down-sampling degradation on five frequently-used benchmark datasets with EDSR-S backbone[2]. As shown in Tab. 2, both ${ \mathcal { L } } _ { \mathrm { E U } } { + } { \mathcal { L } } _ { \mathrm { U D L } }$ and ${ \mathcal { L } } _ { \mathrm { E S U } } { + } { \mathcal { L } } _ { \mathrm { U D L } }$ loss have achieved better performance than original loss. Besides, ${ \mathcal { L } } _ { \mathrm { E S U } } { + } { \mathcal { L } } _ { \mathrm { U D L } }$ loss obtains better results than $\mathcal { L } _ { \mathrm { E U } } { + } \mathcal { L } _ { \mathrm { U D L } }$ due to more accurate uncertainty estimation as shown in Fig. 3 (e) and (f).
+
+# 4.3 Analysis of Different Weighted Loss
+
+There are many different weighted loss guided by different weight maps, such as Error_map, Gradient_map which can also reveal the challenging pixels. We have conducted experiments with a weighted loss function where the weight is a pixel-wise gradient or Error_map. The PSNR results of five benchmark datasets for investigating the influence of different weighted loss functions can be summarized in Tab. 3.
+
+The $H R$ _gradient_map and $L R$ _gradien_map denote calculating gradient map from high-resolution (ground truth) images and low-resolution images respectively. The calculation of gradient can be formulated as
+
+$$
+\begin{array} { r } { V ( i , j ) = I ( i + 1 , j ) - I ( i , j ) , H ( i , j ) = I ( i , j + 1 ) - I ( i , j ) , G ( i , j ) = | | ( V ( i , j ) , H ( i , j ) | | _ { 2 } , } \end{array}
+$$
+
+Table 3: Average PSNR and $\Delta$ PSNR results with BI degradation on five datasets for investigating the influence of different weighted loss functions. The best performance is shown in bold.   
+
+<table><tr><td rowspan=1 colspan=1>Weighted loss</td><td rowspan=1 colspan=1>Set5</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>Set14</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>BSD100</td><td rowspan=1 colspan=1>△</td><td rowspan=1 colspan=1>Urban100</td><td rowspan=1 colspan=1>△</td><td rowspan=1 colspan=1>Manga109</td><td rowspan=1 colspan=1>△</td></tr><tr><td rowspan=1 colspan=1>Baseline</td><td rowspan=1 colspan=1>31.61</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>28.22</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>27.30</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>25.25</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>29.31</td><td rowspan=1 colspan=1>0.00</td></tr><tr><td rowspan=1 colspan=1>Uncertainty(Ours)</td><td rowspan=1 colspan=1>31.90</td><td rowspan=1 colspan=1>0.29个</td><td rowspan=1 colspan=1>28.37</td><td rowspan=1 colspan=1>0.15个</td><td rowspan=1 colspan=1>27.40</td><td rowspan=1 colspan=1>0.10个</td><td rowspan=1 colspan=1>25.54</td><td rowspan=1 colspan=1>0.29个</td><td rowspan=1 colspan=1>29.77</td><td rowspan=1 colspan=1>0.46个</td></tr><tr><td rowspan=1 colspan=1>Error_map</td><td rowspan=1 colspan=1>31.77</td><td rowspan=1 colspan=1>0.16个</td><td rowspan=1 colspan=1>28.30</td><td rowspan=1 colspan=1>0.08个</td><td rowspan=1 colspan=1>27.35</td><td rowspan=1 colspan=1>0.05个</td><td rowspan=1 colspan=1>25.40</td><td rowspan=1 colspan=1>0.15个</td><td rowspan=1 colspan=1>29.57</td><td rowspan=1 colspan=1>0.26个</td></tr><tr><td rowspan=1 colspan=1>HR_gradient_map</td><td rowspan=1 colspan=1>31.68</td><td rowspan=1 colspan=1>0.07个</td><td rowspan=1 colspan=1>28.27</td><td rowspan=1 colspan=1>0.05个</td><td rowspan=1 colspan=1>27.35</td><td rowspan=1 colspan=1>0.05个</td><td rowspan=1 colspan=1>25.42</td><td rowspan=1 colspan=1>0.17个</td><td rowspan=1 colspan=1>29.45</td><td rowspan=1 colspan=1>0.14个</td></tr><tr><td rowspan=1 colspan=1>LR_gradient_map</td><td rowspan=1 colspan=1>31.69</td><td rowspan=1 colspan=1>0.08个</td><td rowspan=1 colspan=1>28.29</td><td rowspan=1 colspan=1>0.07个</td><td rowspan=1 colspan=1>27.35</td><td rowspan=1 colspan=1>0.05个</td><td rowspan=1 colspan=1>25.38</td><td rowspan=1 colspan=1>0.13个</td><td rowspan=1 colspan=1>29.50</td><td rowspan=1 colspan=1>0.19个</td></tr></table>
+
+where $I$ denotes pixels value and $i , j$ denotes position of pixels. Note that we adjust the scaling functions of Error_map, HR_gradient_map and LR_gradient_map to get the best performance.
+
+From the Tab. 3, one can be observed that other weighted loss functions can indeed improve the PSNR results, but only to certain degrees. Comparing four different weight maps, our proposed uncertainty weighted loss function can bring the biggest improvement. Although the Error_map can represent the variance of a single pixel, the Error_map lacks semantic information or local information to capture a more precise estimation of variance comparing uncertainty. With regard to the gradient map of HR or LR images, those gradient maps only well match the edges of images and have a certain correlation to variance. Comparing the visual results of Error_map, HR_gradient_map and $L R$ _gradient_map with uncertainty map, those maps only detect edges of images and fail reflecting complex texture details which are important to final reconstruction performance. Therefore, uncertainty-weighted loss can is still valuable for achieving the best performance among other weighted maps.
+
+# 4.4 Analysis of Different Scaling Functions
+
+We have conducted experiments with several various monotonically increasing functions (including linear and non-linear) and the results can be summarized in Tab. 4.
+
+Table 4: Average PSNR and $\Delta$ PSNR results with BI degradation on five datasets for investigating the influence of different scaling functions. The best performance are shown in bold.   
+
+<table><tr><td rowspan=1 colspan=1>Scaling functions</td><td rowspan=1 colspan=1>Set5</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>Set14</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>BSD100</td><td rowspan=1 colspan=1>△</td><td rowspan=1 colspan=1>Urban100</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>Manga109</td><td rowspan=1 colspan=1>A</td></tr><tr><td rowspan=1 colspan=1>Baseline</td><td rowspan=1 colspan=1>31.61</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>28.22</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>27.30</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>25.25</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>29.31</td><td rowspan=1 colspan=1>0.00</td></tr><tr><td rowspan=1 colspan=1>s-min(s)</td><td rowspan=1 colspan=1>31.90</td><td rowspan=1 colspan=1>0.29个</td><td rowspan=1 colspan=1>28.37</td><td rowspan=1 colspan=1>0.15个</td><td rowspan=1 colspan=1>27.40</td><td rowspan=1 colspan=1>0.10个</td><td rowspan=1 colspan=1>25.54</td><td rowspan=1 colspan=1>0.29个</td><td rowspan=1 colspan=1>29.77</td><td rowspan=1 colspan=1>0.46个</td></tr><tr><td rowspan=1 colspan=1>exp(s)</td><td rowspan=1 colspan=1>31.80</td><td rowspan=1 colspan=1>0.19个</td><td rowspan=1 colspan=1>28.34</td><td rowspan=1 colspan=1>0.12个</td><td rowspan=1 colspan=1>27.40</td><td rowspan=1 colspan=1>0.10个</td><td rowspan=1 colspan=1>25.53</td><td rowspan=1 colspan=1>0.28个</td><td rowspan=1 colspan=1>29.66</td><td rowspan=1 colspan=1>0.35个</td></tr><tr><td rowspan=1 colspan=1>e.xp(s)(1/2)</td><td rowspan=1 colspan=1>31.86</td><td rowspan=1 colspan=1>0.25个</td><td rowspan=1 colspan=1>28.36</td><td rowspan=1 colspan=1>0.14↑</td><td rowspan=1 colspan=1>27.41</td><td rowspan=1 colspan=1>0.11个</td><td rowspan=1 colspan=1>25.55</td><td rowspan=1 colspan=1>0.30↑</td><td rowspan=1 colspan=1>29.71</td><td rowspan=1 colspan=1>0.40↑</td></tr><tr><td rowspan=1 colspan=1>log(s)-min(log(s))</td><td rowspan=1 colspan=1>31.89</td><td rowspan=1 colspan=1>0.28个</td><td rowspan=1 colspan=1>28.39</td><td rowspan=1 colspan=1>0.17个</td><td rowspan=1 colspan=1>27.42</td><td rowspan=1 colspan=1>0.12个</td><td rowspan=1 colspan=1>25.57</td><td rowspan=1 colspan=1>0.32个</td><td rowspan=1 colspan=1>29.74</td><td rowspan=1 colspan=1>0.43个</td></tr></table>
+
+The best and second-best performances are shown in bold. Overall, four various monotonically increasing functions have achieved better results than the baseline method. The best two scaling functions are linear scaling and log scaling with a slight difference as shown in the above table. Since the linear scaling function achieves a comparable performance with low computational cost, we advocate this choice in this paper.
+
+![](images/73cd26da93be776365b72653a8fa3c0ab3ef6d5ec3e615945e501a6f05f46988.jpg)  
+Figure 4: SISR visual quality comparisons of EDSR-S [2] with different loss function on ‘Img_004’ and $\mathrm { \hbar } ^ { 4 } \mathrm { I m g \_ 0 1 } 6 ^ { , }$ from Urban100 [25] (bicubic-downsampling $\times 4 _ { , }$ ). Best viewed in color.
+
+![](images/8c5e414f36b7a9a967963d4eb9e5876347915ad184c571633d9e88046b0f6e74.jpg)  
+Figure 5: SISR visual quality comparisons of DPDNN [3] with different loss function on ‘Img_095 from BSD100 [24] (bicubic-downsampling $\times 4 _ { , }$ ). Best viewed in color.
+
+Table 5: Average PSNR and SSIM results for BI degradation on five benchmark datasets. The best performance is shown in bold. Note that ${ \mathcal { L } } _ { \mathrm { U D L } }$ -Ours denotes adopting $\mathcal { L } _ { \mathrm { E S U } }$ at step1 and ${ \mathcal { L } } _ { \mathrm { U D L } }$ at step2 for simplicity.
