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- .gitattributes +560 -0
- parse/train/-kfLEqppEm_/-kfLEqppEm_.md +660 -0
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.gitattributes
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@@ -12563,3 +12563,563 @@ pdf/dev/GKfNB4BegL.pdf filter=lfs diff=lfs merge=lfs -text
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| 1 |
+
# CONVEX REGULARIZATION IN MONTE-CARLO TREE SEARCH
|
| 2 |
+
|
| 3 |
+
Anonymous authors Paper under double-blind review
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
Monte-Carlo planning and Reinforcement Learning (RL) are essential to sequential decision making. The recent AlphaGo and AlphaZero algorithms have shown how to successfully combine these two paradigms to solve large scale sequential decision problems. These methodologies exploit a variant of the well-known UCT algorithm to trade off the exploitation of good actions and the exploration of unvisited states, but their empirical success comes at the cost of poor sample-efficiency and high computation time. In this paper, we overcome these limitations by studying the benefit of convex regularization in Monte-Carlo Tree Search (MCTS) to drive exploration efficiently and to improve policy updates, as already observed in RL. First, we introduce a unifying theory on the use of generic convex regularizers in MCTS, deriving the first regret analysis of regularized MCTS and showing that it guarantees an exponential convergence rate. Second, we exploit our theoretical framework to introduce novel regularized backup operators for MCTS, based on the relative entropy of the policy update and on the Tsallis entropy of the policy. We provide an intuitive demonstration of the effect of each regularizer empirically verifying the consequence of our theoretical results on a toy problem. Finally, we show how our framework can easily be incorporated in AlphaGo and AlphaZero, and we empirically show the superiority of convex regularization w.r.t. representative baselines, on well-known RL problems across several Atari games.
|
| 8 |
+
|
| 9 |
+
# 1 INTRODUCTION
|
| 10 |
+
|
| 11 |
+
Monte-Carlo Tree Search (MCTS) is a well-known algorithm to solve decision-making problems through the combination of Monte-Carlo planning with an incremental tree structure (Coulom, 2006). Although standard MCTS is only suitable for problems with discrete state and action spaces, recent advances have shown how to enable MCTS in continuous problems (Silver et al., 2016; Yee et al., 2016). Most remarkably, AlphaGo (Silver et al., 2016) and AlphaZero (Silver et al., 2017b;a) couple MCTS with neural networks trained using Reinforcement Learning (RL) (Sutton & Barto, 1998) methods, e.g., Deep $Q$ -Learning (Mnih et al., 2015), to speed up learning of large scale problems with continuous state space. In particular, a neural network is used to compute value function estimates of states as a replacement of time-consuming Monte-Carlo rollouts, and another neural network is used to estimate policies as a probability prior for the therein introduced PUCT action selection method, a variant of well-known UCT sampling strategy commonly used in MCTS for exploration (Kocsis et al., 2006). Despite AlphaGo and AlphaZero achieving state-of-the-art performance in games with high branching factor like Go (Silver et al., 2016) and Chess (Silver et al., 2017a), both methods suffer from poor sample-efficiency, mostly due to the polynomial convergence rate of PUCT (Xiao et al., 2019). This problem, combined with the high computational time to evaluate the deep neural networks, significantly hinder the applicability of both methodologies.
|
| 12 |
+
|
| 13 |
+
In this paper, we provide a unified theory of the use of convex regularization in MCTS, which proved to be an efficient solution for driving exploration and stabilizing learning in RL (Schulman et al., 2015; 2017a; Haarnoja et al., 2018; Buesing et al., 2020). In particular, we show how a regularized objective function in MCTS can be seen as an instance of the Legendre-Fenchel transform, similar to previous findings on the use of duality in RL (Mensch & Blondel, 2018; Geist et al., 2019; Nachum & Dai, 2020) and game theory (Shalev-Shwartz & Singer, 2006; Pavel, 2007). Establishing our theoretical framework, we can derive the first regret analysis of regularized MCTS, and prove that a generic convex regularizer guarantees an exponential convergence rate to the solution of the regularized objective function, which improves on the polynomial rate of PUCT. These results provide a theoretical ground for the use of arbitrary entropy-based regularizers in MCTS until now limited to maximum entropy (Xiao et al., 2019), among which we specifically study the relative entropy of policy updates, drawing on similarities with trust-region and proximal methods in RL (Schulman et al., 2015; 2017b), and the Tsallis entropy, used for enforcing the learning of sparse policies (Lee et al., 2018). Moreover, we provide an empirical analysis of the toy problem introduced in Xiao et al. (2019) to intuitively evince the practical consequences of our theoretical results for each regularizer. Finally, we empirically evaluate the proposed operators in AlphaGo and AlphaZero on problems of increasing complexity, from classic RL problems to an extensive analysis of Atari games, confirming the benefit of our novel operators compared to maximum entropy and, in general, the superiority of convex regularization in MCTS w.r.t. classic methods.
|
| 14 |
+
|
| 15 |
+
# 2 PRELIMINARIES
|
| 16 |
+
|
| 17 |
+
# 2.1 MARKOV DECISION PROCESSES
|
| 18 |
+
|
| 19 |
+
We consider the classical definition of a finite-horizon Markov Decision Process (MDP) as a 5- tuple $\mathcal { M } = \langle \mathcal { S } , \mathcal { A } , \mathcal { R } , \mathcal { P } , \gamma \rangle$ , where $s$ is the state space, $\mathcal { A }$ is the finite discrete action space, $\mathcal { R } : \mathcal { S } \times \mathcal { A } \times \mathcal { S } \to \mathbb { R }$ is the reward function, $\mathcal { P } : \mathcal { S } \times \mathcal { A } \mathcal { S }$ is the transition kernel, and $\gamma \in [ 0 , 1 )$ is the discount factor. A policy $\pi \in \Pi : \mathcal { S } \times \mathcal { A } \mathbb { R }$ is a probability distribution of the event of executing an action $a$ in a state $s$ . A policy $\pi$ induces a value function corresponding to the expected cumulative discounted reward collected by the agent when executing action $a$ in state s, and following the policy π thereafter: Qπ(s, a) , E -P∞k=0 γkri+k+1|si = s, ai = a, π, timal policy $\pi ^ { * }$ , which is the policy that maximizes the expected cumulative discounted reward. The optimal policy corresponds to the one satisfying the optimal Bellman equation (Bellman, 1954) $\begin{array} { r } { Q ^ { * } ( s , a ) \triangleq \int _ { S } \mathcal { P } ( s ^ { \prime } | s , a ) \left[ \mathcal { R } ( s , a , s ^ { \prime } ) + \gamma \operatorname* { m a x } _ { a ^ { \prime } } Q ^ { * } ( s ^ { \prime } , a ^ { \prime } ) \right] d s ^ { \prime } } \end{array}$ , and is the fixed point of the optimal Bellman operator $\begin{array} { r } { \mathcal { T } ^ { * } Q ( s , a ) \triangleq \int _ { S } \mathcal { P } ( s ^ { \prime } | s , a ) \left[ \mathcal { R } ( s , a , s ^ { \prime } ) + \gamma \operatorname* { m a x } _ { a ^ { \prime } } Q ( s ^ { \prime } , a ^ { \prime } ) \right] d s ^ { \prime } } \end{array}$ . Additionally, we define the Bellman operator under the policy $\pi$ as ${ \mathcal { T } } _ { \pi } Q ( s , a )$ , $\begin{array} { r } { \int _ { \mathcal { S } } \mathcal { P } ( s ^ { \prime } | s , a ) \left[ \mathcal { R } ( s , a , s ^ { \prime } ) + \gamma \int _ { \mathcal { A } } \pi ( a ^ { \prime } | s ^ { \prime } ) Q ( s ^ { \prime } , a ^ { \prime } ) d a ^ { \prime } \right] d s ^ { \prime } } \end{array}$ , the optimal value function $V ^ { \ast } ( s )$ , $\operatorname* { m a x } _ { a \in \mathcal { A } } Q ^ { * } ( s , a )$ , and the value function under the policy $\pi$ as $V ^ { \pi } ( s ) \triangleq \operatorname* { m a x } _ { a \in \mathcal { A } } Q ^ { \pi } ( s , a )$ .
|
| 20 |
+
|
| 21 |
+
# 2.2 MONTE-CARLO TREE SEARCH AND UPPER CONFIDENCE BOUNDS FOR TREES
|
| 22 |
+
|
| 23 |
+
Monte-Carlo Tree Search (MCTS) is a planning strategy based on a combination of Monte-Carlo sampling and tree search to solve MDPs. MCTS builds a tree where the nodes are the visited states of the MDP, and the edges are the actions executed in each state. MCTS converges to the optimal policy (Kocsis et al., 2006; Xiao et al., 2019), iterating over a loop composed of four steps:
|
| 24 |
+
|
| 25 |
+
1. Selection: starting from the root node, a tree-policy is executed to navigate the tree until a node with unvisited children, i.e. expandable node, is reached;
|
| 26 |
+
2. Expansion: the reached node is expanded according to the tree policy;
|
| 27 |
+
3. Simulation: run a rollout, e.g. Monte-Carlo simulation, from the visited child of the current node to the end of the episode;
|
| 28 |
+
4. Backup: use the collected reward to update the action-values $Q ( \cdot )$ of the nodes visited in the trajectory from the root node to the expanded node.
|
| 29 |
+
|
| 30 |
+
The tree-policy used to select the action to execute in each node needs to balance the use of already known good actions, and the visitation of unknown states. The Upper Confidence bounds for Trees (UCT) sampling strategy (Kocsis et al., 2006) extends the use of the well-known UCB1 sampling strategy for multi-armed bandits (Auer et al., 2002), to MCTS. Considering each node corresponding to a state $s \in S$ as a different bandit problem, UCT selects an action $a \in { \mathcal { A } }$ applying an upper bound to the action-value function
|
| 31 |
+
|
| 32 |
+
$$
|
| 33 |
+
\mathrm { U C T } ( s , a ) = Q ( s , a ) + \epsilon \sqrt { \frac { \log N ( s ) } { N ( s , a ) } } ,
|
| 34 |
+
$$
|
| 35 |
+
|
| 36 |
+
where $N ( s , a )$ is the number of executions of action $a$ in state $s$ , $\begin{array} { r } { N ( s ) = \sum _ { a } N ( s , a ) } \end{array}$ , and $\epsilon$ is a constant parameter to tune exploration. UCT asymptotically converges to the optimal action-value function $Q ^ { * }$ , for all states and actions, with the probability of executing a suboptimal action at the root node approaching 0 with a polynomial rate $\textstyle { \dot { O } } ( { \frac { 1 } { t } } )$ , for a simulation budget $t$ (Kocsis et al., 2006; Xiao et al., 2019).
|
| 37 |
+
|
| 38 |
+
# 3 REGULARIZED MONTE-CARLO TREE SEARCH
|
| 39 |
+
|
| 40 |
+
The success of RL methods based on entropy regularization comes from their ability to achieve state-of-the-art performance in decision making and control problems, while enjoying theoretical guarantees and ease of implementation (Haarnoja et al., 2018; Schulman et al., 2015; Lee et al., 2018). However, the use of entropy regularization is MCTS is still mostly unexplored, although its advantageous exploration and value function estimation would be desirable to reduce the detrimental effect of high-branching factor in AlphaGo and AlphaZero. To the best of our knowledge, the MENTS algorithm (Xiao et al., 2019) is the first and only method to combine MCTS and entropy regularization. In particular, MENTS uses a maximum entropy regularizer in AlphaGo, proving an exponential convergence rate to the solution of the respective softmax objective function and achieving state-of-the-art performance in some Atari games (Bellemare et al., 2013). In the following, motivated by the success in RL and the promising results of MENTS, we derive a unified theory of regularization in MCTS based on the Legendre-Fenchel transform (Geist et al., 2019), that generalizes the use of maximum entropy of MENTS to an arbitrary convex regularizer. Notably, our theoretical framework enables to rigorously motivate the advantages of using maximum entropy and other entropy-based regularizers, such as relative entropy or Tsallis entropy, drawing connections with their RL counterparts TRPO (Schulman et al., 2015) and Sparse DQN (Lee et al., 2018), as MENTS does with Soft Actor-Critic (SAC) (Haarnoja et al., 2018).
|
| 41 |
+
|
| 42 |
+
# 3.1 LEGENDRE-FENCHEL TRANSFORM
|
| 43 |
+
|
| 44 |
+
Consider an MDP $\mathcal { M } = \langle \mathcal { S } , \mathcal { A } , \mathcal { R } , \mathcal { P } , \gamma \rangle$ , as previously defined. Let $\Omega : \Pi \mathbb { R }$ be a strongly convex function. For a policy $\pi _ { s } = \pi ( \cdot | s )$ and $\bar { Q } _ { s } = Q ( \bar { s } , \cdot ) \in \mathbb { R } ^ { 4 }$ , the Legendre-Fenchel transform (or convex conjugate) of $\Omega$ is $\Omega ^ { * } : \mathbb { R } ^ { A } \xrightarrow [ ] { } \mathbb { R }$ , defined as:
|
| 45 |
+
|
| 46 |
+
$$
|
| 47 |
+
\Omega ^ { \ast } ( Q _ { s } ) \triangleq \operatorname* { m a x } _ { \pi _ { s } \in \Pi _ { s } } \mathcal { T } _ { \pi _ { s } } Q _ { s } - \tau \Omega ( \pi _ { s } ) ,
|
| 48 |
+
$$
|
| 49 |
+
|
| 50 |
+
where the temperature $\tau$ specifies the strength of regularization. Among the several properties of the Legendre-Fenchel transform, we use the following (Mensch & Blondel, 2018; Geist et al., 2019).
|
| 51 |
+
|
| 52 |
+
Proposition 1 Let $\Omega$ be strongly convex.
|
| 53 |
+
|
| 54 |
+
• Unique maximizing argument: $\nabla \Omega ^ { * }$ is Lipschitz and satisfies
|
| 55 |
+
|
| 56 |
+
$$
|
| 57 |
+
\nabla \Omega ^ { * } ( Q _ { s } ) = \arg \operatorname* { m a x } _ { \pi _ { s } \in \Pi _ { s } } \mathcal { T } _ { \pi _ { s } } Q _ { s } - \tau \Omega ( \pi _ { s } ) .
|
| 58 |
+
$$
|
| 59 |
+
|
| 60 |
+
• Boundedness: if there are constants $L _ { \Omega }$ and $U _ { \Omega }$ such that for all $\pi _ { s } \in \Pi _ { s }$ , we have $L _ { \Omega } \leq$ $\Omega ( \pi _ { s } ) \leq U _ { \Omega }$ , then
|
| 61 |
+
|
| 62 |
+
$$
|
| 63 |
+
\operatorname* { m a x } _ { a \in \mathcal { A } } Q _ { s } ( a ) - \tau U _ { \Omega } \le \Omega ^ { * } ( Q _ { s } ) \le \operatorname* { m a x } _ { a \in \mathcal { A } } Q _ { s } ( a ) - \tau L _ { \Omega } .
|
| 64 |
+
$$
|
| 65 |
+
|
| 66 |
+
• Contraction: for any $Q _ { 1 } , Q _ { 2 } \in \mathbb { R } ^ { S \times A }$
|
| 67 |
+
|
| 68 |
+
$$
|
| 69 |
+
\parallel \Omega ^ { * } ( Q _ { 1 } ) - \Omega ^ { * } ( Q _ { 2 } ) \parallel _ { \infty } \leq \gamma \parallel Q _ { 1 } - Q _ { 2 } \parallel _ { \infty } .
|
| 70 |
+
$$
|
| 71 |
+
|
| 72 |
+
Although the Legendre-Fenchel transform $\Omega ^ { * }$ applies to every strongly convex function, for the purpose of this work we only consider a representative set of entropic regularizers.
|
| 73 |
+
|
| 74 |
+
# 3.2 REGULARIZED BACKUP AND TREE POLICY
|
| 75 |
+
|
| 76 |
+
In MCTS, each node of the tree represents a state $s \in S$ and contains a visitation count $N ( s , a )$ . Given a trajectory, we define $n \big ( s _ { T } \big )$ as the leaf node corresponding to the reached state $s _ { T }$ . Let $s _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } . . . , s _ { T }$ be the state action trajectory in a simulation, where $n { \left( { { s _ { T } } } \right) }$ is a leaf node of $\tau$ . Whenever a node $n { \left( { { s _ { T } } } \right) }$ is expanded, the respective action values (Equation 6) are initialized as $Q _ { \Omega } ( s _ { T } , a ) = 0$ , and $\dot { N } ( \dot { s } _ { T } , a ) = 0$ for all $a \in { \mathcal { A } }$ . For all nodes in the trajectory, the visitation count is updated by $N ( s _ { t } , a _ { t } ) = N ( s _ { t } , a _ { t } ) + 1$ , and the action-values by
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+
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+
$$
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+
Q _ { \Omega } ( s _ { t } , a _ { t } ) = \left\{ \begin{array} { l l } { r ( s _ { t } , a _ { t } ) + \gamma \rho } & { \mathrm { i f ~ } t = T } \\ { r ( s _ { t } , a _ { t } ) + \gamma \Omega ^ { * } ( Q _ { \Omega } ( s _ { t + 1 } ) / \tau ) ) } & { \mathrm { i f ~ } t < T } \end{array} \right.
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| 80 |
+
$$
|
| 81 |
+
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+
where $Q _ { \Omega } ( s _ { t + 1 } ) \in \mathbb { R } ^ { A }$ with components $Q _ { \Omega } ( s _ { t + 1 } , a ) , \forall a \in \mathcal { A }$ , and $\rho$ is an estimate returned from an evaluation function computed in $s _ { T }$ , e.g. a discounted cumulative reward averaged over multiple rollouts, or the value-function of node $n ( s _ { T + 1 } )$ returned by a value-function approximator, e.g. a neural network pretrained with deep $Q$ -learning (Mnih et al., 2015), as done in (Silver et al., 2016; Xiao et al., 2019). We revisit the E2W sampling strategy limited to maximum entropy regularization (Xiao et al., 2019) and, through the use of the convex conjugate in Equation (6), we derive a novel sampling strategy that generalizes to any convex regularizer
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+
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+
$$
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+
\pi _ { t } ( a _ { t } | s _ { t } ) = ( 1 - \lambda _ { s _ { t } } ) \nabla \Omega ^ { \ast } ( Q _ { \Omega } ( s _ { t } ) / \tau ) ( a _ { t } ) + \frac { \lambda _ { s _ { t } } } { | \mathcal { A } | } ,
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+
$$
|
| 87 |
+
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+
where $\lambda _ { s _ { t } } = \epsilon | \boldsymbol { \mathcal { A } } | \big / \mathrm { l o g } ( \sum _ { a } N ( s _ { t } , a ) + 1 )$ with $\epsilon > 0$ as an exploration parameter, and $\nabla \Omega ^ { * }$ depends on the measure in use (see Table 1 for maximum, relative, and Tsallis entropy). We call this sampling strategy Extended Empirical Exponential Weight (E3W) to highlight the extension of E2W from maximum entropy to a generic convex regularizer.