+
+<table><tr><td rowspan=2 colspan=1>Base Model</td><td rowspan=2 colspan=1>Scale</td><td rowspan=2 colspan=1>Loss</td><td rowspan=1 colspan=2>Set5[23]</td><td rowspan=1 colspan=1>Set]</td><td rowspan=1 colspan=1>4[8]</td><td rowspan=1 colspan=2>BSD100 [24]</td><td rowspan=1 colspan=1>Urban</td><td rowspan=1 colspan=1>00[25]</td><td rowspan=1 colspan=1>Manga</td><td rowspan=1 colspan=1>109[26]</td></tr><tr><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td></tr><tr><td rowspan=1 colspan=1>EDSR-S [2]</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>37.6637.4837.95</td><td rowspan=1 colspan=1>0.85940.95890.9604</td><td rowspan=1 colspan=1>33.2232.9933.50</td><td rowspan=1 colspan=1>0.91460.91260.9165</td><td rowspan=1 colspan=1>31.9531.7632.13</td><td rowspan=1 colspan=1>0.89690.89460.8991</td><td rowspan=1 colspan=1>30.7130.1131.54</td><td rowspan=1 colspan=1>0.92050.91340.9304</td><td rowspan=1 colspan=1>37.7937.3838.38</td><td rowspan=1 colspan=1>0.97520.97390.9767</td></tr><tr><td rowspan=1 colspan=1>DPDNN [3]</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>37.7537.7438.00</td><td rowspan=1 colspan=1>0.96000.95970.9605</td><td rowspan=1 colspan=1>33.3033.2733.63</td><td rowspan=1 colspan=1>0.91500.91480.9176</td><td rowspan=1 colspan=1>32.0931.9832.16</td><td rowspan=1 colspan=1>0.89900.89730.8995</td><td rowspan=1 colspan=1>31.5030.9731.72</td><td rowspan=1 colspan=1>0.92200.92380.9331</td><td rowspan=1 colspan=1>-38.1438.55</td><td rowspan=1 colspan=1>-0.97580.9769</td></tr><tr><td rowspan=1 colspan=1>EDSR [2]</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>38.1137.8738.29</td><td rowspan=1 colspan=1>0.96020.96040.9615</td><td rowspan=1 colspan=1>33.9233.4334.14</td><td rowspan=1 colspan=1>0.91950.91640.9236</td><td rowspan=1 colspan=1>32.3232.0832.40</td><td rowspan=1 colspan=1>0.90130.89900.9027</td><td rowspan=1 colspan=1>32.9331.4632.99</td><td rowspan=1 colspan=1>0.93510.93010.9446</td><td rowspan=1 colspan=1>39.1037.9139.53</td><td rowspan=1 colspan=1>0.97730.97650.9787</td></tr><tr><td rowspan=1 colspan=1>EDSR-S [2]</td><td rowspan=1 colspan=1>×3</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>33.9033.2734.15</td><td rowspan=1 colspan=1>0.92310.91780.9251</td><td rowspan=1 colspan=1>29.9529.6030.15</td><td rowspan=1 colspan=1>0.83520.82980.8388</td><td rowspan=1 colspan=1>28.8528.6028.99</td><td rowspan=1 colspan=1>0.79960.79360.8021</td><td rowspan=1 colspan=1>27.3026.5227.72</td><td rowspan=1 colspan=1>0.83440.81420.8430</td><td rowspan=1 colspan=1>32.5231.1432.97</td><td rowspan=1 colspan=1>0.93690.92580.9406</td></tr><tr><td rowspan=1 colspan=1>DPDNN [3]</td><td rowspan=1 colspan=1>×3</td><td rowspan=1 colspan=1>OriginalGRAM[21]LUDL-Ours</td><td rowspan=1 colspan=1>33.9333.9234.30</td><td rowspan=1 colspan=1>0.92400.92410.9267</td><td rowspan=1 colspan=1>30.0230.0030.31</td><td rowspan=1 colspan=1>0.83600.83620.8419</td><td rowspan=1 colspan=1>29.0028.8629.10</td><td rowspan=1 colspan=1>0.80100.80000.8047</td><td rowspan=1 colspan=1>27.6127.3728.02</td><td rowspan=1 colspan=1>0.84200.83530.8505</td><td rowspan=1 colspan=1>132.4133.27</td><td rowspan=1 colspan=1>10.93730.9435</td></tr><tr><td rowspan=1 colspan=1>EDSR [2]</td><td rowspan=1 colspan=1>×3</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>34.6534.3434.83</td><td rowspan=1 colspan=1>0.92800.92700.9312</td><td rowspan=1 colspan=1>30.5230.2830.69</td><td rowspan=1 colspan=1>0.84620.84120.8497</td><td rowspan=1 colspan=1>29.2529.0729.28</td><td rowspan=1 colspan=1>0.80930.80440.8109</td><td rowspan=1 colspan=1>28.8027.9828.99</td><td rowspan=1 colspan=1>0.86530.87890.8697</td><td rowspan=1 colspan=1>34.1733.3234.63</td><td rowspan=1 colspan=1>0.94760.94320.9502</td></tr><tr><td rowspan=1 colspan=1>EDSR-S [2]</td><td rowspan=1 colspan=1>×4</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>31.6131.0831.90</td><td rowspan=1 colspan=1>0.88620.87870.8897</td><td rowspan=1 colspan=1>28.2227.8928.37</td><td rowspan=1 colspan=1>0.77210.76700.7755</td><td rowspan=1 colspan=1>27.3027.1227.40</td><td rowspan=1 colspan=1>0.72710.72290.7301</td><td rowspan=1 colspan=1>25.2524.8125.54</td><td rowspan=1 colspan=1>0.75750.74290.7671</td><td rowspan=1 colspan=1>29.3128.1829.77</td><td rowspan=1 colspan=1>0.89070.87620.8967</td></tr><tr><td rowspan=1 colspan=1>DPDNN [3]</td><td rowspan=1 colspan=1>×4</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>31.7231.8932.20</td><td rowspan=1 colspan=1>0.88900.89130.8944</td><td rowspan=1 colspan=1>28.2828.3728.60</td><td rowspan=1 colspan=1>0.77300.77720.7819</td><td rowspan=1 colspan=1>27.4427.4127.56</td><td rowspan=1 colspan=1>0.72900.73140.7356</td><td rowspan=1 colspan=1>25.5325.6326.09</td><td rowspan=1 colspan=1>0.76800.77080.7862</td><td rowspan=1 colspan=1>-29.7030.38</td><td rowspan=1 colspan=1>-0.90030.9082</td></tr><tr><td rowspan=1 colspan=1>EDSR [2]</td><td rowspan=1 colspan=1>×4</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>32.4632.3232.59</td><td rowspan=1 colspan=1>0.89680.89710.8998</td><td rowspan=1 colspan=1>28.8028.7328.87</td><td rowspan=1 colspan=1>0.78760.78580.7889</td><td rowspan=1 colspan=1>27.7127.6627.78</td><td rowspan=1 colspan=1>0.74200.73950.7431</td><td rowspan=1 colspan=1>26.6426.3526.75</td><td rowspan=1 colspan=1>0.80330.79550.8054</td><td rowspan=1 colspan=1>31.0230.7331.24</td><td rowspan=1 colspan=1>0.91480.91250.9167</td></tr></table>
+
+# 4.5 Results with BI Degradation Model
+
+For bicubic downsampling (BI), we have compared proposed ${ \mathcal { L } } _ { \mathrm { U D L } }$ loss function with GRAM [21] and original loss functions such as $M S E$ or $\mathcal { L } _ { 1 }$ on three different SISR networks. The average PSNR and SSIM results in Tab. 5 are cited from corresponding papers or retrained from officially released code. It is easy to see that our proposed ${ \mathcal { L } } _ { \mathrm { U D L } }$ loss function is superior to GRAM [21] and original loss functions such as $M S E$ or $\mathcal { L } _ { 1 }$ in terms of PSNR and SSIM values. Note that the improvements achieved by our proposed method do not bring any additional computation cost during testing time. Comparing EDSR-S ( $. 5 M$ parameters) with EDSR ( $4 3 M$ parameters), our proposed ${ \mathcal { L } } _ { \mathrm { U D L } }$ can bring lightweight networks with more greater performance improvements than big ones. The visual image comparison results are reported in Fig. 4 and Fig. 5. As shown in Fig. 4, our proposed ${ \mathcal { L } } _ { \mathrm { U D L } }$ has recovered with fewer visible artifacts (e.g., the circular pattern of the roof and the lines on the glassy surface) than original loss and GRAM [21]. Fig. 4 (f) depicts the uncertainty learned by our ${ \mathcal { L } } _ { \mathrm { U D L } }$ , revealing the challenging pixels with poor reconstruction performance. From Fig. 5, vertical center-line of window has been recover more clear with precisely estimated uncertainty shown in (e) and (f), while DPDNN and DPDNN-GRAM [21] failed to discern shown in (c) and (d) respectively. More visual comparisons can be found in supplementary material.
+
+Table 6: Average PSNR and SSIM results for BD degradation on five benchmark datasets. The best performance is shown in bold. Note that ${ \mathcal { L } } _ { \mathrm { U D L } }$ -Ours denotes adopting $\mathcal { L } _ { \mathrm { E S U } }$ at step1 and ${ \mathcal { L } } _ { \mathrm { U D L } }$ at step2 for simplicity.   