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+
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# 3.3 CONVERGENCE RATE TO REGULARIZED OBJECTIVE
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We show that the regularized value $V _ { \Omega }$ can be effectively estimated at the root state $s \in { \mathcal { S } }$ , with the assumption that each node in the tree has a $\sigma ^ { 2 }$ -subgaussian distribution. This result extends the analysis provided in (Xiao et al., 2019), which is limited to the use of maximum entropy.
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+
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+
Theorem 1 At the root node s where $N ( s )$ is the number of visitations, with $\epsilon > 0$ , $V _ { \Omega } ( s )$ is the estimated value, with constant $C$ and $\hat { C }$ , we have
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+
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+
$$
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+
\mathbb { P } ( | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | > \epsilon ) \le C \exp \{ - \frac { N ( s ) \epsilon } { \hat { C } \sigma ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \} ,
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+
$$
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+
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+
where $V _ { \Omega } ( s ) = \Omega ^ { * } ( Q _ { s } )$ and $V _ { \Omega } ^ { * } ( s ) = \Omega ^ { * } ( Q _ { s } ^ { * } )$ . From this theorem, we obtain that the convergence rate of choosing the best action $a ^ { * }$ at the root node, when using the E3W strategy, is exponential.
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+
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Theorem 2 Let $a _ { t }$ be the action returned by E3W at step t. For large enough t and constants $C , { \hat { C } }$
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+
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+
$$
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+
\mathbb { P } ( a _ { t } \neq a ^ { * } ) \leq C t \exp \{ - \frac { t } { \hat { C } \sigma ( \log ( t ) ) ^ { 3 } } \} .
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+
$$
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+
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+
# 4 ENTROPY-REGULARIZATION BACKUP OPERATORS
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From the introduction of a unified view of generic strongly convex regularizers as backup operators in MCTS, we narrow the analysis to entropy-based regularizers. For each entropy function, Table 1 shows the Legendre-Fenchel transform and the maximizing argument, which can be respectively replaced in our backup operation (Equation 6) and sampling strategy E3W (Equation 7). Using maximum entropy retrieves the maximum entropy MCTS problem introduced in the MENTS algorithm (Xiao et al., 2019). This approach closely resembles the maximum entropy RL framework used to encourage exploration (Haarnoja et al., 2018; Schulman et al., 2017a). We introduce two novel MCTS algorithms based on the minimization of relative entropy of the policy update, inspired by trust-region (Schulman et al., 2015) and proximal optimization methods (Schulman et al., 2017b) in RL, and on the maximization of Tsallis entropy, which has been more recently introduced in RL as an effective solution to enforce the learning of sparse policies (Lee et al., 2018). We call these algorithms RENTS and TENTS. Contrary to maximum and relative entropy, the definition of the
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+
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Legendre-Fenchel and maximizing argument of Tsallis entropy is non-trivial, being
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+
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+
$$
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+
\begin{array} { r l } & { ~ \Omega ^ { * } ( Q _ { t } ) = \tau \cdot \mathrm { s p m a x } ( Q _ { t } ( s , \cdot ) / \tau ) , } \\ & { ~ \nabla \Omega ^ { * } ( Q _ { t } ) = \operatorname* { m a x } \Bigg ( \displaystyle \frac { Q _ { t } ( s , a ) } { \tau } - \displaystyle \frac { \sum _ { a \in { \mathcal { K } } } Q _ { t } ( s , a ) / \tau - 1 } { | { \mathcal { K } } | } , 0 \Bigg ) , } \end{array}
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+
$$
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+
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+
where spmax is defined for any function $f : \mathcal { S } \times \mathcal { A } \mathbb { R }$ as
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+
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+
$$
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+
\operatorname { s p m a x } ( f ( s , \cdot ) ) \triangleq \sum _ { a \in { \mathcal { K } } } \left( { \frac { f ( s , a ) ^ { 2 } } { 2 } } - { \frac { ( \sum _ { a \in { \mathcal { K } } } f ( s , a ) - 1 ) ^ { 2 } } { 2 | K | ^ { 2 } } } \right) + { \frac { 1 } { 2 } } ,
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+
$$
|
| 123 |
+
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+
and $\kappa$ is the set of actions that satisfy $\begin{array} { r } { 1 + i f ( s , a _ { i } ) > \sum _ { j = 1 } ^ { i } f ( s , a _ { j } ) } \end{array}$ , with $a _ { i }$ indicating the action with the $i$ -th largest value of $f ( s , a )$ (Lee et al., 2018).
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+
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+
Table 1: List of entropy regularizers with Legendre-Fenchel transforms and maximizing arguments.
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+
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+
<table><tr><td>Entropy</td><td>Regularizer Ω2(πs)</td><td>Legendre-Fenchel Ω*(Q s)</td><td> Max argument VΩ*(Qs)</td></tr><tr><td rowspan="2">Maximum</td><td rowspan="2">∑π(a|s)logπ(a|s)</td><td>log∑e Q(s,a) T</td><td>Q(s,a） e T</td></tr><tr><td>Qt(s,a)</td><td>Ωe Q(s,b) T Qt(s,a）</td></tr><tr><td rowspan="2">Relative</td><td rowspan="2">DkL(πt(a|s)llπt-1(a|s))log∑aTt-1(als)e</td><td rowspan="2">T</td><td>Tt-1(a|s)e T</td></tr><tr><td>∑Tt-1(b|s)e Qt(s，b） T</td></tr><tr><td>Tsallis</td><td>( π(a|s) -1)</td><td>Equation (10)</td><td>Equation (11)</td></tr></table>
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+
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+
# 4.1 REGRET ANALYSIS
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+
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+
At the root node, let each children node $i$ be assigned with a random variable $X _ { i }$ , with mean value $V _ { i }$ , while the quantities related to the optimal branch are denoted by $^ *$ , e.g. mean value $V ^ { * }$ . At each nthe root node, at timestep n, is defined as RUCTn = nV ∗ − Pnt=1 Vit . Similarly, we define the regret timestep $n$ , the mean value of variable $X _ { i }$ is $V _ { i _ { n } }$ . The pseudo-regret (Coquelin $\&$ Munos, 2007) at of $\mathrm { E } 3 \mathrm { W }$ at the root node of the tree as
|
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+
|
| 134 |
+
$$
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+
R _ { n } = n V ^ { * } - \sum _ { t = 1 } ^ { n } V _ { i _ { t } } = n V ^ { * } - \sum _ { t = 1 } ^ { n } \mathbb { I } ( i _ { t } = i ) V _ { i _ { t } } = n V ^ { * } - \sum _ { i } V _ { i } \sum _ { t = 1 } ^ { n } { \hat { \pi } } _ { t } ( a _ { i } | s ) ,
|
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+
$$
|
| 137 |
+
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+
where $\hat { \pi } _ { t } ( \cdot )$ is the policy at time step $t$ , and $\mathbb { I } ( \cdot )$ is the indicator function.
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+
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+
Theorem 3 Let $\begin{array} { r } { \kappa _ { i } = \nabla \Omega ^ { * } ( a _ { i } | s ) + \frac { L } { p } \sqrt { \dot { C } \sigma ^ { 2 } \log \frac { C } { \delta } } / 2 n _ { i } } \end{array}$ , and $\begin{array} { r } { \chi _ { i } = \nabla \Omega ^ { * } ( a _ { i } | s ) - \frac { L } { p } \sqrt { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } / 2 n } , } \end{array}$ where $\nabla \Omega ^ { * } ( . | s )$ is the policy with respect to the mean value vector $V ( \cdot )$ at the root node s. For any $\delta > 0$ , with probability at least $1 - \delta$ , ∃ constant $L , p , C , { \hat { C } }$ so that the pseudo regret $R _ { n }$ satisfies
|
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+
|
| 142 |
+
$$
|
| 143 |
+
n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \kappa _ { i } + \frac { L } { p } \big ( \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \gamma } \big ) \Big ) \leq R _ { n } \leq n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \chi _ { i } - \frac { L } { p } \big ( \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \gamma } \big ) \Big ) .
|
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+
$$
|
| 145 |
+
|
| 146 |
+
This theorem provides bounds for the regret of E3W using a generic convex regularizer $\Omega$ ; thus, we can easily retrieve from it the regret bound for each entropy regularizer. Let $\begin{array} { r } { m = \operatorname* { m i n } _ { a } \nabla \Omega ^ { * } ( a | s ) } \end{array}$ .
|
| 147 |
+
|
| 148 |
+
Corollary 1 Maximum entropy: $\begin{array} { r } { n V ^ { * } - \tilde { n } \sum _ { i } V _ { i } \Big ( \kappa _ { i } + L \big ( \frac { \tau \log | A | } { 1 - \gamma } \big ) \Big ) \leq R _ { n } \leq n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \chi _ { i } - L \big ( \frac { \tau \log | A | } { 1 - \gamma } \big ) \Big ) . } \end{array}$
|
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+
|
| 150 |
+
$\begin{array} { r } { n V ^ { * } - \tilde { n } \sum _ { i } V _ { i } \Big ( \kappa _ { i } + L \big ( \frac { \tau ^ { \langle \log | A | - \frac { 1 } { m } \rangle } } { 1 - \gamma } \big ) \Big ) \leq R _ { n } \leq n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \chi _ { i } - L \big ( \frac { \tau ( \log | A | - \frac { 1 } { m } ) } { 1 - \gamma } \big ) \Big ) . } \end{array}$
|
| 151 |
+
|
| 152 |
+
# Corollary 3 Tsallis entropy:
|
| 153 |
+
|
| 154 |
+
$$
|
| 155 |
+
\begin{array} { r } { n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \kappa _ { i } + \frac { L } { 2 } \big ( \frac { | A | - 1 } { 2 | A | } \frac { \tau } { 1 - \gamma } \big ) \Big ) \leq R _ { n } \leq n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \chi _ { i } - \frac { L } { 2 } \big ( \frac { | A | - 1 } { 2 | A | } \frac { \tau } { 1 - \gamma } \big ) \Big ) . } \end{array}
|
| 156 |
+
$$
|
| 157 |
+
|
| 158 |
+
Remarks. The regret bound of UCT and its variance have already been analyzed for nonregularized MCTS with binary tree (Coquelin & Munos, 2007). On the contrary, our regret bound analysis in Theorem 3 applies to generic regularized MCTS. From the specialized bounds in the corollaries, we observe that the maximum and relative entropy share similar results, although the bounds for relative entropy are slightly smaller due to $\textstyle { \frac { 1 } { m } }$ . Remarkably, the bounds for Tsallis entropy become tighter for increasing number of actions, which translates in limited regret in problems with high branching factor. This result establishes the advantage of Tsallis entropy in complex problems w.r.t. to other entropy regularizers, as empirically confirmed by the positive results in several Atari games described in Section 5.
|
| 159 |
+
|
| 160 |
+
# 4.2 ERROR ANALYSIS
|
| 161 |
+
|
| 162 |
+
We analyse the error of the regularized value estimate at the root node $n ( s )$ w.r.t. the optimal value: $\varepsilon _ { \Omega } = \dot { V _ { \Omega } } ( s ) - \dot { V ^ { \ast } } ( s )$ .
|
| 163 |
+
|
| 164 |
+
Theorem 4 For any $\delta > 0$ and generic convex regularizer $\Omega$ , with some constant $C , { \hat { C } }$ , with probability at least $1 - \delta$ , $\varepsilon _ { \Omega }$ satisfies
|
| 165 |
+
|
| 166 |
+
$$
|
| 167 |
+
- \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } - \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \gamma } \leq \varepsilon _ { \Omega } \leq \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } .
|
| 168 |
+
$$
|
| 169 |
+
|
| 170 |
+
To give a better understanding of the effect of each entropy regularizer in Table 1, we specialize the bound in Equation 14 to each of them. From (Lee et al., 2018), we know that for maximum entropy $\begin{array} { r } { \Omega ( { \boldsymbol \pi } _ { t } ) \stackrel { - } { = } \sum _ { a } \pi _ { t } \log \pi _ { t } } \end{array}$ , we have $- \log | \mathcal { A } | \ \leq \ \Omega ( \pi _ { t } ) \ \leq \ 0$ ; for relative entropy $\Omega ( \pi _ { t } ) =$ $\mathrm { K L } ( \pi _ { t } | | \pi _ { t - 1 } )$ , if we define $m = \mathrm { m i n } _ { a } \pi _ { t - 1 } ( a | s )$ , then we can derive $0 \leq \Omega ( \pi _ { t } ) \leq - \log | \mathcal { A } | +$ $\log { \frac { 1 } { m } }$ ; and for Tsallis entropy $\Omega ( \pi _ { t } ) = { \textstyle { \frac { 1 } { 2 } } } ( \parallel \pi _ { t } \parallel _ { 2 } ^ { 2 } - 1 )$ , we have $- \frac { | \ r _ { A } | - 1 } { 2 | \ r _ { A } | } \le \Omega ( \pi _ { t } ) \le 0$ . Then,
|
| 171 |
+
|
| 172 |
+
Corollary 5 relative entropy error: − $- \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log { \frac { C } { \delta } } } { 2 N ( s ) } } - \frac { \tau ( \log | A | - \log \frac { 1 } { m } ) } { 1 - \gamma } \leq \varepsilon _ { \Omega } \leq \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } .$
|
| 173 |
+
|
| 174 |
+
Corollary 6 Tsallis entropy error: $- \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } - \frac { | A | - 1 } { 2 | A | } \frac { \tau } { 1 - \gamma } \leq \varepsilon _ { \Omega } \leq \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } .$
|
| 175 |
+
|
| 176 |
+
These results show that when the number of actions $| { \cal { A } } |$ is large, TENTS enjoys the smallest error;
|
| 177 |
+
moreover, we also see that lower bound of RENTS is always smaller than for MENTS.
|
| 178 |
+
|
| 179 |
+
# 5 EMPIRICAL EVALUATION
|
| 180 |
+
|
| 181 |
+
In this section, we empirically evaluate the benefit of the proposed entropy-based MCTS regularizers. First, we complement our theoretical analysis with an empirical study of the synthetic tree toy problem introduced in Xiao et al. (2019), which serves as a simple scenario to give an interpretable demonstration of the effects of our theoretical results in practice. Second, we compare to AlphaGo and AlphaZero (Silver et al., 2016; 2017a), recently introduced to enable MCTS to solve large scale problems with high branching factor. Our implementation is a simplified version of the original algorithms, where we remove various tricks in favor of better interpretability. For the same reason, we do not compare with the most recent and state-of-the-art variant of AlphaZero known as MuZero (Schrittwieser et al., 2019), as this is a slightly different solution highly tuned to maximize performance, and a detailed description of its implementation is not available.
|
| 182 |
+
|
| 183 |
+
# 5.1 SYNTHETIC TREE
|
| 184 |
+
|
| 185 |
+
This toy problem is introduced in Xiao et al. (2019) to highlight the improvement of MENTS over UCT. It consists of a tree with branching factor $k$ and depth $d$ . Each edge of the tree is assigned a random value between 0 and 1. At each leaf, a Gaussian distribution is used as an evaluation function resembling the return of random rollouts. The mean of the Gaussian distribution is the sum of the values assigned to the edges connecting the root node to the considered leaf, while the standard deviation is $\bar { \sigma } = 0 . 0 5 ^ { 1 }$ . For stability, all the means are normalized between 0 and 1. As in Xiao et al. (2019), we create 5 trees on which we perform 5 different runs in each, resulting in 25 experiments, for all the combinations of branching factor $k = \{ 2 , 4 , 6 , 8 , 1 0 , 1 2 , 1 4 , 1 6 \}$ and depth $d = \{ 1 , 2 , 3 , 4 , 5 \}$ , computing: (i) the value estimation error at the root node w.r.t. the regularized optimal value: $\begin{array} { r } { \varepsilon _ { \Omega } = V _ { \Omega } - V * ; } \end{array}$ (ii) the value estimation error at the root node w.r.t. the unregularized optimal value: $\varepsilon _ { \mathrm { U C T } } = V _ { \Omega } - V * _ { \mathrm { U C T } } $ ; (iii) the regret $R$ as in Equation (13). For a fair comparison, we use fixed $\tau = 0 . 1$ and $\epsilon = 0 . 1$ across all algorithms. Figure 1 and 2 show how UCT and each regularizer behave for different configurations of the tree. We observe that, while RENTS and MENTS converge slower for increasing tree sizes, TENTS is robust w.r.t. the size of the tree and almost always converges faster than all other methods to the respective optimal value. Notably, the optimal value of TENTS seems to be very close to the one of UCT, i.e. the optimal value of the unregularized objective, and also converges faster than the one estimated by UCT, while MENTS and RENTS are considerably further from this value. In terms of regret, UCT explores less than the regularized methods and it is less prone to high regret, at the cost of slower convergence time. Nevertheless, the regret of TENTS is the smallest between the ones of the other regularizers, which seem to explore too much. These results show a general superiority of TENTS in this toy problem, also confirming our theoretical findings about the advantage of TENTS in terms of approximation error (Corollary 6) and regret (Corollary 3), in problems with many actions.
|
| 186 |
+
|
| 187 |
+
|
| 188 |
+
Figure 1: For each algorithm, we show the convergence of the value estimate at the root node to the respective optimal value (top), to the UCT optimal value (middle), and the regret (bottom).
|
| 189 |
+
|
| 190 |
+
|
| 191 |
+
Figure 2: For different branching factor $k$ (rows) and depth $d$ (columns), the heatmaps show: the absolute error of the value estimate at the root node after the last simulation of each algorithm w.r.t. the respective optimal value (a), and w.r.t. the optimal value of UCT (b); regret at the root node (c).
|
| 192 |
+
|
| 193 |
+
|
| 194 |
+
Figure 3: Cumulative rewards of AlphaZero with UCT and entropy-based operators, in CartPole (a) and Acrobot (b). Results are averaged over 5 and 10 seeds and show $9 5 \%$ confidence intervals.
|
| 195 |
+
|
| 196 |
+
# 5.2 ENTROPY-REGULARIZED ALPHAZERO
|
| 197 |
+
|
| 198 |
+
In its standard form, AlphaZero (Silver et al., 2017a) uses the PUCT sampling strategy, a variant of UCT (Kocsis et al., 2006) that samples actions according to the policy
|
| 199 |
+
|
| 200 |
+
$$
|
| 201 |
+
\mathit { P U C T } ( s , a ) = { Q } ( s , a ) + \epsilon { P } ( s , a ) { \frac { \sqrt { N ( s ) } } { 1 + N ( s , a ) } } ,
|
| 202 |
+
$$
|
| 203 |
+
|
| 204 |
+
where $P$ is a prior probability on action selection, and $\epsilon$ is an exploration constant. A value network and a policy network are used to compute, respectively, the action-value function $Q$ and the prior policy $P$ . We use a single neural network, with 2 hidden layers composed of 128 ELU units, and two output layer respectively for the action-value function and the policy. We run 500 AlphaZero episodes, where each episode is composed of 300 steps. A step consists of running 32 MCTS simulations from the root node, as defined in Section 2, using the action-value function computed by the value network instead of using Monte-Carlo rollouts. At the end of each cycle, the average action-value of the root node is computed and stored, the tree is expanded using the given sampling strategy, and the root node is updated with the reached node. At the end of the episode, a minibatch of 32 samples is built from the 300 stored action-values, and the network is trained with one step of gradient descent using RMSProp with learning rate 0.001. The entropy-regularized variants of AlphaZero can be simply derived replacing the average backup operator, with the desired entropy function, and replacing PUCT with E3W using the respective maximizing argument and $\epsilon = 0 . 1$ .
|
| 205 |
+
|
| 206 |
+
Cartpole and Acrobot. Figure 3 shows the cumulative reward of standard AlphaZero based on PUCT, and the three entropy-regularized variants, on the Cartpole and Acrobot discrete control problems (Brockman et al., 2016). While standard AlphaZero clearly lacks good convergence and stability, the entropy-based variants behave differently according to the problem. First, although not significantly superior, RENTS exhibits the most stable learning and faster convergence, confirming the benefit of relative entropy in control problems as already known for trust-region methods in RL (Schulman et al., 2015). Second, considering the small number of discrete actions in the problems, TENTS cannot benefit from the learning of sparse policies and shows slightly unstable learning in Cartpole, even though the overall performance is satisfying in both problems. Last, MENTS solves the problems slightly slower than RENTS, but reaches the same final performance. Although the results on these simple problems are not conclusive to assert the superiority of one method over the other, they definitely confirm the advantage of regularization in MCTS, and hint at the benefit of the use of relative entropy in control problems. Further analysis on more complex control problems will be desirable (e.g. MuJoCo (Todorov et al., 2012)), but the need to account for continuous actions, a non-trivial setting for MCTS, makes it out of the scope of this paper.
|
| 207 |
+
|
| 208 |
+
Table 2: Average score in Atari over 100 seeds per game. Bold denotes no statistically significant difference to the highest mean (t-test, $p < 0 . 0 5$ ). Bottom row shows # no difference to highest mean.
|
| 209 |
+
|
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<table><tr><td></td><td>UCT</td><td>MaxMCTS</td><td>MENTS</td><td>RENTS</td><td>TENTS</td></tr><tr><td>Alien</td><td>1, 486.80</td><td>1,461.10</td><td>1, 508.60</td><td>1,547.80</td><td>1, 568.60</td></tr><tr><td>Amidar</td><td>115.62</td><td>124.92</td><td>123.30</td><td>125.58</td><td>121.84</td></tr><tr><td>Asterix</td><td>4,855.00</td><td>5,484.50</td><td>5,576.00</td><td>5,743.50</td><td>5,647.00</td></tr><tr><td>Asteroids</td><td>873.40</td><td>899.60</td><td>1,414.70</td><td>1,486.40</td><td>1,642.10</td></tr><tr><td>Atlantis</td><td>35,182.00</td><td>35,720.00</td><td>36,277.00</td><td>35,314.00</td><td>35,756.00</td></tr><tr><td>BankHeist</td><td>475.50</td><td>458.60</td><td>622.30</td><td>636.70</td><td>631.40</td></tr><tr><td>BeamRider</td><td>2,616.72</td><td>2,661.30</td><td>2,822.18</td><td>2,558.94</td><td>2,804.88</td></tr><tr><td>Breakout</td><td>303.04</td><td>296.14</td><td>309.03</td><td>300.35</td><td>316.68</td></tr><tr><td>Centipede</td><td>1, 782.18</td><td>1,728.69</td><td>2,012.86</td><td>2,253.42</td><td>2,258.89</td></tr><tr><td>DemonAttack</td><td>579.90</td><td>640.80</td><td>1,044.50</td><td>1,124.70</td><td>1,113.30</td></tr><tr><td>Enduro</td><td>129.28</td><td>124.20</td><td>128.79</td><td>134.88</td><td>132.05</td></tr><tr><td>Frostbite</td><td>1,244.00</td><td>1,332.10</td><td>2,388.20</td><td>2,369.80</td><td>2,260.60</td></tr><tr><td>Gopher</td><td>3,348.40</td><td>3,303.00</td><td>3,536.40</td><td>3,372.80</td><td>3,447.80</td></tr><tr><td>Hero</td><td>3,009.95</td><td>3,010.55</td><td>3,044.55</td><td>3,077.20</td><td>3,074.00</td></tr><tr><td>MsPacman</td><td>1,940.20</td><td>1,907.10</td><td>2,018.30</td><td>2,190.30</td><td>2,094.40</td></tr><tr><td>Phoenix</td><td>2,747.30</td><td>2,626.60</td><td>3,098.30</td><td>2,582.30</td><td>3,975.30</td></tr><tr><td>Qbert</td><td>7,987.25</td><td>8,033.50</td><td>8,051.25</td><td>8,254.00</td><td>8,437.75</td></tr><tr><td>Robotank</td><td>11.43</td><td>11.00</td><td>11.59</td><td>11.51</td><td>11.47</td></tr><tr><td>Seaquest</td><td>3,276.40</td><td>3,217.20</td><td>3,312.40</td><td>3,345.20</td><td>3,324.40</td></tr><tr><td>Solaris</td><td>895.00</td><td>923.20</td><td>1, 118.20</td><td>1,115.00</td><td>1,127.60</td></tr><tr><td>SpaceInvaders</td><td>778.45</td><td>835.90</td><td>832.55</td><td>867.35</td><td>822.95</td></tr><tr><td>WizardOfWor</td><td>685.00</td><td>666.00</td><td>1,211.00</td><td>1,241.00</td><td>1,231.00</td></tr><tr><td>#Highest mean</td><td>6/22</td><td>7/22</td><td>17/22</td><td>16/22</td><td>22/22</td></tr></table>
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# 5.3 ENTROPY-REGULARIZED ALPHAGO
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The learning time of AlphaZero can be slow in problems with high branching factor, due to the need of a large number of MCTS simulations for obtaining good estimates of the randomly initialized action-values. To overcome this problem, AlphaGo (Silver et al., 2016) initializes the action-values using the values retrieved from a pretrained network, which is kept fixed during the training.
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Atari. Atari 2600 (Bellemare et al., 2013) is a popular benchmark for testing deep RL methodologies (Mnih et al., 2015; Van Hasselt et al., 2016; Bellemare et al., 2017) but still relatively disregarded in MCTS. We use a Deep $Q$ -Network, pretrained using the same experimental setting of Mnih et al. (2015), to initialize the action-value function of each node after expansion as $Q _ { i n i t } ^ { - } ( s , a ) = \left( Q ( s , a ) - V ( s ) \right) / \tau$ , for MENTS and TENTS, as done in Xiao et al. (2019). For RENTS we init Qi $\mathbf { \Phi } _ { n i t } ( s , a ) = \log { P _ { \mathrm { p r i o r } } ( a | s ) ) } + \left( Q ( s , a ) - V ( s ) \right) / \tau$ , where $P _ { \mathrm { p r i o r } }$ is the Boltzmann distribution induced by action-values $Q ( s , . )$ computed from the network. Each experimental run consists of 512 MCTS simulations. The temperature $\tau$ is optimized for each algorithm and game via grid-search between 0.01 and 1. The discount factor is $\gamma = 0 . 9 9$ , and for PUCT the exploration constant is $c = 0 . 1$ . Table 2 shows the performance, in terms of cumulative reward, of standard AlphaGo with PUCT and our three regularized versions, on 22 Atari games. Moreover, we test also AlphaGo using the MaxMCTS backup (Khandelwal et al., 2016) for further comparison with classic baselines. We observe that regularized MCTS dominates other baselines, in particular TENTS achieves the highest scores in all the 22 games, showing that sparse policies are more effective in Atari. This can be explained by Corollary 6 which shows that Tsallis entropy can lead to a lower error at the root node even with a high number of actions compared to relative or maximum entropy.
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# 6 CONCLUSION
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We introduced a theory of convex regularization in Monte-Carlo Tree Search (MCTS) based on the Legendre-Fenchel transform. Exploiting this theoretical framework, we studied the regret of MCTS when using a generic strongly convex regularizer, and we proved that it has an exponential convergence rate. We use these results to motivate the use of entropy regularization in MCTS, particularly considering maximum, relative, and Tsallis entropy. Finally, we test regularized MCTS algorithms in discrete control problems and Atari games, showing its advantages over other methods.
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# A RELATED WORK
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Entropy regularization is a common tool for controlling exploration in Reinforcement Learning (RL) and has lead to several successful methods (Schulman et al., 2015; Haarnoja et al., 2018; Schulman et al., 2017a; Mnih et al., 2016). Typically specific forms of entropy are utilized such as maximum entropy (Haarnoja et al., 2018) or relative entropy (Schulman et al., 2015). This approach is an instance of the more generic duality framework, commonly used in convex optimization theory. Duality has been extensively studied in game theory (Shalev-Shwartz & Singer, 2006; Pavel, 2007) and more recently in RL, for instance considering mirror descent optimization (Montgomery & Levine, 2016; Mei et al., 2019), drawing the connection between MCTS and regularized policy optimization (Grill et al., 2020), or formalizing the RL objective via Legendre-Rockafellar duality (Nachum & Dai, 2020). Recently (Geist et al., 2019) introduced regularized Markov Decision Processes, formalizing the RL objective with a generalized form of convex regularization, based on the Legendre-Fenchel transform. In this paper, we provide a novel study of convex regularization in MCTS, and derive relative entropy (KL-divergence) and Tsallis entropy regularized MCTS algorithms, i.e. RENTS and TENTS respectively. Note that the recent maximum entropy MCTS algorithm MENTS (Xiao et al., 2019) is a special case of our generalized regularized MCTS. Unlike MENTS, RENTS can take advantage of any action distribution prior, in the experiments the prior is derived using Deep $Q$ -learning (Mnih et al., 2015). On the other hand, TENTS allows for sparse action exploration and thus higher dimensional action spaces compared to MENTS. In experiments, both RENTS and TENTS outperform MENTS.
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Several works focus on modifying classical MCTS to improve exploration. UCB1-tuned (Auer et al., 2002) modifies the upper confidence bound of UCB1 to account for variance in order to improve exploration. (Tesauro et al., 2012) proposes a Bayesian version of UCT, which obtains better estimates of node values and uncertainties given limited experience. Many heuristic approaches based on specific domain knowledge have been proposed, such as adding a bonus term to value estimates (Gelly & Wang, 2006; Teytaud & Teytaud, 2010; Childs et al., 2008; Kozelek, 2009; Chaslot et al., 2008) or prior knowledge collected during policy search (Gelly & Silver, 2007; Helmbold & Parker-Wood, 2009; Lorentz, 2010; Tom, 2010; Hoock et al., 2010). (Khandelwal et al., 2016) formalizes and analyzes different on-policy and off-policy complex backup approaches for MCTS planning based on RL techniques. (Vodopivec et al., 2017) proposes an approach called SARSAUCT, which performs the dynamic programming backups using SARSA (Rummery, 1995). Both (Khandelwal et al., 2016) and (Vodopivec et al., 2017) directly borrow value backup ideas from RL to estimate the value at each tree node, but they do not provide any proof of convergence.
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# B PROOFS
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Let $\hat { r }$ and $r$ be respectively the average and the the expected reward at the leaf node, and the reward distribution at the leaf node be $\sigma ^ { 2 }$ -sub-Gaussian.
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Lemma 1 For the stochastic bandit problem E3W guarantees that, for $t \geq 4$ ,
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+
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+
$$
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+
\mathbb { P } \big ( \mathrm { \normalfont ~ } r - \hat { r } _ { t } \mathrm { \normalfont ~ } _ { \infty } \ge \frac { 2 \sigma } { \log ( 2 + t ) } \big ) \le 4 | A | \exp \Big ( - \frac { t } { ( \log ( 2 + t ) ) ^ { 3 } } \Big ) .
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+
$$
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+
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+
Proof 1 Let us define $N _ { t } ( a )$ as the number of times action a have been chosen until time $t$ , and Nˆt(a) = Pts=1 πs(a), where πs(a) is the E3W policy at time step s. By choosing λs = l $\begin{array} { r } { \lambda _ { s } = \frac { \left. A \right. } { \log \left( 1 + s \right) } } \end{array}$ og(1+s) , it follows that for all $a$ and $t \geq 4$ ,
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+
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+
$$
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+
\begin{array} { l } { \displaystyle \hat { N } _ { t } ( a ) = \sum _ { s = 1 } ^ { t } \pi _ { s } ( a ) \geq \sum _ { s = 1 } ^ { t } \frac { 1 } { \log ( 1 + s ) } \geq \sum _ { s = 1 } ^ { t } \frac { 1 } { \log ( 1 + s ) } - \frac { s / ( s + 1 ) } { ( \log ( 1 + s ) ) ^ { 2 } } } \\ { \displaystyle \geq \int _ { 1 } ^ { 1 + t } \frac { 1 } { \log ( 1 + s ) } - \frac { s / ( s + 1 ) } { ( \log ( 1 + s ) ) ^ { 2 } } d s = \frac { 1 + t } { \log ( 2 + t ) } - \frac { 1 } { \log 2 } \geq \frac { t } { 2 \log ( 2 + t ) } . } \end{array}
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+
$$
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+
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+
From Theorem $2 . I 9$ in Wainwright (2019), we have the following concentration inequality:
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+
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+
$$
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+
\mathbb { P } ( | N _ { t } ( a ) - \hat { N } _ { t } ( a ) | > \epsilon ) \le 2 \exp \{ - \frac { \epsilon ^ { 2 } } { 2 \sum _ { s = 1 } ^ { t } \sigma _ { s } ^ { 2 } } \} \le 2 \exp \{ - \frac { 2 \epsilon ^ { 2 } } { t } \} ,
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+
$$
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| 349 |
+
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+
where $\sigma _ { s } ^ { 2 } \le 1 / 4$ is the variance of a Bernoulli distribution with $p = \pi _ { s } ( k )$ at time step $s$ . We define the event
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| 351 |
+
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+
$$
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+
E _ { \epsilon } = \{ \forall a \in \mathcal { A } , | \hat { N } _ { t } ( a ) - N _ { t } ( a ) | \leq \epsilon \} ,
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+
$$
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| 355 |
+
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| 356 |
+
and consequently
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+
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+
$$
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+
\mathbb { P } ( | \hat { N } _ { t } ( a ) - N _ { t } ( a ) | \geq \epsilon ) \leq 2 | A | \exp ( - \frac { 2 \epsilon ^ { 2 } } { t } ) .