+
+<table><tr><td rowspan=2 colspan=1>Base Model</td><td rowspan=2 colspan=1>Scale</td><td rowspan=2 colspan=1>Loss</td><td rowspan=1 colspan=2>Set5[23]</td><td rowspan=1 colspan=2>Set14 [8]</td><td rowspan=1 colspan=3>BSD100 [24]</td><td rowspan=1 colspan=1>Urban</td><td rowspan=1 colspan=1>00[25]</td><td rowspan=1 colspan=1>Manga</td><td rowspan=1 colspan=1>09[26]</td></tr><tr><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=2>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td></tr><tr><td rowspan=1 colspan=1>EDSR-S [2]</td><td rowspan=1 colspan=1>×4</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>31.7030.9831.97</td><td rowspan=1 colspan=1>0.89030.87910.8927</td><td rowspan=1 colspan=1>28.3727.8528.45</td><td rowspan=1 colspan=1>0.77780.76670.7793</td><td rowspan=1 colspan=2>27.3727.0527.41</td><td rowspan=1 colspan=1>0.73200.72250.7321</td><td rowspan=1 colspan=1>25.7724.7925.95</td><td rowspan=1 colspan=1>0.77890.74520.7842</td><td rowspan=1 colspan=1>29.8328.1230.18</td><td rowspan=1 colspan=1>0.90140.87730.9053</td></tr><tr><td rowspan=2 colspan=1>DPDNN [3]</td><td rowspan=2 colspan=1>×4</td><td rowspan=2 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=2 colspan=1>31.8631.7532.03</td><td rowspan=2 colspan=1>0.89230.89130.8949</td><td rowspan=2 colspan=1>28.3828.3328.60</td><td rowspan=2 colspan=1>0.77800.77650.7828</td><td rowspan=2 colspan=2>27.3627.3227.48</td><td rowspan=1 colspan=1>0.73110.7302</td><td rowspan=1 colspan=1>25.8225.62</td><td rowspan=1 colspan=1>0.78120.7739</td><td rowspan=1 colspan=1>29.7729.55</td><td rowspan=2 colspan=1>0.90330.90030.9097</td></tr><tr><td rowspan=1 colspan=1>27.48</td><td rowspan=1 colspan=1>0.7355</td><td rowspan=1 colspan=1>26.21</td><td rowspan=1 colspan=1>0.7931</td><td rowspan=1 colspan=1>30.35</td></tr><tr><td rowspan=5 colspan=1>EDSR [2]</td><td rowspan=5 colspan=1>×4</td><td rowspan=5 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=5 colspan=1>32.1732.1332.37</td><td rowspan=5 colspan=1>0.89750.89630.8986</td><td rowspan=5 colspan=1>28.6528.5727.74</td><td rowspan=1 colspan=1>0.7856</td><td rowspan=4 colspan=2>27.5927.49</td><td rowspan=1 colspan=1>0.7400</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td rowspan=3 colspan=1>0.7362</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td rowspan=2 colspan=1>26.5626.19</td><td rowspan=2 colspan=1>0.80430.7916</td><td rowspan=2 colspan=1>30.6630.48</td><td rowspan=3 colspan=1>0.91340.90970.9149</td></tr><tr><td rowspan=1 colspan=1>0.7822</td></tr><tr><td rowspan=1 colspan=1>0.7867</td><td rowspan=1 colspan=2>27.62</td><td rowspan=1 colspan=1>0.7407</td><td rowspan=1 colspan=1>26.65</td><td rowspan=1 colspan=1>0.8065</td><td rowspan=1 colspan=1>30.81</td></tr></table>
+
+# 4.6 Results with BD Degradation Model
+
+For blur downsampling (BD), we have compared proposed ${ \mathcal { L } } _ { \mathrm { U D L } }$ loss function with GRAM [21] and original loss functions such as $M S E$ or $\mathcal { L } _ { 1 }$ on three different baseline networks. The average PSNR and SSIM results in Tab. 6 are retrained from officially released code. It is easy to see that our propose ${ \mathcal { L } } _ { \mathrm { U D L } }$ loss function is superior to GRAM [21] and original loss functions such as $M S E$ or $\mathcal { L } _ { 1 }$ in terms of PSNR and SSIM values. The visual image comparison results of BD degradation are reported in Fig. 6 and Fig. 7. Note that the BD degradation involves Gaussian blur, increasing difficulty in recovering structure patterns. From Fig. 6, we can see that our SR result (Fig. 6 (e)) of $\mathrm { \nabla ^ { \cdot } I m g \ 1 0 9 ^ { \cdot } }$ is the closest to that of the ground-truth. In another challenging image $\mathbf { \dot { \tau } } ^ { \mathrm { \prime } } \mathbf { I m g 0 7 8 } ^ { \prime }$ from Urban100 [25]), our method can recover much more reliable textured details as shown in Fig. 7 (e); while all other methods have severe aliasing artifacts (i.e., distorted tile patterns). The visual quality improvement achieved by ${ \mathcal { L } } _ { \mathrm { U D L } }$ is mainly due to the fact that our proposed method makes full use of the captured uncertainty to train deep networks focusing on the challenging pixels with high uncertainty. More visual comparisons can be found in our supplementary material.
+
+![](images/f9c5c9971afa483363c84d669e494ec595f7f0b61c9b0f74c821154dab28cc4e.jpg)  
+Figure 6: SISR visual quality comparisons of DPDNN [3] with different loss function on ‘Img_109 from Manga109 [26] (blur-downsampling $\times 4 _ { , }$ ). Best viewed in color.
+
+![](images/c30523fd5bb0d22d4e47da96524fc967f6d1e620b06c8a9bb6c42510827f5d53.jpg)  
+Figure 7: SISR visual quality comparisons of EDSR [2] with different loss function on ‘Img_078’ from Urban100 [25] (blur-downsampling $\times 4 _ { , }$ ). Best viewed in color.
+
+# 5 Conclusion
+
+In this paper, we propose a new adaptive weighted loss ${ \mathcal { L } } _ { \mathrm { U D L } }$ for SISR to train SISR networks focusing on challenging pixels with high uncertainty (e.g., textured and edge pixels). Specifically, variance estimation is introduced into SISR so that the high-resolution images (mean) and their corresponding uncertainty (variance) can be learned simultaneously. Moreover, modeling uncertainty under Bayesian framework allows us to leverage sparsity prior for a more precise estimation of uncertainty. Ultimately, pixels with large certainty (e.g., texture and edge pixels) will be prioritized for SISR according to their importance to visual quality. For the first time, we demonstrate that such uncertainty-driven loss can achieve better results than $M S E$ or $\mathcal { L } _ { 1 }$ loss. Experimental results on three popular SISR networks show that our proposed uncertainty-driven loss has achieved better PSNR performance than traditional loss functions without any increased computation during testing.
+
+# Acknowledgement
+
+This work was supported in part by the National Key R&D Program of China under Grant 2018AAA0101400 and the Natural Science Foundation of China under Grant 61991451, Grant 61632019, Grant 61621005, and Grant 61836008. Xin Li’s work is partially supported by the NSF under grants IIS-1951504 and OAC-1940855, the DoJ/NIJ under grant NIJ 2018-75-CX-0032, and the WV Higher Education Policy Commission Grant (HEPC.dsr.18.5).
+
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+        "text": "Uncertainty-Driven Loss for Single Image Super-Resolution ",
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+        "text": "Qian $\\mathbf { N i n g ^ { 1 } }$ , Weisheng $\\mathbf { D o n g } ^ { 1 }$ ∗, Xin Li2, Jinjian $\\mathbf { W } \\mathbf { u } ^ { 1 }$ , Guangming Shi1 1School of Artificial Intelligence, Xidian University, Xi’an 710071, China 2Lane Dep. of CSEE, West Virginia University, Morgantown WV 26506, USA ningqian@stu.xidian.edu.cn, {wsdong,jinjian.wu}@mail.xidian.edu.cn xin.li@mail.wvu.edu, gmshi@xidian.edu.cn ",
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+        "text": "Abstract ",
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+        "text": "In low-level vision such as single image super-resolution (SISR), traditional MSE or $\\mathcal { L } _ { 1 }$ loss function treats every pixel equally with the assumption that the importance of all pixels is the same. However, it has been long recognized that texture and edge areas carry more important visual information than smooth areas in photographic images. How to achieve such spatial adaptation in a principled manner has been an open problem in both traditional model-based and modern learning-based approaches toward SISR. In this paper, we propose a new adaptive weighted loss for SISR to train deep networks focusing on challenging situations such as textured and edge pixels with high uncertainty. Specifically, we introduce variance estimation characterizing the uncertainty on a pixel-by-pixel basis into SISR solutions so the targeted pixels in a high-resolution image (mean) and their corresponding uncertainty (variance) can be learned simultaneously. Moreover, uncertainty estimation allows us to leverage conventional wisdom such as sparsity prior for regularizing SISR solutions. Ultimately, pixels with large certainty (e.g., texture and edge pixels) will be prioritized for SISR according to their importance to visual quality. For the first time, we demonstrate that such uncertainty-driven loss can achieve better results than $M S E$ or $\\mathcal { L } _ { 1 }$ loss for a wide range of network architectures. Experimental results on three popular SISR networks show that our proposed uncertainty-driven loss has achieved better PSNR performance than traditional loss functions without any increased computation during testing. The code is available at https://see.xidian.edu.cn/faculty/wsdong/Projects/UDL-SR.htm ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Single image super-resolution (SISR) aims at reconstructing high-resolution (HR) images from their corresponding degraded low-resolution (LR) images. Since the publication of super-resolution with convolutional neural network (SRCNN) [1], there has been a flurry of works on deep learning-based approaches toward SISR - e.g., EDSR [2], DPDNN [3], RCAN [4], SAN [5], and MoG-DUN [6]. The unifying theme along this line of research appears to be that deeper, bigger, and more complex networks can achieve improved SISR performance by facilitating the reconstruction of high-frequency details such as textures and edges in photographic images. Such improvement has been achieved by novel network architectures (e.g., skip connections [2]), new attention mechanism (e.g., residue channel attention [4]), and closed-loop supervision [7]. Surprisingly, most of these existing methods have adopted $M S E$ or $\\mathcal { L } _ { 1 }$ loss to optimize the parameters of networks. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The commonly used practice, such as $M S E$ or $\\mathcal { L } _ { 1 }$ loss, treats every pixel equally regardless of whether the pixel is in texture/edge regions or smooth areas. The optimality of such non-adaptive loss function has been questioned in the literature of SISR calling for the proposition of perceptual loss function (e.g., [9]). From a Bayesian perspective, the assumption underlying the $M S E$ or $\\bar { \\mathcal { L } } _ { 1 }$ loss is that each pixel obeys the independent and identically distribution with the same variance. Taking $\\mathcal { L } _ { 1 }$ loss as an example, the likelihood of all pixels in an image can be formulated as ",
+        "bbox": [
+            176,
+            852,
+            823,
+            881
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/222bd7ad66e205fc8caca654c095a992888dc8615acaff7cc921e0978b1f6352.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of the difference (d) between HR image (b) and SR image (c) reconstructed by EDSR network [2] on dataset Set14 [8]. The image reconstructed by EDSR network is shown in (c) and (d) shows the absolute difference between the HR image and SR image. Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            88,
+            812,
+            202
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            260,
+            825,
+            316
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f8ff5b4c1d1eaf281a94531ac3e26de8caafd220adec1df7c8f420e8bee77ce3.jpg",
+        "text": "$$\np ( \\pmb { x } \\mid \\pmb { y } , \\pmb { W } ) = \\prod _ { l = 1 } ^ { M } c \\exp ( - \\frac { \\vert \\vert \\pmb { x } ^ { ( l ) } - \\pmb { f } ^ { ( W ) } ( \\pmb { y } ^ { ( l ) } ) \\vert \\vert _ { 1 } } { \\sigma } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            323,
+            673,
+            367