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+
$$
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| 361 |
+
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+
Conditioned on the event $E _ { \epsilon }$ , for $\begin{array} { r } { \epsilon = \frac { t } { 4 \log ( 2 + t ) } } \end{array}$ , we have $\begin{array} { r } { N _ { t } ( a ) \geq \frac { t } { 4 \log ( 2 + t ) } } \end{array}$ . For any action a by the definition of sub-gaussian,
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| 363 |
+
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| 364 |
+
$$
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+
\mathbb { P } \Bigg ( \vert r ( a ) - \hat { r } _ { t } ( a ) \vert > \sqrt { \frac { 8 \sigma ^ { 2 } \log ( \frac { 2 } { \delta } ) \log ( 2 + t ) } { t } } \Bigg ) \leq \mathbb { P } \Bigg ( \vert r ( a ) - \hat { r } _ { t } ( a ) \vert > \sqrt { \frac { 2 \sigma ^ { 2 } \log ( \frac { 2 } { \delta } ) } { N _ { t } ( a ) } } \Bigg ) \leq \delta
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| 366 |
+
$$
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+
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| 368 |
+
by choosing a $\delta$ satisfying $\begin{array} { r } { \log ( \frac { 2 } { \delta } ) = \frac { 1 } { ( \log ( 2 + t ) ) ^ { 3 } } } \end{array}$ , we have
|
| 369 |
+
|
| 370 |
+
$$
|
| 371 |
+
\mathbb { P } \Bigg ( | r ( a ) - \hat { r } _ { t } ( a ) | > \sqrt { \frac { 2 \sigma ^ { 2 } \log \left( \frac { 2 } { \delta } \right) } { N _ { t } ( a ) } } \Bigg ) \leq 2 \exp \Bigg ( - \frac { 1 } { ( \log ( 2 + t ) ) ^ { 3 } } \Bigg ) .
|
| 372 |
+
$$
|
| 373 |
+
|
| 374 |
+
Therefore, for $t \geq 2$
|
| 375 |
+
|
| 376 |
+
$$
|
| 377 |
+
\begin{array} { r l } & { \mathbb { P } \Bigg ( \| r - \hat { r } _ { t } \| _ { \infty } > \frac { 2 \sigma } { \log ( 2 + t ) } \Bigg ) \leq \mathbb { P } \Bigg ( \| r - \hat { r } _ { t } \| _ { \infty } > \frac { 2 \sigma } { \log ( 2 + t ) } \Bigg | E _ { \epsilon } \Bigg ) + \mathbb { P } ( E _ { \epsilon } ^ { C } ) } \\ & { \leq \displaystyle \sum _ { k } \Bigg ( \mathbb { P } \Bigg ( | r ( a ) - \hat { r } _ { t } ( a ) | > \frac { 2 \sigma } { \log ( 2 + t ) } \Bigg ) + \mathbb { P } ( E _ { \epsilon } ^ { C } ) \leq 2 | A | \exp \Bigg ( - \frac { 1 } { ( \log ( 2 + t ) ) ^ { 3 } } \Bigg ) \Bigg ) } \\ & { + 2 | A | \exp \Bigg ( - \frac { t } { ( \log ( 2 + t ) ) ^ { 3 } } \Bigg ) = 4 | A | \exp \Bigg ( - \frac { t } { ( \log ( 2 + t ) ) ^ { 3 } } \Bigg ) . } \end{array}
|
| 378 |
+
$$
|
| 379 |
+
|
| 380 |
+
Lemma 2 Given two policies $\pi ^ { ( 1 ) } = \nabla \Omega ^ { * } ( r ^ { ( 1 ) } )$ and $\pi ^ { ( 2 ) } = \nabla \Omega ^ { * } ( r ^ { ( 2 ) } ) , \exists L ,$ , such that
|
| 381 |
+
|
| 382 |
+
$$
|
| 383 |
+
\parallel \pi ^ { ( 1 ) } - \pi ^ { ( 2 ) } \parallel _ { p } \leq L \parallel r ^ { ( 1 ) } - r ^ { ( 2 ) } \parallel _ { p } .
|
| 384 |
+
$$
|
| 385 |
+
|
| 386 |
+
Proof 2 This comes directly from the fact that $\pi = \nabla \Omega ^ { * } ( r )$ is Lipschitz continuous with $\ell ^ { p \ }$ -norm. Note that $p$ has different values according to the choice of regularizer. Refer to Niculae & Blondel (2017) for a discussion of each norm using Shannon entropy and Tsallis entropy regularizer. Relative entropy shares the same Properties with Shannon Entropy.
|
| 387 |
+
|
| 388 |
+
Lemma 3 Consider the E3W policy applied to a tree. At any node s of the tree with depth $d _ { \mathrm { { z } } }$ , Let us define $N _ { t } ^ { * } ( s , a ) = \pi ^ { * } ( a | s ) . t$ , and $\begin{array} { r } { \hat { N } _ { t } ( s , a ) = \sum _ { s = 1 } ^ { t } \pi _ { s } ( a | s ) } \end{array}$ , where $\pi _ { k } ( a | s )$ is the policy at time step $k$ . There exists some $C$ and $\hat { C }$ such that
|
| 389 |
+
|
| 390 |
+
$$
|
| 391 |
+
\mathbb { P } \big ( | \hat { N } _ { t } ( s , a ) - N _ { t } ^ { * } ( s , a ) | > \frac { C t } { \log t } \big ) \leq \hat { C } | A | t \exp \{ - \frac { t } { ( \log t ) ^ { 3 } } \} .
|
| 392 |
+
$$
|
| 393 |
+
|
| 394 |
+
Proof 3 We denote the following event,
|
| 395 |
+
|
| 396 |
+
$$
|
| 397 |
+
E _ { r _ { k } } = \{ \| r ( s ^ { \prime } , . ) - { \hat { r } } _ { k } ( s ^ { \prime } , . ) \| _ { \infty } < { \frac { 2 \sigma } { \log ( 2 + k ) } } \} .
|
| 398 |
+
$$
|
| 399 |
+
|
| 400 |
+
Thus, conditioned on the event $\textstyle \bigcap _ { i = 1 } ^ { t } E _ { r _ { t } }$ and for $t \geq 4 ,$ , we bound $\vert \hat { N } _ { t } ( s , a ) - N _ { t } ^ { \ast } ( s , a ) \vert$ as
|
| 401 |
+
|
| 402 |
+
$$
|
| 403 |
+
\begin{array} { r l } { \sum _ { k \in \partial _ { s } } \sum _ { i = 1 } ^ { N } \sum _ { j = 1 } ^ { N } \sum _ { k = 0 } ^ { N } \exp _ { i } ^ { - \beta } \sum _ { k = 0 } ^ { N } ( k ) \sum _ { k = 1 } ^ { N } \exp _ { i } ^ { - \beta } ( k ) \sum _ { k = 1 } ^ { N } \exp _ { i } ^ { - \beta } ( k ) \sum _ { k = 1 } ^ { N } } & { } \\ & { \leq \sum _ { k = 1 } ^ { N } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } ( k ) \exp _ { i } ^ { - \beta } \sum _ { k = 1 } ^ { N } } \\ & { \leq \sum _ { k = 1 } ^ { N } \sum _ { i = 0 } ^ { N } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \sum _ { k = 1 } ^ { N } } \\ & { \leq \sum _ { k = 1 } ^ { N } \sum _ { i = 0 } ^ { N } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \sum _ { k = 1 } ^ { N } \sum _ { i = 0 } ^ { N } \exp _ { i } ^ { - \beta } } \\ & \leq \sum _ { k = 1 } ^ { N } \sum _ { i = 0 } ^ { N } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \sum _ { k = 1 } ^ { N } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \sum _ { k = 1 } ^ { N } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \sum _ { k = 1 } ^ { N } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ { - \beta } \exp _ { i } ^ - \beta \end{array}
|
| 404 |
+
$$
|
| 405 |
+
|
| 406 |
+
for some constant $C$ depending on $| A | , p , d , \sigma , L$ , and $\gamma$ . Finally,
|
| 407 |
+
|
| 408 |
+
$$
|
| 409 |
+
\begin{array} { r l } { { \mathbb { P } ( | \hat { N _ { t } } ( s , a ) - N _ { t } ^ { * } ( s , a ) | \geq \frac { C t } { \log t } ) \leq \sum _ { i = 1 } ^ { t } \mathbb { P } ( E _ { r _ { t } } ^ { c } ) = \displaystyle \sum _ { i = 1 } ^ { t } 4 | A | \exp ( - \frac { t } { ( \log ( 2 + t ) ) ^ { 3 } } ) } } \\ & { \leq 4 | A | t \exp ( - \frac { t } { ( \log ( 2 + t ) ) ^ { 3 } } ) } \\ & { = O ( t \exp ( - \frac { t } { ( \log ( t ) ) ^ { 3 } } ) ) . } \end{array}
|
| 410 |
+
$$
|
| 411 |
+
|
| 412 |
+
Lemma 4 Consider the $E 3 W$ policy applied to a tree. $A t$ any node s of the tree, Let us define $N _ { t } ^ { * } ( s , a ) = \pi ^ { * } ( a | s ) . t$ , and $N _ { t } ( s , a )$ as the number of times action $a$ have been chosen until time step $t$ . There exists some $C$ and $\hat { C }$ such that
|
| 413 |
+
|
| 414 |
+
$$
|
| 415 |
+
\mathbb { P } \big ( | N _ { t } ( s , a ) - N _ { t } ^ { * } ( s , a ) | > \frac { C t } { \log t } \big ) \le \hat { C } t \exp \{ - \frac { t } { ( \log t ) ^ { 3 } } \} .
|
| 416 |
+
$$
|
| 417 |
+
|
| 418 |
+
Proof 4 Based on the result from Lemma $^ 3$ , we have
|
| 419 |
+
|
| 420 |
+
$$
|
| 421 |
+
\begin{array} { r l } & { \mathbb { P } \big ( | N _ { t } ( s , a ) - N _ { t } ^ { * } ( s , a ) | > ( 1 + C ) \displaystyle \frac { t } { \log t } \big ) \leq C t \exp \{ - \displaystyle \frac { t } { ( \log t ) ^ { 3 } } \} } \\ & { \leq \mathbb { P } \big ( | \hat { N } _ { t } ( s , a ) - N _ { t } ^ { * } ( s , a ) | > \displaystyle \frac { C t } { \log t } \big ) + \mathbb { P } \big ( | N _ { t } ( s , a ) - \hat { N } _ { t } ( s , a ) | > \displaystyle \frac { t } { \log t } \big ) } \\ & { \leq 4 | A | t \exp \{ - \displaystyle \frac { t } { ( \log ( 2 + t ) ) ^ { 3 } } \} + 2 | A | \exp \{ - \displaystyle \frac { t } { ( \log ( 2 + t ) ) ^ { 2 } } \} ( L e m m a 3 a n d ( \log ( 2 + t ) ) ) } \\ & { \leq O ( t \exp ( - \displaystyle \frac { t } { ( \log t ) ^ { 3 } } ) ) . } \end{array}
|
| 422 |
+
$$
|
| 423 |
+
|
| 424 |
+
Theorem 1 At the root node s of the tree, defining $N ( s )$ as the number of visitations and $V _ { \Omega ^ { * } } ( s )$ as the estimated value at node $s$ , for $\epsilon > 0$ , we have
|
| 425 |
+
|
| 426 |
+
$$
|
| 427 |
+
\mathbb { P } ( | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | > \epsilon ) \le C \exp \{ - \frac { N ( s ) \epsilon } { \hat { C } ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \} .
|
| 428 |
+
$$
|
| 429 |
+
|
| 430 |
+
Proof 5 We prove this concentration inequality by induction. When the depth of the tree is $D = 1$ , from Proposition $^ { l }$ , we get
|
| 431 |
+
|
| 432 |
+
$$
|
| 433 |
+
\left| V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) \right| = \mid \Omega ^ { * } ( Q _ { \Omega } ( s , . ) ) - \Omega ^ { * } ( Q _ { \Omega } ^ { * } ( s , . ) ) \mid \mid _ { \infty } \le \gamma \mid \mid \hat { r } - r ^ { * } \mid _ { \infty } ( C o n t r a c t i o n )
|
| 434 |
+
$$
|
| 435 |
+
|
| 436 |
+
where $\hat { r }$ is the average rewards and $r ^ { * }$ is the mean reward. So that
|
| 437 |
+
|
| 438 |
+
$$
|
| 439 |
+
\mathbb { P } ( | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | > \epsilon ) \le \mathbb { P } ( \gamma \parallel \hat { r } - r ^ { * } \parallel _ { \infty } > \epsilon ) .
|
| 440 |
+
$$
|
| 441 |
+
|
| 442 |
+
From Lemma $^ { l }$ , with $\begin{array} { r } { \epsilon = \frac { 2 \sigma \gamma } { \log ( 2 + N ( s ) ) } } \end{array}$ , we have
|
| 443 |
+
|
| 444 |
+
$$
|
| 445 |
+
\begin{array} { r l } & { \mathbb { P } ( | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | > \epsilon ) \le \mathbb { P } ( \gamma \parallel \hat { r } - r ^ { * } \parallel _ { \infty } > \epsilon ) \le 4 | A | \exp \{ - \frac { N ( s ) \epsilon } { 2 \sigma \gamma ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \} } \\ & { \quad \quad \quad = C \exp \{ - \frac { N ( s ) \epsilon } { \hat { C } ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \} . } \end{array}
|
| 446 |
+
$$
|
| 447 |
+
|
| 448 |
+
Let assume we have the concentration bound at the depth $D - 1$ , Let us define $V _ { \Omega } ( s _ { a } ) = Q _ { \Omega } ( s , a )$ , where $s _ { a }$ is the state reached taking action $a$ from state s. then at depth $D - 1$
|
| 449 |
+
|
| 450 |
+
$$
|
| 451 |
+
\mathbb { P } ( | V _ { \Omega } ( s _ { a } ) - V _ { \Omega } ^ { * } ( s _ { a } ) | > \epsilon ) \le C \exp \{ - \frac { N ( s _ { a } ) \epsilon } { \hat { C } ( \log ( 2 + N ( s _ { a } ) ) ) ^ { 2 } } \} .
|
| 452 |
+
$$
|
| 453 |
+
|
| 454 |
+
Now at the depth $D$ , because of the Contraction Property, we have
|
| 455 |
+
|
| 456 |
+
$$
|
| 457 |
+
\begin{array} { r l } & { | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | \leq \gamma \parallel Q _ { \Omega } ( s , . ) - Q _ { \Omega } ^ { * } ( s , . ) \parallel _ { \infty } } \\ & { \qquad = \gamma | Q _ { \Omega } ( s , a ) - Q _ { \Omega } ^ { * } ( s , a ) | . } \end{array}
|
| 458 |
+
$$
|
| 459 |
+
|
| 460 |
+
So that
|
| 461 |
+
|
| 462 |
+
$$
|
| 463 |
+
\begin{array} { r l r } & { } & { { \mathbb { P } } ( | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | > \epsilon ) \leq { \mathbb { P } } ( \gamma \parallel Q _ { \Omega } ( s , a ) - Q _ { \Omega } ^ { * } ( s , a ) \parallel > \epsilon ) } \\ & { } & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { } & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { } & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { } & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \hat { C } _ { a } ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \end{array}
|
| 464 |
+
$$
|
| 465 |
+
|
| 466 |
+
From $( I 7 )$ , we can have $\begin{array} { r } { \operatorname* { l i m } _ { t \to \infty } N ( s _ { a } ) = \infty } \end{array}$ because if $\exists L , N ( s _ { a } ) \ < \ L$ , we can find $\epsilon > 0$ for which (17) is not satisfied. From Lemma 4, when $N ( s )$ is large enough, we have $N ( s _ { a } ) $ $\pi ^ { * } ( a | s ) N ( s )$ (for example $\begin{array} { r } { N ( s _ { a } ) > \frac { 1 } { 2 } \pi ^ { * } ( a | s ) N ( s ) ) } \end{array}$ , that means we can find $C$ and $\hat { C }$ that satisfy
|
| 467 |
+
|
| 468 |
+
$$
|
| 469 |
+
\mathbb { P } ( | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | > \epsilon ) \le C \exp \{ - \frac { N ( s ) \epsilon } { \hat { C } ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \} .
|
| 470 |
+
$$
|
| 471 |
+
|
| 472 |
+
Lemma 5 At any node s of the tree, $N ( s )$ is the number of visitations. We define the event
|
| 473 |
+
|
| 474 |
+
$$
|
| 475 |
+
E _ { s } = \{ \forall a i n \boldsymbol { A } , | N ( s , a ) - N ^ { * } ( s , a ) | < \frac { N ^ { * } ( s , a ) } { 2 } \} w h e r e \ N ^ { * } ( s , a ) = \pi ^ { * } ( a | s ) N ( s ) ,
|
| 476 |
+
$$
|
| 477 |
+
|
| 478 |
+
where $\epsilon > 0$ and $V _ { \Omega ^ { * } } ( s )$ is the estimated value at node $s$ . We have
|
| 479 |
+
|
| 480 |
+
$$
|
| 481 |
+
\mathbb { P } ( | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | > \epsilon | E _ { s } ) \le C \exp \{ - \frac { N ( s ) \epsilon } { \hat { C } ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \} .
|
| 482 |
+
$$
|
| 483 |
+
|
| 484 |
+
Proof 6 The proof is the same as in Theorem 2. We prove the concentration inequality by induction. When the depth of the tree is $D = 1$ , from Proposition $^ { l }$ , we get
|
| 485 |
+
|
| 486 |
+
$$
|
| 487 |
+
\left| V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) \right| = \left\| \ \Omega ^ { * } ( Q _ { \Omega } ( s , . ) ) - \Omega ^ { * } ( Q _ { \Omega } ^ { * } ( s , . ) ) \ \right\| \leq \gamma \ \left\| \ \hat { r } - r ^ { * } \ \right\| _ { \infty } \left( C o n t r a c t i o n \ P r o p e r t i o n s \right) ,
|
| 488 |
+
$$
|
| 489 |
+
|
| 490 |
+
where $\hat { r }$ is the average rewards and $r ^ { * }$ is the mean rewards. So that
|
| 491 |
+
|
| 492 |
+
$$
|
| 493 |
+
\mathbb { P } ( | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | > \epsilon ) \le \mathbb { P } ( \gamma \parallel \hat { r } - r ^ { * } \parallel _ { \infty } > \epsilon ) .
|
| 494 |
+
$$
|
| 495 |
+
|
| 496 |
+
From Lemma $^ { l }$ , with $\begin{array} { r } { \epsilon = \frac { 2 \sigma \gamma } { \log ( 2 + N ( s ) ) } } \end{array}$ and given $E _ { s }$ , we have
|
| 497 |
+
|
| 498 |
+
$$
|
| 499 |
+
\begin{array} { r l } & { \mathbb { P } ( | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | > \epsilon ) \le \mathbb { P } ( \gamma \parallel \hat { r } - r ^ { * } \parallel _ { \infty } > \epsilon ) \le 4 | A | \exp \{ - \frac { N ( s ) \epsilon } { 2 \sigma \gamma ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \} } \\ & { \quad \quad \quad = C \exp \{ - \frac { N ( s ) \epsilon } { \hat { C } ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \} . } \end{array}
|
| 500 |
+
$$
|
| 501 |
+
|
| 502 |
+
Let assume we have the concentration bound at the depth $D - 1$ , Let us define $V _ { \Omega } ( s _ { a } ) = Q _ { \Omega } ( s , a )$ , where $s _ { a }$ is the state reached taking action a from state $s$ , then at depth $D - 1$
|
| 503 |
+
|
| 504 |
+
$$
|
| 505 |
+
\mathbb { P } ( | V _ { \Omega } ( s _ { a } ) - V _ { \Omega } ^ { * } ( s _ { a } ) | > \epsilon ) \le C \exp \{ - \frac { N ( s _ { a } ) \epsilon } { \hat { C } ( \log ( 2 + N ( s _ { a } ) ) ) ^ { 2 } } \} .