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $_ { \\textbf { \\em x } }$ and $\\textbf {  { y } }$ denote the pair of HR and LR image, $f ^ { ( W ) } ( \\cdot )$ denotes an arbitrary SISR network parameterized by $W$ , and $c , \\sigma$ denote spatially invariant constants. However, such assumption of stationarity or spatial invariance of image prior model is invalid for photographic images in the real world. For instance, if one compares the ground-truth (HR image) and the SR image reconstructed by EDSR [2] as shown in Fig. 1 (c), it can be observed that texture areas (e.g., hair of baboon) are not restored as good as smooth areas (e.g., nose of baboon). Fig. 1 (d) depicts the absolute difference between the HR image and reconstructed SR image, from which we can observe spatial variation of the difference map. Such observation implies that the uncertainty of texture and edge areas as characterized by the variance is much larger than that in smooth areas. How to address such uncertainty-driven loss for SISR sets up the stage for this paper. ",
+        "bbox": [
+            173,
+            375,
+            825,
+            516
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a new adaptive weighted loss (uncertainty-driven loss) for SISR by assigning texture and edge areas with higher weights during the training process. Unlike previous work of perceptual loss [9] focusing on characterizing content and style consistency, we target at explicitly estimating the variance field underlying the unknown HR image in the first step, which can be exploited as an auxiliary signal for guiding the SISR solution in the second step. A direct consequence of our two-step learning approach is that it delivers not only higher visual quality but also improved objective performance such as PSNR and SSIM. Moreover, uncertainty estimation perspective allows us to easily incorporate existing models such as Jefferey’s prior [10, 11] into the proposed SISR solution. It follows that the network training boils down to two sequential steps in which the variance map is estimated from the first step and serves as the attention signal for the second step. The main technical contributions are summarized as follows. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Uncertainty modeling and estimation. We propose to cast SISR into a Bayesian estimation framework under which SR image (mean) and uncertainty (variance) are derived simultaneously. Unlike previous works in which pixels with large uncertainty are attenuated for high-level vision tasks, we advocate to prioritize them for low-level vision tasks such as SISR.   \nUncertainty-driven loss (UDL). The estimation of variance map facilitates the training of SISR network by dividing it into two steps. In the first step, an estimating sparsity uncertainty (ESU) loss function was derived from the classical Jeffrey’s prior to estimate the variance map. In the second step, the estimated variance map serves as the guidance signal leading to adaptive weighted loss named uncertainty-driven loss ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ .   \nUniversality of UDL. The proposed uncertainty loss can easily be employed in any existing SISR network to improve performance and do not increase any additional computation cost during testing.   \n• Experimental results on three different baseline networks show that our proposed uncertaintydriven loss has achieved better PSNR performance than traditional $M S E$ or $\\mathcal { L } _ { 1 }$ loss. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Uncertainty in Deep Learning ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            121,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Many works [12–14] have introduced uncertainty into the regression with input-dependent noises problems, and studied the nature and behavior of uncertainty for a long time. More recently, modeling uncertainty in deep learning have improved the performance and robustness of deep networks in many computer vision tasks [15–17] such as image classification [18], image segmentation [15, 16], and face recognition [17, 19]. The uncertainty in deep learning can be roughly divided into two categories [20]. Epistemic/model uncertainty describes how much the model is uncertain about its predictions. Another type is aleatoric/data uncertainty which refers to noise inherent in observation data. In [15], they presented a Bayesian deep learning framework combining aleatoric uncertainty with epistemic uncertainty for per-pixel semantic segmentation and depth regression tasks. Chang et al.[17] investigated the data uncertainty with estimated mean and variance in face recognition. Those uncertainty-based loss function proposed by those works [15–17] can be summarized as ",
+        "bbox": [
+            173,
+            146,
+            825,
+            299
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5a9148107b16d74d67accf2347094da83861dda5bc839af9af21b808c2b98dbe.jpg",
+        "text": "$$\n\\mathcal { L } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\frac { | | \\pmb { x } _ { i } - \\pmb { f } ( \\pmb { y } _ { i } ) | | _ { 2 } } { 2 \\sigma _ { i } ^ { 2 } } + \\frac { 1 } { 2 } \\ln \\sigma _ { i } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            316,
+            632,
+            359
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $f ( \\pmb { y } _ { i } )$ and $\\sigma _ { i } ^ { 2 }$ denote the learned mean and variance respectively. Using above loss function indeed improved their robustness to noisy data. In those tasks, the pixels with high uncertainty were regarded as unreliable pixels which would bear loss attenuation. On the contrary, in SISR tasks, the pixels with high uncertainty (e.g., complex texture or edge areas) should be prioritized since those regions visually more important than pixels in smooth areas. That can explain why applying above loss into SISR directly leads performance decline. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Modeling Uncertainty for SISR ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            481
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To the best of our knowledge, only two works [21, 22] have studied the behavior of uncertainty for SISR in the open literature. [22] used batch-normalization uncertainty to analyze SISR uncertainty, improving the robustness of the network against adversarial attack. The most recent advance related to our work is Gradient Rescaling Attention Model (GRAM) [21], which analyses the effect of aleatoric/data uncertainty on SISR reconstruction. By decreasing the loss attenuation of large variance pixels, GRAM achieves better results than applying above uncertainty loss into SISR directly. However, GRAM [21] loss remains attenuated when the variance of pixels is high, which contradicts the intuition of prioritizing texture and edge pixels. Thus, GRAM [21] is still inferior to baseline methods since the proposed method fails to prioritize the pixels of large variance. Different from GRAM, we propose a novel uncertainty-driven loss (UDL) to enforce the network concentrating more on the pixels with large variance aiming at better reconstruction of texture and edge regions. By quantifying the uncertainty in SISR under deep Bayesian framework, our proposed method has achieved better results than baseline methods. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Methodology ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            691,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unlike traditional $M S E$ or $\\mathcal { L } _ { 1 }$ loss treating every pixel equally, the proposed new adaptive weighted loss for SISR aims at prioritizing texture and edge pixels that are visually more important than pixels in smooth areas. Toward this objective, we first introduce an approach of estimating intermediate results of SR image (mean) and uncertainty (variance) simultaneously in SISR. Then, with Jeffrey’s prior term, a regularized approach of estimating sparse uncertainty is proposed for more accurate uncertainty estimation. An important new insight brought by this paper is that unlike high-level vision tasks where pixels with large uncertainty are assigned lower weights to attenuate their impact $I I 5 J ,$ one should prioritize these pixels in low-level vision tasks such as SISR. Such observation implies that the attenuation of weighting coefficients in loss function needs to be properly translated into the attention mechanism given the specific vision problem as the context. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In previous study [15], it has been shown that explicitly representing aleatoric uncertainty can lead to performance and robustness improvement to noise data in high-level vision tasks such as image segmentation. Such improvement can be explained away by attenuating the weights of pixels with large uncertainty. However, attenuation has to go the opposite direction in low-level vision tasks such as SISR - i.e., larger weights should be assigned to the pixels with high uncertainty (e.g., texture and edge pixels) because they are visually more important than pixels in smooth regions. It should be noted that existing work such as gradient rescaling strategy in GRAM [21] fails to recognize such difference and does not prioritize pixels with high uncertainty. In this paper, we propose a new adaptive weighted loss named uncertainty-driven loss (UDL) for properly turning attenuation into attention for SISR. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/84f5f28ea0e80d3f72e497ee576be39823ce4c585fb5ad50377ffdc6a519dba9.jpg",
+        "image_caption": [
+            "Figure 2: The overview of training SISR network with proposed ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ loss. The whole training process can divided into two steps; the first step estimates the uncertainty $\\pmb \\theta$ precisely and the second step generates the final mean value $f ( \\boldsymbol { y } )$ . In step1 shown in (a), the mean value $f ( \\boldsymbol { y } )$ and variance $\\pmb \\theta$ are pretrained by $\\mathcal { L } _ { \\mathrm { E S U } }$ loss. During step2, as shown in (b), the mean value $f ( y )$ network is trained by ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ loss, while the network of inferring variance $\\pmb \\theta$ is fixed. Note that the mean value $f ( \\boldsymbol { y } )$ network of step2 starts training from the pretrained network of step1. The Nearest Upsampling denotes interpolation operator. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 Estimating Uncertainty (EU) in SISR. ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            551,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As discussed in [15], there are two classes of uncertainty in Bayesian modeling: aleatoric uncertainty capturing noise inherent in observation data and epistemic uncertainty accounting for uncertainty of model about its predictions. We opt to study the former (aleatoric uncertainty) and explore its application into SISR by designing new uncertainty-driven loss (UDL) functions in this paper. In order to better quantify aleatoric uncertainty in SISR, we use ${ \\mathbf { } } _ { \\mathbf { } } \\mathbf { } _ { \\mathbf { } } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\Psi \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\Psi \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\Psi \\mathbf { } \\mathbf { } \\mathbf { } \\Psi \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf { } \\Psi \\mathbf { } \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf { } \\Psi \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf \\Psi \\Psi \\Psi \\mathbf { } \\mathbf \\Psi \\Psi \\Psi \\mathbf { } \\mathbf \\Psi \\Psi \\Psi \\mathbf { }$ to denote the low-resolution (LR) image and the corresponding high-resolution (HR) image respectively. Let $f ( \\cdot )$ denotes an arbitrary SISR network and the aleatoric uncertainty can be denoted by an additive term $\\theta _ { i }$ . This way, the overall observation model can be formulated as ",