|
| 506 |
+
$$
|
| 507 |
+
|
| 508 |
+
Now at depth $D$ , because of the Contraction Property and given $E _ { s }$ , we have
|
| 509 |
+
|
| 510 |
+
$$
|
| 511 |
+
\begin{array} { r l } & { | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | \leq \gamma \parallel Q _ { \Omega } ( s , . ) - Q _ { \Omega } ^ { * } ( s , . ) \parallel _ { \infty } } \\ & { \qquad = \gamma | Q _ { \Omega } ( s , a ) - Q _ { \Omega } ^ { * } ( s , a ) | ( \exists a , s a t i s f t e d ) . } \end{array}
|
| 512 |
+
$$
|
| 513 |
+
|
| 514 |
+
So that
|
| 515 |
+
|
| 516 |
+
$$
|
| 517 |
+
\begin{array} { r l } & { \mathbb { P } ( | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | > \epsilon ) \leq \mathbb { P } ( \gamma \parallel Q _ { \Omega } ( s , a ) - Q _ { \Omega } ^ { * } ( s , a ) \parallel > \epsilon ) } \\ & { \qquad \leq C _ { a } \exp \{ - \frac { N ( s _ { a } ) \epsilon } { \hat { C } _ { a } ( \log ( 2 + N ( s _ { a } ) ) ) ^ { 2 } } \} } \\ & { \qquad \leq C _ { a } \exp \{ - \frac { N ( s _ { a } ) \epsilon } { \hat { C } _ { a } ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \} } \\ & { \qquad \leq C \exp \{ - \frac { N ( s ) \epsilon } { \hat { C } ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \} ( b e c a u s e o f E _ { s } ) } \end{array}
|
| 518 |
+
$$
|
| 519 |
+
|
| 520 |
+
Theorem 2 Let $a _ { t }$ be the action returned by algorithm E3W at iteration t. Then for t large enough, with some constants $C , { \hat { C } }$ ,
|
| 521 |
+
|
| 522 |
+
$$
|
| 523 |
+
\mathbb { P } ( a _ { t } \neq a ^ { * } ) \leq C t \exp \{ - \frac { t } { \hat { C } \sigma ( \log ( t ) ) ^ { 3 } } \} .
|
| 524 |
+
$$
|
| 525 |
+
|
| 526 |
+
Proof 7 Let us define event $E _ { s }$ as in Lemma 5. Let $a ^ { * }$ be the action with largest value estimate at the root node state $s$ . The probability that $E 3 W$ selects a sub-optimal arm at s is
|
| 527 |
+
|
| 528 |
+
$$
|
| 529 |
+
\begin{array} { r l } & { \displaystyle \mathbb { P } ( a _ { t } \neq a ^ { * } ) \leq \sum _ { a } \mathbb { P } ( V _ { \Omega } ( s _ { a } ) ) > V _ { \Omega } ( s _ { a ^ { * } } ) | E _ { s } ) + \mathbb { P } ( E _ { s } ^ { c } ) } \\ & { \displaystyle = \sum _ { a } \mathbb { P } ( ( V _ { \Omega } ( s _ { a } ) - V _ { \Omega } ^ { * } ( s _ { a } ) ) - ( V _ { \Omega } ( s _ { a ^ { * } } ) - V _ { \Omega } ^ { * } ( s _ { a ^ { * } } ) ) \geq V _ { \Omega } ^ { * } ( s _ { a ^ { * } } ) - V _ { \Omega } ^ { * } ( s _ { a } ) | E _ { s } ) + \mathbb { P } ( E _ { s } ^ { c } ) . } \end{array}
|
| 530 |
+
$$
|
| 531 |
+
|
| 532 |
+
Let us define $\Delta = V _ { \Omega } ^ { * } ( s _ { a ^ { * } } ) - V _ { \Omega } ^ { * } ( s _ { a } )$ , therefore for $\Delta > 0$ , we have
|
| 533 |
+
|
| 534 |
+
$$
|
| 535 |
+
\begin{array} { r l } & { \displaystyle \mathbb { P } ( a _ { t } \neq a ^ { * } ) \leq \sum _ { a } \mathbb { P } ( ( V _ { \Omega } ( s _ { a } ) - V _ { \Omega } ^ { * } ( s _ { a } ) ) - ( V _ { \Omega } ( s _ { a ^ { * } } ) - V _ { \Omega } ^ { * } ( s _ { a ^ { * } } ) ) \geq \Delta | E _ { s } ) + \mathbb { P } ( E _ { s } ^ { c } ) } \\ & { \leq \displaystyle \sum _ { a } \mathbb { P } ( | V _ { \Omega } ( s _ { a } ) - V _ { \Omega } ^ { * } ( s _ { a } ) | \geq \alpha \Delta | E _ { s } ) + \mathbb { P } ( | V _ { \Omega } ( s _ { a ^ { * } } ) - V _ { \Omega } ^ { * } ( s _ { a ^ { * } } ) | \geq \beta \Delta | E _ { s } ) + \mathbb { P } ( E _ { s } ^ { c } ) } \\ & { \leq \displaystyle \sum _ { a } C _ { a } \exp \{ - \frac { N ( s ) ( \alpha \Delta ) } { \hat { C } _ { a } ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \} + C _ { a ^ { * } } \exp \{ - \frac { N ( s ) ( \beta \Delta ) } { \hat { C } _ { a ^ { * } } ( \log ( 2 + N ( s ) ) ) ^ { 2 } } \} + \mathbb { P } ( E _ { s } ^ { c } ) , } \end{array}
|
| 536 |
+
$$
|
| 537 |
+
|
| 538 |
+
where $\alpha + \beta = 1 , \alpha > 0 , \beta > 0 ,$ , and $N ( s )$ is the number of visitations the root node s. Let us define $\begin{array} { r } { \frac { 1 } { \hat { C } } = \operatorname* { m i n } \{ \frac { ( \alpha \Delta ) } { { C } _ { a } } , \frac { ( \beta \Delta ) } { { C } _ { a ^ { * } } } \} } \end{array}$ , and $\begin{array} { r } { C = \frac { 1 } { | A | } \operatorname* { m a x } \{ C _ { a } , C _ { a ^ { * } } \} } \end{array}$ we have
|
| 539 |
+
|
| 540 |
+
$$
|
| 541 |
+
\mathbb { P } ( a \neq a ^ { * } ) \leq C \exp \{ - \frac { t } { \hat { C } \sigma ( \log ( 2 + t ) ) ^ { 2 } } \} + \mathbb { P } ( E _ { s } ^ { c } ) .
|
| 542 |
+
$$
|
| 543 |
+
|
| 544 |
+
From Lemma 4, $\exists C ^ { ' } , \hat { C } ^ { \prime }$ for which
|
| 545 |
+
|
| 546 |
+
$$
|
| 547 |
+
\mathbb { P } ( E _ { s } ^ { c } ) \le C ^ { ' } t \exp \{ - \frac { t } { \hat { C } ^ { ' } ( \log ( t ) ) ^ { 3 } } \} ,
|
| 548 |
+
$$
|
| 549 |
+
|
| 550 |
+
so that
|
| 551 |
+
|
| 552 |
+
$$
|
| 553 |
+
\mathbb { P } ( a \neq a ^ { * } ) \leq O ( t \exp \{ - \frac { t } { ( \log ( t ) ) ^ { 3 } } \} ) .
|
| 554 |
+
$$
|
| 555 |
+
|
| 556 |
+
Theorem 3 Consider an $E 3 W$ policy applied to the tree. Let $\begin{array} { r } { \kappa _ { i } = \nabla \Omega ^ { * } ( a _ { i } | s ) + \frac { L } { p } \sqrt { { \hat { C } } \sigma ^ { 2 } \log \frac { C } { \delta } / 2 n } , } \end{array}$ $\begin{array} { r } { \chi _ { i } = \nabla \Omega ^ { * } ( a _ { i } | s ) - \frac { L } { p } \sqrt { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } / 2 n } , } \end{array}$ , where $\nabla \Omega ^ { * } ( . | s )$ is the policy with respect to the mean value vector $V ( \cdot )$ at the root node s. For any $\delta > 0$ , with probability at least $1 - \delta$ , ∃ constant $L , p , C , { \hat { C } }$ so that the pseudo regret $R _ { n }$ satisfies
|
| 557 |
+
|
| 558 |
+
$$
|
| 559 |
+
n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \kappa _ { i } + \frac { L } { p } \big ( \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \gamma } \big ) \Big ) \leq R _ { n } \leq n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \chi _ { i } - \frac { L } { p } \big ( \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \gamma } \big ) \Big ) .
|
| 560 |
+
$$
|
| 561 |
+
|
| 562 |
+
Proof 8 From Lemma 2 given two policies $\pi ^ { ( 1 ) } = \nabla \Omega ^ { * } ( r ^ { ( 1 ) } )$ and $\pi ^ { ( 2 ) } = \nabla \Omega ^ { * } ( r ^ { ( 2 ) } ) , \exists L ,$ , such that
|
| 563 |
+
|
| 564 |
+
$$
|
| 565 |
+
\parallel \pi ^ { ( 1 ) } - \pi ^ { ( 2 ) } \parallel _ { p } \leq L \parallel r ^ { ( 1 ) } - r ^ { ( 2 ) } \parallel _ { p } \leq L \frac { 1 } { p } \parallel r ^ { ( 1 ) } - r ^ { ( 2 ) } \parallel _ { \infty } .
|
| 566 |
+
$$
|
| 567 |
+
|
| 568 |
+
From (13), we have the regret
|
| 569 |
+
|
| 570 |
+
$$
|
| 571 |
+
R _ { n } = n V ^ { * } - \sum _ { i } V _ { i } \sum _ { t = 1 } ^ { n } { \hat { \pi } } _ { t } ( a _ { i } | s ) ,
|
| 572 |
+
$$
|
| 573 |
+
|
| 574 |
+
where $\hat { \pi } _ { t } ( \cdot )$ is the policy at time step $t$ , and $\mathbb { I } ( \cdot )$ is the indicator function. $V ^ { * }$ is the optimal branch at the root node, $V _ { i }$ is the mean value function of the branch with respect to action $i$ , $V ( \cdot )$ is the $| A |$
|
| 575 |
+
|
| 576 |
+
vector of value function at the root node. $\hat { V } ( \cdot )$ is the $| A |$ estimation vector of value function at the root node. $\pi ( . | s ) = \nabla \Omega ^ { * } ( V ( \cdot ) )$ is the policy with respect to the $V ( \cdot )$ vector at the root node.
|
| 577 |
+
|
| 578 |
+
Then for any $\delta > 0$ , with probability at least $1 - \delta$ , we have
|
| 579 |
+
|
| 580 |
+
$$
|
| 581 |
+
\begin{array} { l } { \displaystyle \lvert \pi ( a _ { i } \vert s ) - \hat { \pi } _ { t } ( a _ { i } \vert s ) \rvert \le \parallel \pi ( . \vert s ) - \hat { \pi } _ { t } ( . \vert s ) \parallel _ { \infty } \le \displaystyle \frac { L } { p } \parallel V ( \cdot ) - \hat { V } ( \cdot ) \parallel _ { \infty } ( L e m m a 2 ) } \\ { \displaystyle \le \displaystyle \frac { L } { p } \lvert V ( \cdot ) - \hat { V } ( \cdot ) \rvert \le \displaystyle \frac { L } { p } \biggl ( \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \delta } + \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } \biggr ) ( T h e o r e m 4 ) } \end{array}
|
| 582 |
+
$$
|
| 583 |
+
|
| 584 |
+
So that
|
| 585 |
+
|
| 586 |
+
$$
|
| 587 |
+
\tau ( a _ { i } | s ) - \frac { L } { p } \Bigg ( \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \delta } + \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } \Bigg ) \leq \hat { \pi } _ { t } ( a _ { i } | s ) \leq \pi ( a _ { i } | s ) + \frac { L } { p } \Bigg ( \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \delta } + \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } \Bigg ) .
|
| 588 |
+
$$
|
| 589 |
+
|
| 590 |
+
so that
|
| 591 |
+
|
| 592 |
+
$$
|
| 593 |
+
\begin{array} { l } { \displaystyle \mathfrak { l } _ { n } = n V ^ { * } - \sum _ { i } V _ { i } \sum _ { t = 1 } ^ { n } \hat { \pi } _ { t } ( a _ { i } | s ) \le n V ^ { * } - \sum _ { i } V _ { i } \sum _ { t = 1 } ^ { n } \left( \pi ( a _ { i } | s ) - \displaystyle \frac { L } { p } \big ( \displaystyle \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \delta } + \sqrt { \displaystyle \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 n } } \big ) \right) } \\ { \displaystyle \mathfrak { l } _ { n } \le n V ^ { * } - \sum _ { i } V _ { i } \sum _ { t = 1 } ^ { n } \left( \pi ( a _ { i } | s ) - \displaystyle \frac { L } { p } \big ( \displaystyle \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \delta } + \sqrt { \displaystyle \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 n } } \big ) \right) } \\ { \displaystyle \mathfrak { l } _ { n } \le n V ^ { * } - n \sum _ { i } V _ { i } \left( \pi ( a _ { i } | s ) - \displaystyle \frac { L } { p } \big ( \displaystyle \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \delta } + \sqrt { \displaystyle \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 n } } \big ) \right) } \end{array}
|
| 594 |
+
$$
|
| 595 |
+
|
| 596 |
+
And
|
| 597 |
+
|
| 598 |
+
$$
|
| 599 |
+
\begin{array} { l } { { R _ { n } \geq n V ^ { \ast } - \displaystyle \sum _ { i } V _ { i } \sum _ { t = 1 } ^ { n } \left( \pi ( a _ { i } | s ) + \frac { L } { p } \big ( \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \delta } + \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 n } } \big ) \right) } } \\ { { R _ { n } \geq n V ^ { \ast } - n \displaystyle \sum _ { i } V _ { i } \Big ( \pi ( a _ { i } | s ) + \frac { L } { p } \big ( \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \delta } + \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 n } } \big ) \Big ) } } \end{array}
|
| 600 |
+
$$
|
| 601 |
+
|
| 602 |
+
In case of Maximum Entropy and Relative Entropy $p = 1$ , because
|
| 603 |
+
|
| 604 |
+
$$
|
| 605 |
+
\parallel \pi ^ { ( 1 ) } - \pi ^ { ( 2 ) } \parallel _ { \infty } \leq L \parallel r ^ { ( 1 ) } - r ^ { ( 2 ) } \parallel _ { \infty } .
|
| 606 |
+
$$
|
| 607 |
+
|
| 608 |
+
So that we have for MENTS
|
| 609 |
+
|
| 610 |
+
$$
|
| 611 |
+
n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \kappa _ { i } + L \big ( \frac { \tau \log | A | } { 1 - \gamma } \big ) \Big ) \leq R _ { n } \leq n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \chi _ { i } - L \big ( \frac { \tau \log | A | } { 1 - \gamma } \big ) \Big ) .
|
| 612 |
+
$$
|
| 613 |
+
|
| 614 |
+
For RENTS, we have
|
| 615 |
+
|
| 616 |
+
$$
|
| 617 |
+
\imath V ^ { * } - n \sum _ { i } V _ { i } \Big ( \kappa _ { i } + L \big ( \frac { \tau ( \log | A | - \frac { 1 } { m } ) } { 1 - \gamma } \big ) \Big ) \leq R _ { n } \leq n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \chi _ { i } - L \big ( \frac { \tau ( \log | A | - \frac { 1 } { m } ) } { 1 - \gamma } \big ) \Big )
|
| 618 |
+
$$
|
| 619 |
+
|
| 620 |
+
where $\begin{array} { r } { m = \operatorname* { m i n } _ { a } \pi ( a | s ) } \end{array}$
|
| 621 |
+
|
| 622 |
+
In case of Tsallis Entropy $p = 2$ ( Niculae & Blondel (2017)), so that
|
| 623 |
+
|
| 624 |
+
$$
|
| 625 |
+
n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \kappa _ { i } + \frac { L } { 2 } \big ( \frac { | A | - 1 } { 2 | A | } \frac { \tau } { 1 - \gamma } \big ) \Big ) \leq R _ { n } \leq n V ^ { * } - n \sum _ { i } V _ { i } \Big ( \chi _ { i } - \frac { L } { 2 } \big ( \frac { | A | - 1 } { 2 | A | } \frac { \tau } { 1 - \gamma } \big ) \Big )
|
| 626 |
+
$$
|
| 627 |
+
|
| 628 |
+
Before derive the next theorem, we state here the Theorem 2 in Geist et al. (2019)
|
| 629 |
+
|
| 630 |
+
• Boundedness: for two constants $L _ { \Omega }$ and $U _ { \Omega }$ such that for all $\pi \in \Pi$ , we have $L _ { \Omega } \leq \Omega ( \pi ) \leq$ $U _ { \Omega }$ , then
|
| 631 |
+
|
| 632 |
+
$$
|
| 633 |
+
V ^ { \ast } ( s ) - \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \gamma } \leq V _ { \Omega } ^ { \ast } ( s ) \leq V ^ { \ast } ( s ) .
|
| 634 |
+
$$
|
| 635 |
+
|
| 636 |
+
Where $\tau$ is the temperature and $\gamma$ is the discount constant.
|
| 637 |
+
|
| 638 |
+
Theorem 4 For any $\delta > 0$ , with probability at least $1 - \delta$ , the $\varepsilon _ { \Omega }$ satisfies
|
| 639 |
+
|
| 640 |
+
$$
|
| 641 |
+
- \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } - \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \gamma } \leq \varepsilon _ { \Omega } \leq \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } .
|
| 642 |
+
$$
|
| 643 |
+
|
| 644 |
+
Proof 9 From Theorem 2, let us define $\begin{array} { r } { \delta = C \exp \{ - \frac { 2 N ( s ) \epsilon ^ { 2 } } { \hat { C } \sigma ^ { 2 } } \} } \end{array}$ , so that $\begin{array} { r } { \epsilon = \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log { \frac { C } { \delta } } } { 2 N ( s ) } } } \end{array}$ then for any $\delta > 0$ , we have
|
| 645 |
+
|
| 646 |
+
$$
|
| 647 |
+
\mathbb { P } ( | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | \le \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } ) \ge 1 - \delta .
|
| 648 |
+
$$
|
| 649 |
+
|
| 650 |
+
Then, for any $\delta > 0$ , with probability at least $1 - \delta$ , we have
|
| 651 |
+
|
| 652 |
+
$$
|
| 653 |
+
\begin{array} { l } { \displaystyle | V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) | \leq \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } } \\ { - \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } \leq V _ { \Omega } ( s ) - V _ { \Omega } ^ { * } ( s ) \leq \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } } \\ { - \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } + V _ { \Omega } ^ { * } ( s ) \leq V _ { \Omega } ( s ) \leq \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } + V _ { \Omega } ^ { * } ( s ) . } \end{array}
|
| 654 |
+
$$
|
| 655 |
+
|
| 656 |
+
From Proposition $I$ , we have
|
| 657 |
+
|
| 658 |
+
$$
|
| 659 |
+
- \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } + V ^ { * } ( s ) - \frac { \tau ( U _ { \Omega } - L _ { \Omega } ) } { 1 - \gamma } \leq V _ { \Omega } ( s ) \leq \sqrt { \frac { \hat { C } \sigma ^ { 2 } \log \frac { C } { \delta } } { 2 N ( s ) } } + V ^ { * } ( s ) .
|
| 660 |
+
$$
|
parse/train/-kfLEqppEm_/-kfLEqppEm__content_list.json
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parse/train/-kfLEqppEm_/-kfLEqppEm__middle.json
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parse/train/-kfLEqppEm_/-kfLEqppEm__model.json
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|
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parse/train/9QffERDO_rJ/9QffERDO_rJ.md
ADDED
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| 1 |
+
# Auxiliary Learning Induced Graph Convolutional Networks
|
| 2 |
+
|
| 3 |
+
Anonymous Author(s)
|
| 4 |
+
Affiliation
|
| 5 |
+
Address
|
| 6 |
+
email
|
| 7 |
+
|
| 8 |
+
# Abstract
|
| 9 |
+
|
| 10 |
+
1 Graph convolutional networks (GCNs) have recently achieved great success in
|
| 11 |
+
2 many applications. However they suffer from an incomplete annotation problem
|
| 12 |
+
3 for complex graph-structured data. In this paper, we introduce a novel auxiliary
|
| 13 |
+
4 learning method for GCNs in a multi-task fashion, which can efficiently enrich the
|
| 14 |
+
5 data annotations. Specifically, both link prediction and label generation are used as
|
| 15 |
+
6 two auxiliary tasks to complement the primary task of node classification. These
|
| 16 |
+
7 two auxiliary tasks are jointly trained with the primary node classification task
|
| 17 |
+
8 via a graph meta-learning strategy. The experimental results demonstrate that the
|
| 18 |
+
9 proposed method consistently and significantly outperforms existing methods and
|
| 19 |
+
10 achieves state-of-the-art results on several benchmark citation network datasets.
|
| 20 |
+
|
| 21 |
+
# 11 1 Introduction
|
| 22 |
+
|
| 23 |
+
12 Graph-structured data is ubiquitous in real-world applications. However, general deep learning
|
| 24 |
+
13 methods, such as convolutional neural networks (CNNs), cannot adapt to graph-structured data
|
| 25 |
+
14 directly, because the nodes in a graph have different numbers of neighbors, which often lose the
|
| 26 |
+
15 ranking information. To handle the graph data effectively, graph convolutional networks (GCNs)
|
| 27 |
+
16 have recently been proposed and used in many applications such as biomolecular prediction [1] and
|
| 28 |
+
17 recommendation systems [2].
|
| 29 |
+
18 Previous methods focus on designing models that can extract information from both the graph
|
| 30 |
+
19 topology and node features. Specifically, existing GCN methods typically design different propagation
|
| 31 |
+
20 strategies [3, 4, 5] for each network layer and stack more network layers [6, 7, 8] to derive larger
|
| 32 |
+
21 receptive fields. However, the neighborhood aggregation is essentially a type of Laplacian smoothing
|
| 33 |
+
22 and stacking too many layers may result in over-smoothing [9]. These drawbacks of existing methods
|
| 34 |
+
23 limit further performance enhancement.
|
| 35 |
+
24 In this paper, we try to explore the bottleneck of node classification from another point of view, i.e.,
|
| 36 |
+
25 from the training data itself. As shown in Fig. 1, graph-structured data has different properties from
|
| 37 |
+
26 grid-like data such as an image. The most obvious difference is that the nodes in a graph are connected
|
| 38 |
+
27 by edges. This causes two main issues whien it comes to annotating graph-structured data, resulting in
|
| 39 |
+
28 that existing methods cannot fully leverage the graph-structured information. First, the edges in most
|
| 40 |
+
29 graph data for semi-supervised node classification are unweighted. This arbitrary edge indication
|
| 41 |
+
30 setting cannot effectively reflect the detailed graph structures. Besides, graph-structured data may be
|
| 42 |
+
31 contaminated with noisy edges. These noisy edges cannot represent the true pairwise relationships
|
| 43 |
+
32 between nodes. Second, using one-hot labels to train a graph-based model is inappropriate. One-hot
|
| 44 |
+
33 labels are widely used in various machine learning tasks, assigning a training sample to a single class.
|
| 45 |
+
34 However, nodes in a graph are connected; even nodes with different classes may have relations. In
|
| 46 |
+
35 this scenario, it is more suitable to use soft labels to assign a node to multiple classes, with different
|
| 47 |
+
36 probabilities indicating which class the node possibly belongs to.
|
| 48 |
+
37 To fully leverage the graph-structured
|
| 49 |
+
38 information for enhancing the node
|
| 50 |
+
39 classification performance of GCNs,
|
| 51 |
+
40 we introduce an auxiliary learning
|
| 52 |
+
41 scheme to the GCN framework in
|
| 53 |
+
42 a multi-task fashion. To this end,
|
| 54 |
+
43 we add two auxiliary tasks to enrich
|
| 55 |
+
44 the topology information of a graph
|
| 56 |
+
45 by softening the node labels and re
|
| 57 |
+
46 weighting the edges. Experimental
|
| 58 |
+
47 results show that our model achieves
|
| 59 |
+
48 state-of-the-art node classification per
|
| 60 |
+
49 formance on several benchmark cita
|
| 61 |
+
50 tion network datasets. Our contribu
|
| 62 |
+
51 tions are as follows:
|
| 63 |
+
52 (1) We propose two auxiliary tasks to capture more accurate graph information and enhance the
|
| 64 |
+
53 model performance. The auxiliary link prediction task ensures that the model captures more graph
|
| 65 |
+
54 topology information and generates probabilistic edges. The auxiliary label generation task softens
|
| 66 |
+
55 the one-hot labels and generates pseudo-labels for unlabeled nodes.
|
| 67 |
+
56 (2) The reconstructed edges and pseudo sudo labels derived via the two auxiliary tasks are iteratively
|
| 68 |
+
57 updated with a node classifier of the primary task based on a meta auxiliary learning strategy, resulting
|
| 69 |
+
8 in state-of-the-art node classification performance.
|
| 70 |
+
|
| 71 |
+
|
| 72 |
+
Figure 1: The difference between image data and graphstructured data. The nodes in graph-structured data are connected. Further, in most graph data for the node classification task, edges are unweighted and noisy.
|
| 73 |
+
|
| 74 |
+
# 59 2 Related Work
|
| 75 |
+
|
| 76 |
+
Over the past few years, GCNs have achieved significant breakthroughs in graph data representation.
|
| 77 |
+
Generally, existing GCNs can be divided into spectral-based methods and spatial-based methods.
|
| 78 |
+
|
| 79 |
+
62 The spectral-based methods use graph spectral theory to define the graph convolutional operation
|
| 80 |
+
63 in a graph Fourier domain. Spectral CNN [10] follows these mathematical foundations, assuming
|
| 81 |
+
64 that a convolutional filter is a set of learnable parameters. To reduce computational complexity,
|
| 82 |
+
65 ChebNet [11] approximates a graph convolutional filter as Chebyshev polynomials of the eigenvalues.
|
| 83 |
+
66 GCN [12] introduces a first-order approximation of ChebNet and proposes a renormalization trick to
|
| 84 |
+
67 alleviate numerical instabilities and exploding/vanishing gradients. DualGCN [13] introduces a dual
|
| 85 |
+
68 GCN architecture with two graph convolutional layers in parallel to encode both local and global
|
| 86 |
+
69 structural information.
|
| 87 |
+
70 The spatial-based methods define feature aggregation in the spatial domain directly, which is more
|
| 88 |
+
71 efficient, general, and flexible [14]. The key challenge for these spatial-based methods is to apply
|
| 89 |
+
72 the convolution operation for different-sized neighborhoods, while at the same time maintaining the
|
| 90 |
+
73 weight sharing property. Neural network for graphs (NN4G) [15] is the first spatial-based method,
|
| 91 |
+
74 applying the graph convolutional operation in the spatial space. Diffusion convolutional neural
|
| 92 |
+
75 networks (DCNN) [16] consider graph convolutions as diffusion processes to efficiently learn features
|
| 93 |
+
76 that are invariant under isomorphism. Message passing neural networks (MPNN) [17] model graph
|
| 94 |
+
77 convolution as a message passing process among the nodes. The graph attention network (GAT)
|
| 95 |
+
78 [3] introduce masked self-attentional layers to assign different weights to adjacent nodes, leading to
|
| 96 |
+
79 learnable filter weights. The mixture model network (MoNet) [18] introduces pseudo-coordinates to
|
| 97 |
+
80 assign different weights to the neighbors of each node. To achieve weight sharing across different
|
| 98 |
+
81 nodes, some spatial-based models attempt to rank a node’s neighbors via certain criteria or metrics,
|
| 99 |
+
82 which transforms the graph-structured data into grid data for further processing. The large-scale
|
| 100 |
+
83 graph convolutional network (LGCN) [19] ranks a node’s neighbors via the node feature values. Then,
|
| 101 |
+
84 multiple 1D convolutional layers are stacked for feature aggregation. Approximate personalized
|
| 102 |
+
85 propagation of neural predictions (APPNP) [4] takes the personalized PageRank algorithm as the
|
| 103 |
+
86 model propagation method to avoid over-smoothing when stacking more layers or increasing the size
|
| 104 |
+
87 of the neighborhood.
|
| 105 |
+
88 Multi-task learning is designed to simultaneously learn a set of related but different tasks for ensuring
|
| 106 |
+
89 that a learning model can derive the best performance across all tasks. Different from multi-task
|
| 107 |
+
90 learning, auxiliary learning is only concerned with model performance on the primary task. For
|
| 108 |
+
91 instance, Deepstereo [20] leverages auxiliary learning to predict the relative poses of multiple cameras
|
| 109 |
+
92 for unsupervised monocular depth estimation. To improve the performance of conversational speech
|
| 110 |
+
93 recognition, auxiliary learning [21] is applied to low-level representations. Compared to the common
|
| 111 |
+
94 learning scheme, meta auxiliary learning can enhance learning performance. For instance, MAXL
|
| 112 |
+
95 [22] adopts meta-learning to automatically generate the auxiliary task labels. Pseudo Label [23] is
|
| 113 |
+
96 a semi-supervised learning method, where a deep neural network is trained using both labeled and
|
| 114 |
+
97 unlabeled data. For unlabeled data, the model picks up the class that has the maximum predicted
|
| 115 |
+
98 probability as the true label to train itself. MPL [24] extends the Pseudo Label [23] via a meta-learning
|
| 116 |
+
99 strategy, where the pseudo-labels are not generated by itself, but by a teacher network.
|
| 117 |
+
100 The existing GCNs are designed for a single task, where the properties of graph-structured data
|
| 118 |
+
101 are not fully explored. We introduce the auxiliary learning scheme to leverage more detailed graph
|
| 119 |
+
102 topology information for enhancing node classification performance.
|
| 120 |
+
|
| 121 |
+
|
| 122 |
+
Figure 2: Network architecture of the vanilla GCN model. The vanilla GCN contains multiple GCN layers. Each layer captures the graph structure to generate the hidden embeddings from the previous layer (For the first layer, it is the original feature of the node) as input, and obtains the output through the message calculation, aggregation and update step. The last layer uses a softmax function to generate classification probabilities for each node.
|
| 123 |
+
|
| 124 |
+
# 3 Method
|
| 125 |
+
|
| 126 |
+
# 3.1 Preliminaries
|
| 127 |
+
|
| 128 |
+
05 Given a graph $\mathcal { G } = \{ \nu , \varepsilon , \mathbf { X } \}$ , $\nu$ is a set of nodes and $\mathcal { E }$ is a set of the edges connecting the related nodes. 06 $\mathbf { X } \in \mathbb { R } ^ { N \times d }$ represents the features matrix of the nodes, where $d$ is the dimension of the node 107 features and $N = | \nu |$ is the number of nodes.
|
| 129 |
+
|
| 130 |
+
108 The proposed method adopts the vanilla GCN [12] as the backbone network, taking the graph
|
| 131 |
+
109 adjacency matrix A, labeled training set $\mathbf { Y } _ { \mathrm { t r a i n } }$ , and original features $\mathbf { X }$ of the nodes as inputs to
|
| 132 |
+
110 perform the semi-supervised node classification task. Based on the MPNN framework [17], each
|
| 133 |
+
111 layer of the vanilla GCN is defined in three parts:
|
| 134 |
+
112 (1) Message computation: The message of node $v _ { i }$ and its neighbor node $v _ { j }$ is calculated, where
|
| 135 |
+
113 $j \in \mathcal { N } ( i )$ , as:
|
| 136 |
+
|
| 137 |
+
$$
|
| 138 |
+
\mathbf { m } _ { j } ^ { ( k ) } = \frac { 1 } { \sqrt { \deg ( i ) \deg ( j ) } } \mathbf { h } _ { j } ^ { ( k - 1 ) } \mathbf { W } ^ { ( k ) } , \ j \in \{ i \} \bigcup \mathcal { N } ( i ) .
|
| 139 |
+
$$
|
| 140 |
+
|
| 141 |
+
Here, 114 $\deg ( i )$ is the degree of node $v _ { i }$ and $\mathbf { W } ^ { ( k ) } \in \mathbb { R } ^ { c _ { k - 1 } \times c _ { k } }$ are the learnable parameters of the 115 $k ^ { \mathrm { t h } }$ -layer, where $c _ { k }$ is the size of the hidden embedding.
|
| 142 |
+
|
| 143 |
+
116 (2) Aggregation: The messages of node $v _ { i }$ and its neighbors are aggregated by summing them up:
|
| 144 |
+
|
| 145 |
+
$$
|
| 146 |
+
\mathbf { m } _ { \mathcal { N } ( i ) } ^ { ( k ) } = \sum _ { j \in \{ i \} \cup \mathcal { N } ( i ) } \mathbf { m } _ { j } ^ { ( k ) } ,
|
| 147 |
+
$$
|
| 148 |
+
|
| 149 |
+
|
| 150 |
+
Figure 3: The overall network architecture. Our method consists of three networks: a backbone network for the primary task and two auxiliary task networks. (1) The backbone network is a vanilla GCN, which predicts the classification results of each node. (2) The first auxiliary task network is a link predictor, which focuses on the link prediction task and generates a probabilistic edge structure as the input of the backbone network. (3) The second auxiliary task network is a label generator, which employs the label generation task to generate the pseudo soft labels for supervising the node classifier. The parts connected by the dashed arrow share the same parameters, and the red slash on the arrow indicates a stop-gradient (detach) operation.
|
| 151 |
+
|
| 152 |
+
where m(k)(i)117 denotes the aggregated message.
|
| 153 |
+
|
| 154 |
+
18 (3) Feature updating: Finally, the hidden representation is updated with the aggregated message. For
|
| 155 |
+
19 the vanilla GCN, the update function can be considered as applying a non-linear operation to the
|
| 156 |
+
120 aggregated message:
|
| 157 |
+
|
| 158 |
+
$$
|
| 159 |
+
\mathbf { h } _ { i } ^ { k } = \left\{ \begin{array} { l l } { \mathrm { s o f t m a x } ( \mathbf { m } _ { \mathcal { N } ( i ) } ^ { ( k ) } ) , } & { \mathrm { i f ~ } k = K } \\ { \mathrm { R e L U } ( \mathbf { m } _ { \mathcal { N } ( i ) } ^ { ( k ) } ) , } & { \mathrm { o t h e r w i s e } . } \end{array} \right.
|
| 160 |
+
$$
|
| 161 |
+
|
| 162 |
+
121 Here, $K$ is the number of model layers. The last layer of the vanilla GCN should output the
|
| 163 |
+
122 classification probability via a softmax function. Otherwise, a ReLU operation is used.
|
| 164 |
+
|
| 165 |
+
# 3.2 Multi-Task Network Architecture
|
| 166 |
+
|
| 167 |
+
In this paper, we propose an auxiliary learning induced GCN for semi-supervised node classification (see Fig. 3). To enhance the node classification performance of the backbone vanilla GCN, we design two auxiliary tasks: link prediction and pseudo-label generation.
|
| 168 |
+
|
| 169 |
+
127 The first $K - 1$ layers of a $K$ -layer vanilla GCN model can be considered as a feature extractor, while
|
| 170 |
+
128 the last layer can be considered as a classifier. Denoting the feature extractor and node classifier of
|
| 171 |
+
129 the backbone model as $h _ { \theta _ { 2 } }$ and $f _ { \theta _ { 1 } }$ , respectively, where $\theta _ { 1 }$ and $\theta _ { 2 }$ are the learnable parameters, the
|
| 172 |
+
130 proposed two auxiliary task networks are defined as follows.
|
| 173 |
+
|
| 174 |
+
# 3.2.1 Link Predictor
|
| 175 |
+
|
| 176 |
+
132 To enrich the edge information, we design an auxiliary link prediction task to infer the missing edges
|
| 177 |
+
133 and present the probability of edge existence. The link predictor we propose contains a decoder $\bar { R } ( \cdot )$
|
| 178 |
+
134 and the feature extractor $h _ { \theta _ { 2 } }$ of the backbone model.
|
| 179 |
+
|
| 180 |
+
The feature extractor 135 $h _ { \theta _ { 2 } } ( \cdot )$ takes reduced adjacency matrix ${ \bf A } _ { s }$ which corresponds to the sampled 136 edge set $\mathcal { E } _ { s } \subset \mathcal { E }$ , and node features $\mathbf { X }$ as inputs to generate the hidden embedding:
|
| 181 |
+
|
| 182 |
+
$$
|
| 183 |
+
\begin{array} { r } { { \bf H } _ { s } = h _ { \theta _ { 2 } } ( { \bf X } , { \bf A } _ { s } ) . } \end{array}
|
| 184 |
+
$$
|
| 185 |
+
|
| 186 |
+
137 Then, the decoder computes the similarity between each node based on $\mathbf { H } _ { s }$ to predict the edge
|
| 187 |
+
138 existence probabilities. We simply use an inner-product calculator with a sigmoid function as the
|
| 188 |
+
139 implementation of decoder $R ( \cdot )$ . The similarity of two hidden embeddings can be computed as
|
| 189 |
+
|
| 190 |
+
$$
|
| 191 |
+
r _ { i j } = R ( \mathbf { h } _ { i } , \mathbf { h } _ { j } ) = \sigma ( \mathbf { h } _ { i } \mathbf { h } _ { j } ^ { T } ) ,
|
| 192 |
+
$$
|
| 193 |
+
|
| 194 |
+
140 where $\mathbf { h } _ { i }$ and $\mathbf { h } _ { j }$ denote the hidden embeddings of node $v _ { i }$ and node $v _ { j }$ in $\mathbf { H } _ { s }$ , respectively.