+        "bbox": [
+            173,
+            577,
+            825,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c6917cd10cf72428dd34951975d8941fec0f0cc73f5ccfdf600236b03dec6789.jpg",
+        "text": "$$\n\\pmb { x } _ { i } = f ( \\pmb { y } _ { i } ) + \\epsilon \\pmb { \\theta } _ { i } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
+            689,
+            562,
+            705
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\epsilon$ represents the Laplace distribution with zero-mean and unit-variance. Existing deep-learning based SISR methods target at training a network to learn the SR image (mean) $f ( \\pmb { y } _ { i } )$ only. To more accurately characterize aleatoric uncertainty for SISR, we propose to estimate not only the SR image (mean) $\\dot { f } ( \\pmb { y } _ { i } )$ but also the uncertainty (variance) $\\theta _ { i }$ simultaneously. ",
+        "bbox": [
+            174,
+            707,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For a given LR image $\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }$ and corresponding HR image $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ , a Laplace distribution 2 is assumed for characterizing the likelihood function by ",
+        "bbox": [
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+            767,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/78b05099e0681ce2829097ba6446197f9636335c2ab1b2d131fbe1ec7778e47d.jpg",
+        "text": "$$\np ( \\pmb { x } _ { i } , \\pmb { \\theta } _ { i } | \\pmb { y } _ { i } ) = \\frac { 1 } { 2 \\pmb { \\theta } _ { i } } \\exp ( - \\frac { | | \\pmb { x } _ { i } - \\pmb { f } ( \\pmb { y } _ { i } ) | | _ { 1 } } { \\pmb { \\theta } _ { i } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            797,
+            645,
+            830
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $f ( \\pmb { y } _ { i } )$ and $\\theta _ { i }$ denote the SR image (mean) and the uncertainty (variance) which are learned by deep neural networks (DNNs) respectively. Then, the log likelihood can be formulated as follows, ",
+        "bbox": [
+            179,
+            830,
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+            858
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/45b9fd5fe364dded5c85f0a1e7258bc992231247e95a9ca632f625ed9d681b88.jpg",
+        "text": "$$\n\\ln p ( { \\pmb x } _ { i } , \\pmb \\theta _ { i } | { \\pmb y } _ { i } ) = - \\frac { | | { \\pmb x } _ { i } - f ( { \\pmb y } _ { i } ) | | _ { 1 } } { \\pmb \\theta _ { i } } - \\ln \\pmb \\theta _ { i } - \\ln 2\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            859,
+            666,
+            892
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/84479f818a49b71b89a280268aba9b89fc85314b12af5b6bb3ff38e05c6f146e.jpg",
+        "image_caption": [
+            "Figure 3: SISR visual quality comparisons of EDSR-S [2] with different loss function on ‘Img_005’ from Set5 [23] (bicubic-downsampling $\\times 4 )$ ). Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            88,
+            838,
+            217
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For numerical stability, we train the networks to estimate log variance $\\begin{array} { r } { s _ { i } = \\ln \\theta _ { i } } \\end{array}$ as shown in Fig. 2 (a). At last, the maximum likelihood estimation of (5) can be reformulated as the minimization of following loss function for estimating uncertainty (EU) in SISR. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8d5427d7bf970b529b8313f01d7d7a959dee085057dbb6f23cef09b6a35f00b7.jpg",
+        "text": "$$\n\\mathcal { L } _ { E U } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\exp ( - s _ { i } ) \\vert \\vert \\pmb { x } _ { i } - f ( \\pmb { y } _ { i } ) \\vert \\vert _ { 1 } + s _ { i }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            324,
+            655,
+            368
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Jeffrey’s Prior for Estimating Sparse Uncertainty (ESU) in SISR. The loss function $\\mathcal { L } _ { \\mathrm { E U } }$ includes two terms; the first one is associated with fidelity term and the second one prevents the network from predicting infinite uncertainty for all pixels. Those two terms reach equilibrium but there is no prior that imposed on the uncertainty estimation. Therefore, based on the observation that the uncertainty is sparse in view of the whole image as shown in Fig. 2, we propose to impose Jeffrey’s prior [10] $\\begin{array} { r } { p ( \\dot { \\boldsymbol { w } } ) \\propto \\frac { 1 } { w } } \\end{array}$ on uncertainty $\\theta _ { i }$ , which can be expressed as ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/09843a6651767fa7253a3ee84c2301b54eb062e5f25ba75a9540d158ea592fcd.jpg",
+        "text": "$$\n\\nu ( x _ { i } , \\theta _ { i } | y _ { i } ) = p ( x _ { i } | y _ { i } , \\theta _ { i } ) p ( \\theta _ { i } ) \\propto \\frac { 1 } { 2 \\theta _ { i } } \\exp ( - \\frac { \\left| | x _ { i } - f ( y _ { i } ) | \\right| _ { 1 } } { \\theta _ { i } } ) \\frac { 1 } { \\theta _ { i } } = \\frac { 1 } { 2 \\theta _ { i } ^ { 2 } } \\exp ( - \\frac { \\left| | x _ { i } - f ( y _ { i } ) | \\right| _ { 1 } } { \\theta _ { i } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            474,
+            830,
+            510
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Then the log likelihood and loss function can be separately formulated as follows, ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8e23f52039d3e483b03ab9b73800d0eeebc8dbc75c09cde1590340cdc979cd62.jpg",
+        "text": "$$\n\\ln p ( \\pmb { x } _ { i } | \\pmb { y } _ { i } ) = - \\frac { | | \\pmb { x } _ { i } - \\pmb { f } ( \\pmb { y } _ { i } ) | | _ { 1 } } { \\pmb { \\theta } _ { i } } - 2 \\ln \\pmb { \\theta } _ { i } - \\ln 2\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            542,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9c0d6a10fc44bd4593431de928ddd2519cd5e08ecac097def37aad9b507940a6.jpg",
+        "text": "$$\n\\mathcal { L } _ { E S U } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\exp ( - s _ { i } ) | | x _ { i } - f ( \\pmb { y } _ { i } ) | | _ { 1 } + 2 s _ { i }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            643
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The limitations of $\\mathcal { L } _ { \\bf E U }$ and ${ \\mathcal { L } } _ { \\mathbf { E S U } }$ loss. Applying $\\mathcal { L } _ { \\mathrm { E U } }$ and $\\mathcal { L } _ { \\mathrm { E S U } }$ loss leads to more accurate estimation of uncertainty (variance field), but counter-intuitively, they do not directly improve the performance of SISR. We have conducted experiments comparing those three different loss functions to verify the above claim. As shown in Tab. 1, the average PSNR and SSIM results of $\\mathcal { L } _ { \\mathrm { E S U } }$ and $\\mathcal { L } _ { \\mathrm { E U } }$ are notably lower than the original results. The reason behind this observation is that both ${ \\mathcal { L } } _ { \\mathrm { E U } }$ and $\\mathcal { L } _ { \\mathrm { E S U } }$ loss functions have incorporated the variance term $( \\pmb \\theta _ { i } )$ into the divisor of the absolution difference term. Consequently, a pixel with a large variance will be penalized after the division and has less impact on the overall loss function. Note that such attenuation of pixels with large uncertainty is preferred for high-level vision tasks, as demonstrated in previous works [15–17] on image classification [18], image segmentation [15, 16], and face recognition [17, 19]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Low-level vision tasks such as SISR are much different. As shown in Fig. 1, pixels with large uncertainty carry visually important information such as textured and edges. They need to be prioritized (opposite to attenuation) and given larger instead of smaller weights. To verify such claim, we have presented a simple example comparing the visual results between $\\mathcal { L } _ { \\mathrm { E U } }$ and $\\mathcal { L } _ { \\mathrm { E S U } }$ as shown in Fig. 3. It can be seen that the uncertainty captured by $\\mathcal { L } _ { \\mathrm { E S U } }$ loss is better than $\\mathcal { L } _ { \\mathrm { E U } }$ loss. The improvement of $\\mathcal { L } _ { \\mathrm { E S U } }$ in Eq. (9) over $\\mathcal { L } _ { \\mathrm { E U } }$ in Eq. (6) is attributed to the prioritization of pixels with large uncertainty ( $\\boldsymbol { s } _ { i }$ values). Fig. 3 (f) clearly demonstrate superiority of exploiting the sparsity constraint with the uncertainty estimation. ",
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+    {
+        "type": "table",
+        "img_path": "images/d25e9e5dfdc0d653094ce57d604ba7724cc19d3ed1d3ea9f585ab2214503b291.jpg",
+        "table_caption": [
+            "Table 1: Average PSNR and SSIM results for BI degradation on five datasets for investigating three different loss. The best performance is shown in bold. We record the results in $1 . 2 \\times 1 0 ^ { \\overline { { 5 } } }$ iterations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Base Model</td><td rowspan=\"2\">Scale</td><td rowspan=\"2\">Loss</td><td colspan=\"2\">Set5[23]</td><td colspan=\"2\">Set14 [8]</td><td colspan=\"2\">BSD100[24]</td><td colspan=\"2\">Urban100 [25]</td><td colspan=\"2\">Manga109 [26]</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td rowspan=\"3\">EDSR-S[2]</td><td rowspan=\"3\">×4</td><td>Original</td><td>30.93</td><td>0.8740</td><td>27.80</td><td>0.7627</td><td>27.05</td><td>0.7190</td><td>24.71</td><td>0.7351</td><td>28.14</td><td>0.8693</td></tr><tr><td>LEU</td><td>30.19</td><td>0.8627</td><td>27.29</td><td>0.7538</td><td>26.78</td><td>0.7120</td><td>24.21</td><td>0.7179</td><td>26.78</td><td>0.8481</td></tr><tr><td>LESU</td><td>30.31</td><td>0.8637</td><td>27.39</td><td>0.7543</td><td>26.83</td><td>0.7124</td><td>24.27</td><td>0.7192</td><td>26.92</td><td>0.8496</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2 Uncertainty-Driven Loss (UDL) for SISR ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Improvement of $\\mathcal { L } _ { \\mathrm { E S U } }$ over $\\mathcal { L } _ { \\mathrm { E U } }$ inspired us to go one step further. To better prioritize pixels with large uncertainty, we propose a new adaptive weighted loss named uncertainty-driven loss (UDL) for SISR. Unlike $\\mathcal { L } _ { \\mathrm { E S U } }$ loss putting a larger weight to the second term than $\\mathcal { L } _ { \\mathrm { E U } }$ , we suggest that the first term can also be modified to directly associate the aleatoric/data uncertainty of $f ( \\pmb { y } _ { i } )$ . That is, instead of using $e x p ( - s _ { i } )$ to attenuate the importance of pixels with large uncertainty, we need to use a monotonically increasing function to prioritize them. Linear scaling would be a natural option, which leads to the following loss function ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a0b94a8e9a5c457fea447cf6b3f9d363eea891b94c75a8266a1511a3456e937f.jpg",
+        "text": "$$\n\\mathcal { L } _ { U D L } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\hat { s _ { i } } | | \\pmb { x } _ { i } - f ( \\pmb { y } _ { i } ) | | _ { 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\hat { s } } _ { i } = s _ { i } - \\operatorname* { m i n } ( s _ { i } )$ is a non-negative linear scaling function. To prevent uncertainty value from degenerating into zeros, the result of uncertainty estimation network in the first step will be passed to the second step as the attention signal $\\displaystyle s = \\ln \\theta$ ), as shown in Fig. 2. By leveraging the log variance to represent the challenging and cumbersome pixels with higher uncertainty, we propose a new weighted loss named uncertainty-driven loss ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ . In ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ loss, texture and edge pixels with higher uncertainty tend to have larger weights than those in smooth regions. In summary, the uncertainty estimation $\\pmb \\theta$ serves as the bridge connecting two steps: it is the output of the first step; but passed on to the second step as the guidance required for calculating ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ loss. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.3 Two-step Training of Dual Networks ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As shown in Fig. 2, the whole training process can be divided into two steps; the first step estimates the uncertainty $\\pmb { \\theta }$ precisely and the second step generates the final mean value $f ( \\boldsymbol { y } )$ with the aid from the estimated uncertainty $\\pmb \\theta$ from step1. More specifically, the mean value $f ( y )$ and variance $\\theta$ are pre-trained by $\\mathcal { L } _ { \\mathrm { E S U } }$ loss during step1 as shown in Fig. 2 (a). After the uncertainty has been estimated, the mean value $f ( \\boldsymbol { y } )$ network is trained by ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ loss with variance $\\theta$ as shown in Fig. 2 (b), while the network of inferring variance $\\pmb { \\theta }$ is fixed. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Note that the mean value $f ( \\boldsymbol { y } )$ network of step2 starts training from the pre-trained network of step1. Such partial parameter sharing is a salient property of our proposed dual networks with parallel symmetric attention [27]. In theory, we can extend the two-step training into multiple-step training by alternating between the estimation of uncertainty (variance $\\pmb \\theta$ ) and mean value $f ( \\boldsymbol { y } )$ . Conceptually, an improved estimation of unknown HR image can leads to an improved estimation of aleatoric uncertainty and vice versa. This line of reasoning will lead to the pursuit of a deep equilibrium model [28] for SISR; but it is beyond the scope of this paper. ",
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+    {
+        "type": "text",
+        "text": "3.4 Discussions: Why UDL Outperforms GRAM? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To the best of our knowledge, only one work GRAM [21] has studied data uncertainty in SISR, which is the most related to our work. We will discuss connections and differences between proposed UDL and GRAM [21] here. First, both GRAM [21] and our work has found out that applying the traditional uncertainty loss designed for high-level computer vision tasks into SISR task directly results in performance decline. For high-level computer vision tasks, the pixels with higher uncertainty indicates less confidence in final inference, which needs loss attenuation. However, for SISR tasks, the pixels with higher uncertainty (e.g., texture and edge pixels) should be prioritized with larger weights because they are visually more important than pixels in smooth regions. To solve this problem, GRAM [21] proposes to use uncertainty to generate an attention mask that decreases loss attenuation. However, GRAM [21] loss still is attenuated when the variance of pixels is high. Thus, GRAM [21] is still inferior to baseline method since it still does not prioritize pixels with high uncertainty. ",
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+        "type": "text",
+        "text": "Different from GRAM, we propose an uncertainty-driven loss to assign the pixels with high variance more weight to prioritize them. Besides, modeling uncertainty under Bayesian framework allows us to leverage sparsity prior for a more precise estimation of uncertainty. Ultimately, our proposed method consists of those two technical contributions that achieve better results than baseline methods and outperform GRAM. ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 Experimental Settings ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Datasets and Metrics. 800 high-quality (2K resolution) images from the DIV2K dataset [29] have been used for training. Following EDSR [2], five standard benchmark datasets: Set5 [23], Set14 [8], BSD100[24], Urban100 [25], Manga109 [26] are used for testing. Performance evaluation in terms of of PSNR and SSIM [30] metrics is conducted on the luminance (Y) channel only. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Training Setting. We randomly select 16 RGB LR patches sized by $4 8 \\times 4 8$ as the inputs. The image patches are randomly rotated by $9 0 ^ { \\circ }$ , $1 8 0 ^ { \\circ }$ , $2 7 0 ^ { \\circ }$ and flipped horizontally. The ADAM algorithm [31] with $\\beta _ { 1 } = 0 . 9 , \\beta _ { 2 } = 0 . 9 9 9$ , $\\epsilon \\overset { \\cdot } { = } 1 0 ^ { - 8 }$ is adopted to optimize the network. The initial learning rate is $1 0 ^ { - 4 }$ and decreases by half for every $2 \\times 1 0 ^ { 5 }$ minibatch updates. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Degradation models. To demonstrate the effectiveness of our proposed uncertainty-driven loss in varying degradation scenarios, we have designed the following experiments with two different degradation models. Let BI denotes bicubic downsampling. The second one is BD which uses Gaussian blur followed by nearest downsampling to generate LR images. Specifically, we apply $1 1 \\times 1 1$ sized Gaussian kernel with a standard deviation 1.6 for blurring in our experiments. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "SISR Networks. We choose three different networks to verify the effectiveness of proposed ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ loss. The first one is EDSR-S or called baseline network in [2]. EDSR-S [2] mainly consists of 16 Resblock with 64 channels, having $1 . 5 M$ parameters. The second one is DPDNN[3] where denoiser network is U-net under model-guided framework. The last one is a big network EDSR[2], consisting of 32 Resblock with 256 channels, having $4 3 M$ parameters. The analysis of training cost can be found in our supplementary material. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 Ablation Study ",
+        "text_level": 1,
+        "bbox": [
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+            532,
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7f4faf6fa56774c602a3db7381801db8142a4661b906e94ef28d76c4d78284c7.jpg",
+        "table_caption": [
+            "Table 2: Average PSNR and SSIM results for BI degradation on five datasets for investigating three different loss. The best performance is shown in bold. We record the results in $4 \\times 1 0 ^ { 5 }$ iterations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Base Model</td><td rowspan=\"2\">Scale</td><td rowspan=\"2\">Loss</td><td colspan=\"2\">Set5[23]</td><td colspan=\"2\">Set14[8]</td><td colspan=\"2\">BSD100 [24]</td><td colspan=\"2\">Urban100[25]</td><td colspan=\"2\">Manga109 [26]</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td rowspan=\"3\">EDSR-S[2]</td><td rowspan=\"3\">×4</td><td>Original</td><td>31.61</td><td>0.8862</td><td>28.22</td><td>0.7721</td><td>27.30</td><td>0.7271</td><td>25.25</td><td>0.7575</td><td>29.31</td><td>0.8907</td></tr><tr><td>LEU+LUDL</td><td>31.83</td><td>0.8895</td><td>28.33</td><td>0.7754</td><td>27.37</td><td>0.7297</td><td>25.49</td><td>0.7665</td><td>29.70</td><td>0.8959</td></tr><tr><td>LESU+LUDL</td><td>31.90</td><td>0.8897</td><td>28.37</td><td>0.7755</td><td>27.40</td><td>0.7301</td><td>25.54</td><td>0.7671</td><td>29.77</td><td>0.8967</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To further verify the effectiveness of sparse uncertainty estimation at step1, we have conducted an ablation study to compare the final PSNR/SSIM results of ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ with $\\mathcal { L } _ { \\mathrm { E U } }$ or with $\\mathcal { L } _ { \\mathrm { E S U } }$ at step1. In our ablation study, we have used $\\times 4$ bicubic down-sampling degradation on five frequently-used benchmark datasets with EDSR-S backbone[2]. As shown in Tab. 2, both ${ \\mathcal { L } } _ { \\mathrm { E U } } { + } { \\mathcal { L } } _ { \\mathrm { U D L } }$ and ${ \\mathcal { L } } _ { \\mathrm { E S U } } { + } { \\mathcal { L } } _ { \\mathrm { U D L } }$ loss have achieved better performance than original loss. Besides, ${ \\mathcal { L } } _ { \\mathrm { E S U } } { + } { \\mathcal { L } } _ { \\mathrm { U D L } }$ loss obtains better results than $\\mathcal { L } _ { \\mathrm { E U } } { + } \\mathcal { L } _ { \\mathrm { U D L } }$ due to more accurate uncertainty estimation as shown in Fig. 3 (e) and (f). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3 Analysis of Different Weighted Loss ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "There are many different weighted loss guided by different weight maps, such as Error_map, Gradient_map which can also reveal the challenging pixels. We have conducted experiments with a weighted loss function where the weight is a pixel-wise gradient or Error_map. The PSNR results of five benchmark datasets for investigating the influence of different weighted loss functions can be summarized in Tab. 3. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The $H R$ _gradient_map and $L R$ _gradien_map denote calculating gradient map from high-resolution (ground truth) images and low-resolution images respectively. The calculation of gradient can be formulated as ",
+        "bbox": [
+            176,
+            852,
+            821,
+            893
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/933627fb1f46e3922e9c70e3f7a2a7a03802ea5b549e79f256f85e23783ab457.jpg",
+        "text": "$$\n\\begin{array} { r } { V ( i , j ) = I ( i + 1 , j ) - I ( i , j ) , H ( i , j ) = I ( i , j + 1 ) - I ( i , j ) , G ( i , j ) = | | ( V ( i , j ) , H ( i , j ) | | _ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            895,
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+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/f9dfcd48667cd9ff0446485f622af7d1d1c816ae32f369528195a4bed4556595.jpg",
+        "table_caption": [