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+
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# 141 3.2.2 Label Generator
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Although using the one-hot label is inappropriate, it is difficult to obtain soft labels with manual annotations for real graph-structured data. To tackle this issue, we introduce an auxiliary label generation task to generate soft labels that reflect the tendency of different classes each node belongs to. Label generator $g _ { \varphi } ( { \bf X } , { \bf A } )$ is a vanilla GCN, where $\varphi$ are the learnable parameters and A is the adjacency matrix. The label generation network predicts the label distribution of each node based on the graph structure $\mathbf { A }$ and the raw features $\mathbf { X }$ of the nodes, as:
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+
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$$
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+
\hat { \mathbf { Y } } ^ { g } = g _ { \varphi } ( \mathbf { X } , \mathbf { A } ) ,
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+
$$
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+
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+
where 48 $\mathbf { Y } ^ { g }$ are the predicted pseudo labels for guiding the training of the backbone network and the 49 label generator.
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+
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# 3.2.3 Node classifier
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151 The node classifier carries out the primary semi-supervised node classification task in our model.
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152 Compared to the vanilla GCN model, we add a graph reconstruction step at the beginning. The graph
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153 adjacency matrix reconstructed with the hidden embeddings contains richer edge information since
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154 the proposed auxiliary link prediction task can enhance the graph topology capturing ability of the
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155 feature extractor $h _ { \theta _ { 2 } } ( \cdot )$ . However, the hidden embeddings derived by the feature extractor change
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156 rapidly in the first few iterations, resulting in a changing reconstructed adjacency matrix. Directly
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157 applying the reconstructed adjacency matrix to the entire backbone network increases the training
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158 instability. Thus, we only apply it as the input of the feature extractor $h _ { \theta _ { 2 } } ( \cdot )$ , while the classifier
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159 $f _ { \theta _ { 1 } } ( \cdot )$ still adopts the original adjacency matrix as input.
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160 In each training iteration, the feature extractor $h _ { \theta _ { 2 } } ( \cdot )$ first generates the hidden embeddings of the
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161 nodes $\mathbf { H } = h _ { \theta _ { 2 } } ( \mathbf { X } , \mathbf { A } )$ . The decoder uses $\mathbf { H }$ to reconstruct the adjacency matrix $\mathbf { A } _ { \mathrm { r e c o n } } = R ( \mathbf { H } , \mathbf { H } )$ .
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162 Then, the feature extractor takes the reconstructed adjacency matrix $\mathbf { A } _ { \mathrm { r e c o n } }$ to compute the hidden
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163 embeddings $\mathbf { H } _ { \mathrm { r e c o n } } = h _ { \theta _ { 2 } } ( \mathbf { X } , \mathbf { A } _ { \mathrm { r e c o n } } )$ .
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+
64 Finally, the computed hidden embeddings $\mathbf { H } _ { \mathrm { r e c o n } }$ and the original graph adjacency matrix A are fed
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65 to the classifier to obtain the final classification results:
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+
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| 224 |
+
$$
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| 225 |
+
\hat { \mathbf { Y } } ^ { f } = f _ { \theta _ { 1 } } ( \mathbf { H } _ { \mathrm { r e c o n } } , \mathbf { A } ) .
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+
$$
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+
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+
# 166 3.3 Auxiliary Training Phases
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+
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| 230 |
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67 In addition to the backbone network, the proposed method contains two auxiliary task networks. In
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68 the following, we will introduce the objectives for the node classifier $f _ { \theta _ { 1 } }$ , link predictor $h _ { \theta _ { 2 } }$ , and
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+
69 label generator $g _ { \varphi }$ in order, and then leverage a meta auxiliary learning scheme to train the proposed
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70 multi-task network.
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+
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| 235 |
+
# 3.3.1 Training the Node Classifier $f _ { \theta _ { 1 } }$
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+
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| 237 |
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172 The purpose of the node classifier $f _ { \theta _ { 1 } } ( \cdot )$ is to carry out the graph-based semi-supervised node
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| 238 |
+
173 classification task, which yields the final prediction results. Naturally, it is necessary to use the
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| 239 |
+
174 classification loss between the predicted result and the real categories of nodes to supervise the
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| 240 |
+
175 training process. At the same time, to reflect the tendency of the class a node belongs to, the training
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+
176 should make the prediction labels of the classifier be close to the pseudo soft labels generated by
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+
177 label generator $g _ { \varphi } ( \cdot )$ .
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+
178 In the $t ^ { \mathrm { t h } }$ iteration, denoting the pseudo soft labels as $\hat { \mathbf { Y } } ^ { g ( t ) } = g _ { \varphi ^ { ( t ) } } ( \mathbf { X } , \mathbf { A } )$ , the real labels used in
|
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+
179 training as $\mathbf { Y } _ { \mathrm { t r a i n } }$ , and the prediction results of the node classifier of the backbone network $f _ { \theta _ { 1 } }$ as
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+
180 $\hat { \mathbf { Y } } ^ { f ( t ) } = f _ { \theta _ { 1 } ^ { ( t ) } } \big ( h _ { \theta _ { 2 } ^ { ( t ) } } \big ( \mathbf { X } , \mathbf { A } _ { \mathrm { r e c o n } } ^ { ( t ) } \big ) , \mathbf { A } \big )$ , the objective for the node classifier $f _ { \theta _ { 1 } } ( \cdot )$ is defined as
|
| 246 |
+
|
| 247 |
+
$$
|
| 248 |
+
\mathcal { L } _ { \theta _ { 1 } } ^ { ( t ) } = \mathcal { L } _ { \mathrm { C E } } ( \hat { \mathbf { Y } } _ { \mathrm { t r a i n } } ^ { f ( t ) } , \mathbf { Y } _ { \mathrm { t r a i n } } ) + \mathcal { L } _ { \mathrm { M S E } } ( \hat { \mathbf { Y } } ^ { f ( t ) } , \hat { \mathbf { Y } } ^ { g ( t ) } ) .
|
| 249 |
+
$$
|
| 250 |
+
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| 251 |
+
181 The objective contains two parts: the loss on the real training labels, and the loss on the generated
|
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+
182 pseudo soft labels. $\mathcal { L } _ { \mathrm { C E } }$ denotes the cross-entropy loss and $\mathcal { L } _ { \mathrm { M S E } }$ denotes the mean squared loss.
|
| 253 |
+
183 Although this objective can be used to update the learnable parameters $\theta _ { 2 }$ of the feature extractor
|
| 254 |
+
184 $h _ { \theta _ { 2 } } ( \cdot )$ , as it generates the hidden embeddings used for node classification, we only employ it to
|
| 255 |
+
185 supervise the learning of the node parameters $\theta _ { 1 }$ of the classifier $f _ { \theta _ { 1 } } ( \cdot )$ . To avoid adding unnecessary
|
| 256 |
+
186 supervision for generating the reconstructed graph adjacency matrix $\mathbf { A } _ { \mathrm { r e c o n } }$ , we add a stop-gradient
|
| 257 |
+
187 operation (detach) which is formulated as follows:
|
| 258 |
+
|
| 259 |
+
$$
|
| 260 |
+
{ \bf A } _ { \mathrm { r e c o n } } ^ { ( t ) } = { \tt d e t a c h } ( R ( { \bf H } , { \bf H } ) ) .
|
| 261 |
+
$$
|
| 262 |
+
|
| 263 |
+
# 3.3.2 Auxiliary Training for the Feature Extractor $h _ { \theta _ { 2 } }$
|
| 264 |
+
|
| 265 |
+
189 The feature extractor $h _ { \theta _ { 2 } }$ is shared by the link prediction module and the backbone network. In each
|
| 266 |
+
190 iteration, we first randomly sample a certain percentage of edges from the real edge set $\mathcal { E }$ to form a
|
| 267 |
+
191 sampled edge set $\mathcal { E } _ { s } \subset \mathcal { E }$ . Denoting the reduced graph adjacency matrix as ${ \bf A } _ { s }$ , which corresponds to
|
| 268 |
+
192 the sampled edge set $\mathcal { E } _ { s }$ , and applying a message passing operation with ${ \bf A } _ { s }$ as
|
| 269 |
+
|
| 270 |
+
$$
|
| 271 |
+
\begin{array} { r } { \mathbf { H } _ { s } ^ { ( t ) } = h _ { \theta _ { 2 } ^ { ( t ) } } ( \mathbf { X } , \mathbf { A } _ { s } ^ { ( t ) } ) , } \end{array}
|
| 272 |
+
$$
|
| 273 |
+
|
| 274 |
+
193 then the objective for the feature extractor is defined as
|
| 275 |
+
|
| 276 |
+
$$
|
| 277 |
+
\mathcal { L } _ { \theta _ { 2 } } ^ { ( t ) } = \mathcal { L } _ { \mathrm { C E } } ( R ( \mathbf { H } _ { s } ^ { ( t ) } , \mathbf { H } _ { s } ^ { ( t ) } ) , \mathbf { A } ) .
|
| 278 |
+
$$
|
| 279 |
+
|
| 280 |
+
Here $R ( \mathbf { H } _ { s } ^ { ( t ) } , \mathbf { H } _ { s } ^ { ( t ) } )$ represents the correlation between the hidden embeddings of each pair of nodes.
|
| 281 |
+
|
| 282 |
+
# 3.3.3 Auxiliary Training for the Label Generator $g _ { \varphi }$
|
| 283 |
+
|
| 284 |
+
The label generator $g _ { \varphi }$ is a vanilla GCN model used to predict the label distribution for each node, which naturally needs to be supervised by a classification loss. In the $t ^ { \mathrm { t h } }$ iteration, a label generator first generates the prediction results using the original graph adjacency matrix A and the node features $\mathbf { X }$ , which is formulated as $\hat { \mathbf { Y } } ^ { g ( t ) } = g _ { \varphi ^ { ( t ) } } ( \mathbf { X } , \mathbf { A } )$ . Denoting the training labels as $\mathbf { Y } _ { \mathrm { t r a i n } }$ , the objective of the label generator $g _ { \varphi }$ can be formulated as
|
| 285 |
+
|
| 286 |
+
$$
|
| 287 |
+
\mathcal { L } _ { \varphi } ^ { ( t ) } = \mathcal { L } _ { \mathrm { C E } } ( \hat { \mathbf { Y } } _ { \mathrm { t r a i n } } ^ { g ( t ) } , \mathbf { Y } _ { \mathrm { t r a i n } } ) .
|
| 288 |
+
$$
|
| 289 |
+
|
| 290 |
+
# 201 3.3.4 Meta-Learning Based Training Strategy
|
| 291 |
+
|
| 292 |
+
The final node classification results only depend on the primary task, while the performance of the other modules, including the label generator and link predictor are not our ultimate concern. Simply using the classification loss to train the label generator $g _ { \varphi }$ or using the link prediction loss to train the feature extractor $h _ { \theta _ { 2 } }$ cannot provide the effects of auxiliary learning. Thus, it is crucial to make the node classifier $f _ { \theta _ { 1 } }$ perform better after it is trained with the pseudo soft labels while taking the hidden embeddings derived by the feature extractor $h _ { \theta _ { 2 } }$ as inputs. To this end, we use meta auxiliary learning to update the model parameters.
|
| 293 |
+
|
| 294 |
+
209 For the auxiliary label generation task, we consider not only the classification performance of the
|
| 295 |
+
210 label generator $g _ { \varphi }$ , but also the auxiliary effect on the node classifier. We assume that the classifier
|
| 296 |
+
211 parameters $\theta _ { 1 }$ are updated with gradient descent based on the pseudo soft labels in the $t ^ { \mathrm { t h } }$ iteration,
|
| 297 |
+
|
| 298 |
+
$$
|
| 299 |
+
\boldsymbol { \theta } _ { 1 } ^ { \prime } = \boldsymbol { \theta } _ { 1 } ^ { ( t ) } - \eta \nabla _ { \boldsymbol { \theta } _ { 1 } } \mathcal { L } _ { \mathrm { M S E } } ( \hat { \mathbf { Y } } ^ { f ( t ) } , \hat { \mathbf { Y } } ^ { g ( t ) } ) .
|
| 300 |
+
$$
|
| 301 |
+
|
| 302 |
+
212 A direct way to evaluate the auxiliary effect of the pseudo soft labels is to compute the classification loss of the prediction given by the node classifier using the updated parameters 213 $\theta _ { 1 } ^ { \prime }$ , as
|
| 303 |
+
|
| 304 |
+
$$
|
| 305 |
+
\begin{array} { r } { \mathcal { L } ^ { f \prime } = \mathcal { L } _ { \mathrm { C E } } \big ( f _ { \theta _ { 1 } ^ { \prime } } \big ( h _ { \theta _ { 2 } ^ { ( t ) } } \big ( \mathbf { X } , \mathbf { A } _ { \mathrm { r e c o n } } \big ) , \mathbf { A } \big ) _ { \mathrm { t r a i n } } , \mathbf { Y } _ { \mathrm { t r a i n } } \big ) . } \end{array}
|
| 306 |
+
$$
|
| 307 |
+
|
| 308 |
+
214 This loss can quantify how much performance improvement the classifier gains from the two auxiliary
|
| 309 |
+
215 tasks. Because $\theta _ { 1 } ^ { \prime }$ is updated with the pseudo soft labels generated by $g _ { \varphi } ( \cdot )$ , the loss $\mathcal { L } ^ { f \prime }$ is also a
|
| 310 |
+
216 function of $\varphi$ . This means that the objective could be used to supervise the learning of $\varphi$ . Note
|
| 311 |
+
217 that $\nabla _ { \varphi } \mathcal { L } ^ { f \prime }$ requires the gradient of the gradient to be computed [25], which can be considered as a
|
| 312 |
+
218 meta-learning strategy. The final objective of the label generator $g _ { \varphi }$ with meta-learning is formulated
|
| 313 |
+
219 as
|
| 314 |
+
|
| 315 |
+
$$
|
| 316 |
+
\mathcal { L } _ { \varphi - \mathrm { { m e t a } } } ^ { ( t ) } = \mathcal { L } _ { \varphi } ^ { ( t ) } + \mathcal { L } ^ { f \prime } .
|
| 317 |
+
$$
|
| 318 |
+
|
| 319 |
+
220 Similar to the label generator $g _ { \varphi } ( \cdot )$ , the link predictor $h _ { \theta _ { 2 } } ^ { b }$ should derive the effective node embed
|
| 320 |
+
221 dings to enhance the node classification performance. Since the hidden embeddings used for node
|
| 321 |
+
|
| 322 |
+
# Algorithm 1 AL-GCN
|
| 323 |
+
|
| 324 |
+
Input: Graph adjacency matrix A, the node features $\mathbf { X }$ , the data labels $\mathbf { Y } _ { \mathrm { t r a i n } }$ of a training set. Output: A feature extractor $h _ { \theta _ { 2 } }$ , a node classifier $f _ { \theta _ { 1 } }$
|
| 325 |
+
|
| 326 |
+
1: Initialize learnable parameters $\theta _ { 1 } , \theta _ { 2 }$ , $\varphi$
|
| 327 |
+
2: while not converged do
|
| 328 |
+
3: # node classifier training phase
|
| 329 |
+
4: $\begin{array} { r l } & { \hat { \mathbf { Y } } ^ { g } \gets g _ { \varphi } ( \mathbf { X } , \mathbf { A } ) } \\ & { \mathbf { H } \gets h _ { \theta _ { 2 } } ( \mathbf { X } , \mathbf { A } ) } \\ & { \mathbf { A } _ { \mathrm { r e c o n } } = \mathsf { d e t a c h } ( R ( \mathbf { H } , \mathbf { H } ) ) } \\ & { \mathbf { H } _ { \mathrm { r e c o n } } \gets h _ { \theta _ { 2 } } ( \mathbf { X } , \mathbf { A } _ { \mathrm { r e c o n } } ) } \\ & { \hat { \mathbf { Y } } ^ { f } \gets f _ { \theta _ { 1 } } ( \mathbf { H } _ { \mathrm { r e c o n } } , \mathbf { A } ) } \\ & { \mathcal { L } _ { \theta _ { 1 } } \gets \mathcal { L } _ { \mathrm { C E } } ( \hat { \mathbf { Y } } _ { \mathrm { t r a i n } } ^ { f } , \mathbf { Y } _ { \mathrm { t r a i n } } ) + \mathcal { L } _ { \mathrm { M S E } } ( \hat { \mathbf { Y } } ^ { f } , \hat { \mathbf { Y } } ^ { g } ) } \\ & { \mathbf { \Delta } _ { \mathrm { r e } } ^ { \mathsf { x } } , \quad \mathbf { \Delta } _ { \mathrm { L } } ^ { \mathsf { x } } , } \end{array}$
|
| 330 |
+
5:
|
| 331 |
+
6:
|
| 332 |
+
7:
|
| 333 |
+
8:
|
| 334 |
+
9:
|
| 335 |
+
10: Update: $\theta _ { 1 } \gets \mathrm { A d a m } ( \mathcal { L } _ { \theta _ { 1 } } , \theta _ { 1 } )$
|
| 336 |
+
11: # meta-learning preparation
|
| 337 |
+
12: Compute: $-$
|
| 338 |
+
13: $\begin{array} { r l } { } & { { } \mathcal { L } _ { \mathrm { C E } } ( f ^ { b } ( h _ { \theta _ { 2 } } ^ { b } ( \mathbf { X } , \mathbf { A } _ { \mathrm { r e c o n } } ) , \mathbf { A } ) _ { \mathrm { t r a i n } } , \mathbf { Y } _ { \mathrm { t r a i n } } ) } \end{array}$
|
| 339 |
+
14: # label generator training phase
|
| 340 |
+
15: $\begin{array} { r l } & { \hat { \mathbf Y } ^ { g } \gets g _ { \varphi } ( \mathbf X , \mathbf A ) } \\ & { \mathcal { L } _ { \varphi } \gets \mathcal { L } _ { \mathrm { C E } } ( \hat { \mathbf Y } _ { \mathrm { t r a i n } } ^ { g } , \mathbf Y _ { \mathrm { t r a i n } } ) _ { , } + } \end{array}$
|
| 341 |
+
16:
|
| 342 |
+
17: Update: $\varphi \gets \mathrm { A d a m } ( \mathcal { L } _ { \varphi } , \varphi )$
|
| 343 |
+
18: # feature extractor training phase
|
| 344 |
+
19: ${ \bf A } _ { s } \gets$ RandomSample(A)
|
| 345 |
+
20: Hs ← hθ (X, As)
|
| 346 |
+
21: $\mathcal { L } _ { \boldsymbol { \theta } _ { 2 } } \gets \mathcal { L } _ { \mathrm { C E } } ( R ( \mathbf { H } _ { s } , \mathbf { H } _ { s } ) , \mathbf { A } ) + $
|
| 347 |
+
22: Update: $\theta _ { 2 } \gets \mathrm { A d a m } ( \mathcal { L } _ { \theta _ { 2 } } , \theta _ { 2 } )$
|
| 348 |
+
23: end while
|
| 349 |
+
222 classification are derived from the feature extractor $h _ { \theta _ { 2 } }$ , the objective defined in Eq. 14 can also be
|
| 350 |
+
223 considered as a function of $\theta _ { 2 }$ . Thus, the objective of the feature extractor $h _ { \theta _ { 2 } }$ with meta-learning is
|
| 351 |
+
224 defined as
|
| 352 |
+
|
| 353 |
+
$$
|
| 354 |
+
\mathcal { L } _ { \theta _ { 2 } \mathrm { - m e t a } } ^ { ( t ) } = \mathcal { L } _ { \theta _ { 2 } } ^ { ( t ) } + \mathcal { L } ^ { f ^ { \prime } } .