+            "Table 3: Average PSNR and $\\Delta$ PSNR results with BI degradation on five datasets for investigating the influence of different weighted loss functions. The best performance is shown in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Weighted loss</td><td rowspan=1 colspan=1>Set5</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>Set14</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>BSD100</td><td rowspan=1 colspan=1>△</td><td rowspan=1 colspan=1>Urban100</td><td rowspan=1 colspan=1>△</td><td rowspan=1 colspan=1>Manga109</td><td rowspan=1 colspan=1>△</td></tr><tr><td rowspan=1 colspan=1>Baseline</td><td rowspan=1 colspan=1>31.61</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>28.22</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>27.30</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>25.25</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>29.31</td><td rowspan=1 colspan=1>0.00</td></tr><tr><td rowspan=1 colspan=1>Uncertainty(Ours)</td><td rowspan=1 colspan=1>31.90</td><td rowspan=1 colspan=1>0.29个</td><td rowspan=1 colspan=1>28.37</td><td rowspan=1 colspan=1>0.15个</td><td rowspan=1 colspan=1>27.40</td><td rowspan=1 colspan=1>0.10个</td><td rowspan=1 colspan=1>25.54</td><td rowspan=1 colspan=1>0.29个</td><td rowspan=1 colspan=1>29.77</td><td rowspan=1 colspan=1>0.46个</td></tr><tr><td rowspan=1 colspan=1>Error_map</td><td rowspan=1 colspan=1>31.77</td><td rowspan=1 colspan=1>0.16个</td><td rowspan=1 colspan=1>28.30</td><td rowspan=1 colspan=1>0.08个</td><td rowspan=1 colspan=1>27.35</td><td rowspan=1 colspan=1>0.05个</td><td rowspan=1 colspan=1>25.40</td><td rowspan=1 colspan=1>0.15个</td><td rowspan=1 colspan=1>29.57</td><td rowspan=1 colspan=1>0.26个</td></tr><tr><td rowspan=1 colspan=1>HR_gradient_map</td><td rowspan=1 colspan=1>31.68</td><td rowspan=1 colspan=1>0.07个</td><td rowspan=1 colspan=1>28.27</td><td rowspan=1 colspan=1>0.05个</td><td rowspan=1 colspan=1>27.35</td><td rowspan=1 colspan=1>0.05个</td><td rowspan=1 colspan=1>25.42</td><td rowspan=1 colspan=1>0.17个</td><td rowspan=1 colspan=1>29.45</td><td rowspan=1 colspan=1>0.14个</td></tr><tr><td rowspan=1 colspan=1>LR_gradient_map</td><td rowspan=1 colspan=1>31.69</td><td rowspan=1 colspan=1>0.08个</td><td rowspan=1 colspan=1>28.29</td><td rowspan=1 colspan=1>0.07个</td><td rowspan=1 colspan=1>27.35</td><td rowspan=1 colspan=1>0.05个</td><td rowspan=1 colspan=1>25.38</td><td rowspan=1 colspan=1>0.13个</td><td rowspan=1 colspan=1>29.50</td><td rowspan=1 colspan=1>0.19个</td></tr></table>",
+        "bbox": [
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+            125,
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+            198
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $I$ denotes pixels value and $i , j$ denotes position of pixels. Note that we adjust the scaling functions of Error_map, HR_gradient_map and LR_gradient_map to get the best performance. ",
+        "bbox": [
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+            226,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "From the Tab. 3, one can be observed that other weighted loss functions can indeed improve the PSNR results, but only to certain degrees. Comparing four different weight maps, our proposed uncertainty weighted loss function can bring the biggest improvement. Although the Error_map can represent the variance of a single pixel, the Error_map lacks semantic information or local information to capture a more precise estimation of variance comparing uncertainty. With regard to the gradient map of HR or LR images, those gradient maps only well match the edges of images and have a certain correlation to variance. Comparing the visual results of Error_map, HR_gradient_map and $L R$ _gradient_map with uncertainty map, those maps only detect edges of images and fail reflecting complex texture details which are important to final reconstruction performance. Therefore, uncertainty-weighted loss can is still valuable for achieving the best performance among other weighted maps. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 Analysis of Different Scaling Functions ",
+        "text_level": 1,
+        "bbox": [
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+            420,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We have conducted experiments with several various monotonically increasing functions (including linear and non-linear) and the results can be summarized in Tab. 4. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/ad9b99813ae2fe92a9e75c952a517af32d4258c1528b052af0d2abdfb09f2581.jpg",
+        "table_caption": [
+            "Table 4: Average PSNR and $\\Delta$ PSNR results with BI degradation on five datasets for investigating the influence of different scaling functions. The best performance are shown in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Scaling functions</td><td rowspan=1 colspan=1>Set5</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>Set14</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>BSD100</td><td rowspan=1 colspan=1>△</td><td rowspan=1 colspan=1>Urban100</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>Manga109</td><td rowspan=1 colspan=1>A</td></tr><tr><td rowspan=1 colspan=1>Baseline</td><td rowspan=1 colspan=1>31.61</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>28.22</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>27.30</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>25.25</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>29.31</td><td rowspan=1 colspan=1>0.00</td></tr><tr><td rowspan=1 colspan=1>s-min(s)</td><td rowspan=1 colspan=1>31.90</td><td rowspan=1 colspan=1>0.29个</td><td rowspan=1 colspan=1>28.37</td><td rowspan=1 colspan=1>0.15个</td><td rowspan=1 colspan=1>27.40</td><td rowspan=1 colspan=1>0.10个</td><td rowspan=1 colspan=1>25.54</td><td rowspan=1 colspan=1>0.29个</td><td rowspan=1 colspan=1>29.77</td><td rowspan=1 colspan=1>0.46个</td></tr><tr><td rowspan=1 colspan=1>exp(s)</td><td rowspan=1 colspan=1>31.80</td><td rowspan=1 colspan=1>0.19个</td><td rowspan=1 colspan=1>28.34</td><td rowspan=1 colspan=1>0.12个</td><td rowspan=1 colspan=1>27.40</td><td rowspan=1 colspan=1>0.10个</td><td rowspan=1 colspan=1>25.53</td><td rowspan=1 colspan=1>0.28个</td><td rowspan=1 colspan=1>29.66</td><td rowspan=1 colspan=1>0.35个</td></tr><tr><td rowspan=1 colspan=1>e.xp(s)(1/2)</td><td rowspan=1 colspan=1>31.86</td><td rowspan=1 colspan=1>0.25个</td><td rowspan=1 colspan=1>28.36</td><td rowspan=1 colspan=1>0.14↑</td><td rowspan=1 colspan=1>27.41</td><td rowspan=1 colspan=1>0.11个</td><td rowspan=1 colspan=1>25.55</td><td rowspan=1 colspan=1>0.30↑</td><td rowspan=1 colspan=1>29.71</td><td rowspan=1 colspan=1>0.40↑</td></tr><tr><td rowspan=1 colspan=1>log(s)-min(log(s))</td><td rowspan=1 colspan=1>31.89</td><td rowspan=1 colspan=1>0.28个</td><td rowspan=1 colspan=1>28.39</td><td rowspan=1 colspan=1>0.17个</td><td rowspan=1 colspan=1>27.42</td><td rowspan=1 colspan=1>0.12个</td><td rowspan=1 colspan=1>25.57</td><td rowspan=1 colspan=1>0.32个</td><td rowspan=1 colspan=1>29.74</td><td rowspan=1 colspan=1>0.43个</td></tr></table>",
+        "bbox": [
+            176,
+            527,
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+            599
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The best and second-best performances are shown in bold. Overall, four various monotonically increasing functions have achieved better results than the baseline method. The best two scaling functions are linear scaling and log scaling with a slight difference as shown in the above table. Since the linear scaling function achieves a comparable performance with low computational cost, we advocate this choice in this paper. ",
+        "bbox": [
+            173,
+            625,
+            825,
+            695
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/73cd26da93be776365b72653a8fa3c0ab3ef6d5ec3e615945e501a6f05f46988.jpg",
+        "image_caption": [
+            "Figure 4: SISR visual quality comparisons of EDSR-S [2] with different loss function on ‘Img_004’ and $\\mathrm { \\hbar } ^ { 4 } \\mathrm { I m g \\_ 0 1 } 6 ^ { , }$ from Urban100 [25] (bicubic-downsampling $\\times 4 _ { , }$ ). Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            710,
+            828,
+            876
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/8c5e414f36b7a9a967963d4eb9e5876347915ad184c571633d9e88046b0f6e74.jpg",
+        "image_caption": [
+            "Figure 5: SISR visual quality comparisons of DPDNN [3] with different loss function on ‘Img_095 from BSD100 [24] (bicubic-downsampling $\\times 4 _ { , }$ ). Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            87,
+            831,
+            202
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Average PSNR and SSIM results for BI degradation on five benchmark datasets. The best performance is shown in bold. Note that ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ -Ours denotes adopting $\\mathcal { L } _ { \\mathrm { E S U } }$ at step1 and ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ at step2 for simplicity. ",
+        "bbox": [
+            176,
+            237,
+            820,
+            279
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/dcde4d3847dba2ec8aac40849dc76a146d46d7b9546cf0f61b9796726685bd24.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=1>Base Model</td><td rowspan=2 colspan=1>Scale</td><td rowspan=2 colspan=1>Loss</td><td rowspan=1 colspan=2>Set5[23]</td><td rowspan=1 colspan=1>Set]</td><td rowspan=1 colspan=1>4[8]</td><td rowspan=1 colspan=2>BSD100 [24]</td><td rowspan=1 colspan=1>Urban</td><td rowspan=1 colspan=1>00[25]</td><td rowspan=1 colspan=1>Manga</td><td rowspan=1 colspan=1>109[26]</td></tr><tr><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td></tr><tr><td rowspan=1 colspan=1>EDSR-S [2]</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>37.6637.4837.95</td><td rowspan=1 colspan=1>0.85940.95890.9604</td><td rowspan=1 colspan=1>33.2232.9933.50</td><td rowspan=1 colspan=1>0.91460.91260.9165</td><td rowspan=1 colspan=1>31.9531.7632.13</td><td rowspan=1 colspan=1>0.89690.89460.8991</td><td rowspan=1 colspan=1>30.7130.1131.54</td><td rowspan=1 colspan=1>0.92050.91340.9304</td><td rowspan=1 colspan=1>37.7937.3838.38</td><td rowspan=1 colspan=1>0.97520.97390.9767</td></tr><tr><td rowspan=1 colspan=1>DPDNN [3]</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>37.7537.7438.00</td><td rowspan=1 colspan=1>0.96000.95970.9605</td><td rowspan=1 colspan=1>33.3033.2733.63</td><td rowspan=1 colspan=1>0.91500.91480.9176</td><td rowspan=1 colspan=1>32.0931.9832.16</td><td rowspan=1 colspan=1>0.89900.89730.8995</td><td rowspan=1 colspan=1>31.5030.9731.72</td><td rowspan=1 colspan=1>0.92200.92380.9331</td><td rowspan=1 colspan=1>-38.1438.55</td><td rowspan=1 colspan=1>-0.97580.9769</td></tr><tr><td rowspan=1 colspan=1>EDSR [2]</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>38.1137.8738.29</td><td rowspan=1 colspan=1>0.96020.96040.9615</td><td rowspan=1 colspan=1>33.9233.4334.14</td><td rowspan=1 colspan=1>0.91950.91640.9236</td><td rowspan=1 colspan=1>32.3232.0832.40</td><td rowspan=1 colspan=1>0.90130.89900.9027</td><td rowspan=1 colspan=1>32.9331.4632.99</td><td rowspan=1 colspan=1>0.93510.93010.9446</td><td rowspan=1 colspan=1>39.1037.9139.53</td><td rowspan=1 colspan=1>0.97730.97650.9787</td></tr><tr><td rowspan=1 colspan=1>EDSR-S [2]</td><td rowspan=1 colspan=1>×3</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>33.9033.2734.15</td><td rowspan=1 colspan=1>0.92310.91780.9251</td><td rowspan=1 colspan=1>29.9529.6030.15</td><td rowspan=1 colspan=1>0.83520.82980.8388</td><td rowspan=1 colspan=1>28.8528.6028.99</td><td rowspan=1 colspan=1>0.79960.79360.8021</td><td rowspan=1 colspan=1>27.3026.5227.72</td><td rowspan=1 colspan=1>0.83440.81420.8430</td><td rowspan=1 colspan=1>32.5231.1432.97</td><td rowspan=1 colspan=1>0.93690.92580.9406</td></tr><tr><td rowspan=1 colspan=1>DPDNN [3]</td><td rowspan=1 colspan=1>×3</td><td rowspan=1 colspan=1>OriginalGRAM[21]LUDL-Ours</td><td rowspan=1 colspan=1>33.9333.9234.30</td><td rowspan=1 colspan=1>0.92400.92410.9267</td><td rowspan=1 colspan=1>30.0230.0030.31</td><td rowspan=1 colspan=1>0.83600.83620.8419</td><td rowspan=1 colspan=1>29.0028.8629.10</td><td rowspan=1 colspan=1>0.80100.80000.8047</td><td rowspan=1 colspan=1>27.6127.3728.02</td><td rowspan=1 colspan=1>0.84200.83530.8505</td><td rowspan=1 colspan=1>132.4133.27</td><td rowspan=1 colspan=1>10.93730.9435</td></tr><tr><td rowspan=1 colspan=1>EDSR [2]</td><td rowspan=1 colspan=1>×3</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>34.6534.3434.83</td><td rowspan=1 colspan=1>0.92800.92700.9312</td><td rowspan=1 colspan=1>30.5230.2830.69</td><td rowspan=1 colspan=1>0.84620.84120.8497</td><td rowspan=1 colspan=1>29.2529.0729.28</td><td rowspan=1 colspan=1>0.80930.80440.8109</td><td rowspan=1 colspan=1>28.8027.9828.99</td><td rowspan=1 colspan=1>0.86530.87890.8697</td><td rowspan=1 colspan=1>34.1733.3234.63</td><td rowspan=1 colspan=1>0.94760.94320.9502</td></tr><tr><td rowspan=1 colspan=1>EDSR-S [2]</td><td rowspan=1 colspan=1>×4</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>31.6131.0831.90</td><td rowspan=1 colspan=1>0.88620.87870.8897</td><td rowspan=1 colspan=1>28.2227.8928.37</td><td rowspan=1 colspan=1>0.77210.76700.7755</td><td rowspan=1 colspan=1>27.3027.1227.40</td><td rowspan=1 colspan=1>0.72710.72290.7301</td><td rowspan=1 colspan=1>25.2524.8125.54</td><td rowspan=1 colspan=1>0.75750.74290.7671</td><td rowspan=1 colspan=1>29.3128.1829.77</td><td rowspan=1 colspan=1>0.89070.87620.8967</td></tr><tr><td rowspan=1 colspan=1>DPDNN [3]</td><td rowspan=1 colspan=1>×4</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>31.7231.8932.20</td><td rowspan=1 colspan=1>0.88900.89130.8944</td><td rowspan=1 colspan=1>28.2828.3728.60</td><td rowspan=1 colspan=1>0.77300.77720.7819</td><td rowspan=1 colspan=1>27.4427.4127.56</td><td rowspan=1 colspan=1>0.72900.73140.7356</td><td rowspan=1 colspan=1>25.5325.6326.09</td><td rowspan=1 colspan=1>0.76800.77080.7862</td><td rowspan=1 colspan=1>-29.7030.38</td><td rowspan=1 colspan=1>-0.90030.9082</td></tr><tr><td rowspan=1 colspan=1>EDSR [2]</td><td rowspan=1 colspan=1>×4</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>32.4632.3232.59</td><td rowspan=1 colspan=1>0.89680.89710.8998</td><td rowspan=1 colspan=1>28.8028.7328.87</td><td rowspan=1 colspan=1>0.78760.78580.7889</td><td rowspan=1 colspan=1>27.7127.6627.78</td><td rowspan=1 colspan=1>0.74200.73950.7431</td><td rowspan=1 colspan=1>26.6426.3526.75</td><td rowspan=1 colspan=1>0.80330.79550.8054</td><td rowspan=1 colspan=1>31.0230.7331.24</td><td rowspan=1 colspan=1>0.91480.91250.9167</td></tr></table>",
+        "bbox": [
+            176,
+            287,
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+            592
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.5 Results with BI Degradation Model ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            626,
+            459,
+            641
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For bicubic downsampling (BI), we have compared proposed ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ loss function with GRAM [21] and original loss functions such as $M S E$ or $\\mathcal { L } _ { 1 }$ on three different SISR networks. The average PSNR and SSIM results in Tab. 5 are cited from corresponding papers or retrained from officially released code. It is easy to see that our proposed ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ loss function is superior to GRAM [21] and original loss functions such as $M S E$ or $\\mathcal { L } _ { 1 }$ in terms of PSNR and SSIM values. Note that the improvements achieved by our proposed method do not bring any additional computation cost during testing time. Comparing EDSR-S ( $. 5 M$ parameters) with EDSR ( $4 3 M$ parameters), our proposed ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ can bring lightweight networks with more greater performance improvements than big ones. The visual image comparison results are reported in Fig. 4 and Fig. 5. As shown in Fig. 4, our proposed ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ has recovered with fewer visible artifacts (e.g., the circular pattern of the roof and the lines on the glassy surface) than original loss and GRAM [21]. Fig. 4 (f) depicts the uncertainty learned by our ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ , revealing the challenging pixels with poor reconstruction performance. From Fig. 5, vertical center-line of window has been recover more clear with precisely estimated uncertainty shown in (e) and (f), while DPDNN and DPDNN-GRAM [21] failed to discern shown in (c) and (d) respectively. More visual comparisons can be found in supplementary material. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/53cac63c8f32e7b4c0a2ea046a1b913d91cea8f26e0b8c0066bd68e8ab038eeb.jpg",
+        "table_caption": [
+            "Table 6: Average PSNR and SSIM results for BD degradation on five benchmark datasets. The best performance is shown in bold. Note that ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ -Ours denotes adopting $\\mathcal { L } _ { \\mathrm { E S U } }$ at step1 and ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ at step2 for simplicity. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=1>Base Model</td><td rowspan=2 colspan=1>Scale</td><td rowspan=2 colspan=1>Loss</td><td rowspan=1 colspan=2>Set5[23]</td><td rowspan=1 colspan=2>Set14 [8]</td><td rowspan=1 colspan=3>BSD100 [24]</td><td rowspan=1 colspan=1>Urban</td><td rowspan=1 colspan=1>00[25]</td><td rowspan=1 colspan=1>Manga</td><td rowspan=1 colspan=1>09[26]</td></tr><tr><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=2>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td><td rowspan=1 colspan=1>PSNR</td><td rowspan=1 colspan=1>SSIM</td></tr><tr><td rowspan=1 colspan=1>EDSR-S [2]</td><td rowspan=1 colspan=1>×4</td><td rowspan=1 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=1 colspan=1>31.7030.9831.97</td><td rowspan=1 colspan=1>0.89030.87910.8927</td><td rowspan=1 colspan=1>28.3727.8528.45</td><td rowspan=1 colspan=1>0.77780.76670.7793</td><td rowspan=1 colspan=2>27.3727.0527.41</td><td rowspan=1 colspan=1>0.73200.72250.7321</td><td rowspan=1 colspan=1>25.7724.7925.95</td><td rowspan=1 colspan=1>0.77890.74520.7842</td><td rowspan=1 colspan=1>29.8328.1230.18</td><td rowspan=1 colspan=1>0.90140.87730.9053</td></tr><tr><td rowspan=2 colspan=1>DPDNN [3]</td><td rowspan=2 colspan=1>×4</td><td rowspan=2 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=2 colspan=1>31.8631.7532.03</td><td rowspan=2 colspan=1>0.89230.89130.8949</td><td rowspan=2 colspan=1>28.3828.3328.60</td><td rowspan=2 colspan=1>0.77800.77650.7828</td><td rowspan=2 colspan=2>27.3627.3227.48</td><td rowspan=1 colspan=1>0.73110.7302</td><td rowspan=1 colspan=1>25.8225.62</td><td rowspan=1 colspan=1>0.78120.7739</td><td rowspan=1 colspan=1>29.7729.55</td><td rowspan=2 colspan=1>0.90330.90030.9097</td></tr><tr><td rowspan=1 colspan=1>27.48</td><td rowspan=1 colspan=1>0.7355</td><td rowspan=1 colspan=1>26.21</td><td rowspan=1 colspan=1>0.7931</td><td rowspan=1 colspan=1>30.35</td></tr><tr><td rowspan=5 colspan=1>EDSR [2]</td><td rowspan=5 colspan=1>×4</td><td rowspan=5 colspan=1>OriginalGRAM [21]LUDL-Ours</td><td rowspan=5 colspan=1>32.1732.1332.37</td><td rowspan=5 colspan=1>0.89750.89630.8986</td><td rowspan=5 colspan=1>28.6528.5727.74</td><td rowspan=1 colspan=1>0.7856</td><td rowspan=4 colspan=2>27.5927.49</td><td rowspan=1 colspan=1>0.7400</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td rowspan=3 colspan=1>0.7362</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td rowspan=2 colspan=1>26.5626.19</td><td rowspan=2 colspan=1>0.80430.7916</td><td rowspan=2 colspan=1>30.6630.48</td><td rowspan=3 colspan=1>0.91340.90970.9149</td></tr><tr><td rowspan=1 colspan=1>0.7822</td></tr><tr><td rowspan=1 colspan=1>0.7867</td><td rowspan=1 colspan=2>27.62</td><td rowspan=1 colspan=1>0.7407</td><td rowspan=1 colspan=1>26.65</td><td rowspan=1 colspan=1>0.8065</td><td rowspan=1 colspan=1>30.81</td></tr></table>",
+        "bbox": [
+            176,
+            785,
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+            904
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "4.6 Results with BD Degradation Model ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            261,
+            464,
+            275
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "For blur downsampling (BD), we have compared proposed ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ loss function with GRAM [21] and original loss functions such as $M S E$ or $\\mathcal { L } _ { 1 }$ on three different baseline networks. The average PSNR and SSIM results in Tab. 6 are retrained from officially released code. It is easy to see that our propose ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ loss function is superior to GRAM [21] and original loss functions such as $M S E$ or $\\mathcal { L } _ { 1 }$ in terms of PSNR and SSIM values. The visual image comparison results of BD degradation are reported in Fig. 6 and Fig. 7. Note that the BD degradation involves Gaussian blur, increasing difficulty in recovering structure patterns. From Fig. 6, we can see that our SR result (Fig. 6 (e)) of $\\mathrm { \\nabla ^ { \\cdot } I m g \\ 1 0 9 ^ { \\cdot } }$ is the closest to that of the ground-truth. In another challenging image $\\mathbf { \\dot { \\tau } } ^ { \\mathrm { \\prime } } \\mathbf { I m g 0 7 8 } ^ { \\prime }$ from Urban100 [25]), our method can recover much more reliable textured details as shown in Fig. 7 (e); while all other methods have severe aliasing artifacts (i.e., distorted tile patterns). The visual quality improvement achieved by ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ is mainly due to the fact that our proposed method makes full use of the captured uncertainty to train deep networks focusing on the challenging pixels with high uncertainty. More visual comparisons can be found in our supplementary material. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/f9c5c9971afa483363c84d669e494ec595f7f0b61c9b0f74c821154dab28cc4e.jpg",
+        "image_caption": [
+            "Figure 6: SISR visual quality comparisons of DPDNN [3] with different loss function on ‘Img_109 from Manga109 [26] (blur-downsampling $\\times 4 _ { , }$ ). Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/c30523fd5bb0d22d4e47da96524fc967f6d1e620b06c8a9bb6c42510827f5d53.jpg",
+        "image_caption": [
+            "Figure 7: SISR visual quality comparisons of EDSR [2] with different loss function on ‘Img_078’ from Urban100 [25] (blur-downsampling $\\times 4 _ { , }$ ). Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "text": "5 Conclusion ",
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+        "text": "In this paper, we propose a new adaptive weighted loss ${ \\mathcal { L } } _ { \\mathrm { U D L } }$ for SISR to train SISR networks focusing on challenging pixels with high uncertainty (e.g., textured and edge pixels). Specifically, variance estimation is introduced into SISR so that the high-resolution images (mean) and their corresponding uncertainty (variance) can be learned simultaneously. Moreover, modeling uncertainty under Bayesian framework allows us to leverage sparsity prior for a more precise estimation of uncertainty. Ultimately, pixels with large certainty (e.g., texture and edge pixels) will be prioritized for SISR according to their importance to visual quality. For the first time, we demonstrate that such uncertainty-driven loss can achieve better results than $M S E$ or $\\mathcal { L } _ { 1 }$ loss. Experimental results on three popular SISR networks show that our proposed uncertainty-driven loss has achieved better PSNR performance than traditional loss functions without any increased computation during testing. ",
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+        "text": "Acknowledgement ",
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+        "text": "This work was supported in part by the National Key R&D Program of China under Grant 2018AAA0101400 and the Natural Science Foundation of China under Grant 61991451, Grant 61632019, Grant 61621005, and Grant 61836008. Xin Li’s work is partially supported by the NSF under grants IIS-1951504 and OAC-1940855, the DoJ/NIJ under grant NIJ 2018-75-CX-0032, and the WV Higher Education Policy Commission Grant (HEPC.dsr.18.5). ",
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+        "text": "References ",
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