|
| 355 |
+
$$
|
| 356 |
+
|
| 357 |
+
225 This objective means that, except for the link prediction task, the feature extractor $h _ { \theta _ { 2 } }$ should ensure
|
| 358 |
+
226 that the generated hidden embedding enhance the classification accuracy of the node classifier $f _ { \theta _ { 1 } }$
|
| 359 |
+
227 updated with the pseudo soft labels. The overall training process is shown in Alg. 1.
|
| 360 |
+
|
| 361 |
+
# 4 Experiments
|
| 362 |
+
|
| 363 |
+
# 4.1 Experimental Settings and Compared Methods
|
| 364 |
+
|
| 365 |
+
We demonstrate the classification performance of our method via semi-supervised document classification on three citation network datasets, including Cora, Citeseer, and Pubmed [26], where nodes represent the documents and edges are citation links. Dataset statistics are summarized in Table 1.
|
| 366 |
+
|
| 367 |
+
233 The proposed method is a novel GCN, which is leveraged to carry out a node classification task.
|
| 368 |
+
234 We compare our method with several popular graph-based node classification methods including
|
| 369 |
+
235 GCN [12], GAT [3], DualGCN [13], SGC [27], and APPNP [4]. As we use a three-layer GCN as
|
| 370 |
+
236 the backbone in our proposed model, we compare both two-layer and three-layer GCNs, denoted as
|
| 371 |
+
237 GCN2 and GCN3, respectively.
|
| 372 |
+
|
| 373 |
+
For the semi-supervised node classification, we use all node features but only 20 labels per class for training and 500 nodes as the validation set. We train the proposed method for a maximum of 400 epochs using Adam [28] with a learning rate of 0.01. We use the well-trained parameters, which achieve the best performance on the validation set during the training phases, to evaluate classification accuracy on a test set of 1,000 labeled examples. We run each method 100 times and compute the average classification accuracy on a single NVIDIA GTX 1080Ti GPU. To implement all the compared methods more conveniently, we use PyG [29] as the graph-based learning framework.
|
| 374 |
+
|
| 375 |
+
Table 1: Dataset statistics.
|
| 376 |
+
|
| 377 |
+
<table><tr><td>Dataset</td><td>Nodes</td><td>Edges</td><td>Features</td><td>Classes</td></tr><tr><td>Cora</td><td>2,708</td><td>5,278</td><td>1,433</td><td>7</td></tr><tr><td>CiteSeer</td><td>3,327</td><td>4,552</td><td>3,703</td><td>6</td></tr><tr><td>Cora</td><td>19,717</td><td>44,324</td><td>500</td><td>3</td></tr></table>
|
| 378 |
+
|
| 379 |
+
Table 2: Classification results on the datasets (bold: best, underline: runner-up).
|
| 380 |
+
|
| 381 |
+
<table><tr><td>Method</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td></tr><tr><td>GAT [3] DualGCN[13]</td><td>82.5 ±0.8%</td><td>71.4±0.7%</td><td>78.4± 0.4%</td></tr><tr><td rowspan="3">SGC [27] APPNP [4]</td><td>83.4 ± 0.5%</td><td>72.6 ± 0.6%</td><td>79.9 ± 0.3%</td></tr><tr><td>81.3 ± 0.7%</td><td>70.9 ± 0.6%</td><td>78.2 ± 0.5%</td></tr><tr><td>83.2 ± 0.4%</td><td>71.6 ± 0.5%</td><td>79.8 ± 0.3%</td></tr><tr><td rowspan="2">GCN2 [12] AL-GCN2 (ours)</td><td>81.5±0.7%</td><td>71.5 ± 0.5%</td><td>79.2 ± 0.4%</td></tr><tr><td>82.3 ± 0.4%</td><td>72.6 ± 0.5%</td><td>79.6± 0.5%</td></tr><tr><td rowspan="2">GCN3[12] AL-GCN3 (ours)</td><td>80.7±1.2%</td><td>68.0±1.4%</td><td>77.7 ± 0.5%</td></tr><tr><td>84.7 ± 0.4%</td><td>72.3± 0.5%</td><td>81.4± 0.6%</td></tr></table>
|
| 382 |
+
|
| 383 |
+
# 245 4.2 Experimental Results
|
| 384 |
+
|
| 385 |
+
46 In this section, we provide the experimental results of the node classification, an ablation study, and
|
| 386 |
+
47 the visualization of hidden embeddings. More experimental results, such as parameter and model
|
| 387 |
+
48 robustness studies can be found in the supplementary materials.
|
| 388 |
+
|
| 389 |
+
We conduct the node classification task on three citation network datasets. As shown in Table 2, our method consistently and significantly enhances the learning performance compared to the other methods. In particular, for the Cora dataset, the proposed method is superior to GCN by $4 . 9 \%$ Compared to the other methods, our model considers more graph-structured information via auxiliary learning. Thus, it is consistently and significantly superior to the compared methods, achieving state-of-the-art results.
|
| 390 |
+
|
| 391 |
+
# 4.3 Ablation Studies
|
| 392 |
+
|
| 393 |
+
# 4.3.1 On the Auxiliary Learning Modules
|
| 394 |
+
|
| 395 |
+
To determine how the link predictor $( P )$ and the pseudo label generator $( G )$ affect the node classification performance, we apply the following two ablation models: (1) Vanilla GCN with the link predictor, termed $\mathrm { G C N } { + } P$ . (2) Vanilla GCN with the lable generator, termed ${ \mathrm { G C N } } { + } G$ . We compare ${ \mathrm { G C N } } { \mathrm { + } } G$ and $\mathrm { G C N } { + } P$ with the proposed method, AL-GCN $( \mathbf { G } \mathbf { C } \mathbf { N } { + } \mathbf { } \mathbf { } P { + } G )$ , and the original GCN. Table 3 lists the results. As can be seen, the proposed method consistently outperforms $\mathrm { G C N } { + } P$ and ${ \mathrm { G C N } } { + } G$ . Specifically, compared to the link predictor, the label generator has more effect on the Citeseer dataset. However, for the Pubmed dataset, the link predictor has much more effect on the learning performance.
|
| 396 |
+
|
| 397 |
+
# 4.3.2 On the Reconstructed Graph Adjacency Matrix
|
| 398 |
+
|
| 399 |
+
For the backbone network, a reconstructed graph adjacency matrix via a link prediction task is used as input. To determine how the reconstructed graph adjacency matrix affects the node classification performance, we compare the proposed model with the following two models: (1) A model that takes the original graph adjacency matrix without the reconstructed one as the input of the backbone network, termed w/o-recG; (2) A model that takes only the reconstructed graph adjacency matrix as the input of the backbone network, termed w/o-oriG.
|
| 400 |
+
|
| 401 |
+
272 As shown in Table 4, our method consistently outperforms w/o-recG and w/o-oriG. This can be
|
| 402 |
+
273 attributed to the fact that the reconstructed graph adjacency matrix via the link predictor can capture
|
| 403 |
+
274 the detailed topology information of a graph and the fixed original graph adjacency matrix increases
|
| 404 |
+
275 the training stability.
|
| 405 |
+
|
| 406 |
+
Table 3: The ablation experiment results.
|
| 407 |
+
|
| 408 |
+
<table><tr><td>Method</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td></tr><tr><td>GCN</td><td>80.7 ± 1.2%</td><td>68.0 ± 1.4%</td><td>77.7 ± 0.5%</td></tr><tr><td>GCN+P</td><td>83.0 ± 0.7%</td><td>70.6 ± 0.8%</td><td>81.3 ± 0.6%</td></tr><tr><td>GCN+G</td><td>83.0 ± 0.6%</td><td>71.7 ± 0.8%</td><td>78.8± 0.6%</td></tr><tr><td>GCN+P+G (AL-GCN)</td><td>84.7 ± 0.4%</td><td>72.3± 0.5%</td><td>81.4 ± 0.6%</td></tr></table>
|
| 409 |
+
|
| 410 |
+
Table 4: The ablation experiment results in terms of classification accuracy (in percent).
|
| 411 |
+
|
| 412 |
+
<table><tr><td>Method</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td></tr><tr><td>GCN</td><td>80.7±1.2%</td><td>68.0±1.4%</td><td>77.7 ± 0.5%</td></tr><tr><td>w/o-recG</td><td>84.5 ± 0.5%</td><td>71.9 ± 0.6%</td><td>80.1 ± 0.5%</td></tr><tr><td>W/0-oriG</td><td>84.1 ±0.6%</td><td>71.4 ± 0.9%</td><td>80.6 ± 1.5%</td></tr><tr><td>AL-GCN</td><td>84.7 ±0.4%</td><td>72.3 ± 0.5%</td><td>81.4± 0.6%</td></tr></table>
|
| 413 |
+
|
| 414 |
+
|
| 415 |
+
Figure 4: Visualization of the hidden embedding obtained by different methods via t-SNE algorithm.
|
| 416 |
+
|
| 417 |
+
# 276 4.4 Visualization of Hidden Embeddings
|
| 418 |
+
|
| 419 |
+
To determine how the hidden embeddings affect the learning performance, we use the visualization tool t-SNE [30] to observe their distribution. As shown in Fig. 4, the embedding results of GCN and GAT are denser, and the separation of different clusters is not obvious. In contrast, the node distributions learned by our proposed method are more separate, with most of the nodes from the same classes being close to each other, resulting in obvious cluster structures. These experimental results demonstrate that the proposed method can capture more detailed structure information of a graph, including the nodes and edges, resulting in more effective hidden embeddings.
|
| 420 |
+
|
| 421 |
+
# 5 Conclusion
|
| 422 |
+
|
| 423 |
+
We have proposed a novel graph convolutional network for semi-supervised node classification. Different from existing methods, the proposed model focuses on enriching the graph data and adopts meta auxiliary learning to enhance the representations of nodes and edges in a graph. To enrich node label information, an auxiliary label generator is used to generate pseudo probabilistic labels. Meanwhile, an auxiliary link predictor is used to generate probabilistic edges to enrich the graph structure information. The enriched node and edge information can iteratively enhance the performance of the node classification task. Experimental results on several benchmark citation datasets show that the proposed model is superior to the existing methods. For future work, we note that real-world data is usually contaminated by noise, which results in a robustness problem for graph learning methods. We plan to extend our model to handle noisy data by designing a more robust learning method for graph-structured data.
|
| 424 |
+
|
| 425 |
+
# References
|
| 426 |
+
|
| 427 |
+
[1] David Duvenaud, Dougal Maclaurin, Jorge Aguilera-Iparraguirre, Rafael Gómez-Bombarelli, Timothy Hirzel, Alán Aspuru-Guzik, and Ryan P. Adams. Convolutional networks on graphs for learning molecular fingerprints. In Advances in Neural Information Processing Systems, pages 2224–2232, 2015.
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| 428 |
+
[2] Rex Ying, Ruining He, Kaifeng Chen, Pong Eksombatchai, William L. Hamilton, and Jure Leskovec. Graph convolutional neural networks for web-scale recommender systems. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 974–983, 2018. [3] Petar Velickovic, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph attention networks. In 6th International Conference on Learning Representations, 2018. [4] Johannes Klicpera, Aleksandar Bojchevski, and Stephan Günnemann. Predict then propagate: Graph neural networks meet personalized pagerank. In 7th International Conference on Learning Representations, 2019. [5] Xiao Wang, Meiqi Zhu, Deyu Bo, Peng Cui, Chuan Shi, and Jian Pei. AM-GCN: adaptive multichannel graph convolutional networks. In Proceedings of the 26t h ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 1243–1253, 2020. [6] Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Stefanie Jegelka. Representation learning on graphs with jumping knowledge networks. In Proceedings of the 35th International Conference on Machine Learning, volume 80, pages 5449–5458, 2018. [7] Lingxiao Zhao and Leman Akoglu. Pairnorm: Tackling oversmoothing in gnns. In 8th International Conference on Learning Representations, 2020.
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[8] Ming Chen, Zhewei Wei, Zengfeng Huang, Bolin Ding, and Yaliang Li. Simple and deep graph convolutional networks. In Proceedings of the 37th International Conference on Machine Learning, volume 119, pages 1725–1735, 2020. [9] Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into graph convolutional networks for semi-supervised learning. In Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, pages 3538–3545, 2018.
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[10] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. In 2nd International Conference on Learning Representations, 2014.
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| 431 |
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[11] Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In Advances in Neural Information Processing Systems, pages 3837–3845, 2016.
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| 432 |
+
[12] Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In 5th International Conference on Learning Representations, 2017.
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| 433 |
+
[13] Chenyi Zhuang and Qiang Ma. Dual graph convolutional networks for graph-based semisupervised classification. In Proceedings of the 2018 World Wide Web Conference on World Wide Web, pages 499–508, 2018.
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| 434 |
+
[14] Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S. Yu. A comprehensive survey on graph neural networks. CoRR, abs/1901.00596, 2019.
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| 435 |
+
[15] Alessio Micheli. Neural network for graphs: A contextual constructive approach. IEEE Trans. Neural Networks, 20(3):498–511, 2009.
|
| 436 |
+
[16] James Atwood and Don Towsley. Diffusion-convolutional neural networks. In Advances in Neural Information Processing Systems, pages 1993–2001, 2016.
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| 437 |
+
[17] Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural message passing for quantum chemistry. In Proceedings of the 34th International Conference on Machine Learning, volume 70, pages 1263–1272, 2017.
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| 438 |
+
[18] Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodolà, Jan Svoboda, and Michael M. Bronstein. Geometric deep learning on graphs and manifolds using mixture model cnns. In 2017 IEEE Conference on Computer Vision and Pattern Recognition, pages 5425–5434, 2017.
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| 439 |
+
[19] Hongyang Gao, Zhengyang Wang, and Shuiwang Ji. Large-scale learnable graph convolutional networks. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 1416–1424, 2018.
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| 440 |
+
[20] John Flynn, Ivan Neulander, James Philbin, and Noah Snavely. Deepstereo: Learning to predict new views from the world’s imagery. CoRR, abs/1506.06825, 2015.
|
| 441 |
+
[21] Shubham Toshniwal, Hao Tang, Liang Lu, and Karen Livescu. Multitask learning with low-level auxiliary tasks for encoder-decoder based speech recognition. In 18th Annual Conference of the International Speech Communication Association, pages 3532–3536, 2017.
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| 442 |
+
[22] Timothy M. Hospedales, Antreas Antoniou, Paul Micaelli, and Amos J. Storkey. Meta-learning in neural networks: A survey. CoRR, abs/2004.05439, 2020.
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| 443 |
+
[23] Dong-Hyun Lee. Pseudo-label: The simple and efficient semi-supervised learning method for deep neural networks. In Workshop on Challenges in Representation Learning, volume 3, 2013.
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| 444 |
+
[24] Hieu Pham, Qizhe Xie, Zihang Dai, and Quoc V. Le. Meta pseudo labels. CoRR, abs/2003.10580, 2020.
|
| 445 |
+
[25] Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In Proceedings of the 34th International Conference on Machine Learning, volume 70, pages 1126–1135, 2017.
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| 446 |
+
[26] Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina EliassiRad. Collective classification in network data. AI Magazine, 29(3):93, September 2008.
|
| 447 |
+
[27] Felix Wu, Amauri H. Souza Jr., Tianyi Zhang, Christopher Fifty, Tao Yu, and Kilian Q. Weinberger. Simplifying graph convolutional networks. In Proceedings of the 36th International Conference on Machine Learning, volume 97, pages 6861–6871, 2019.
|
| 448 |
+
[28] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In 3rd International Conference on Learning Representations, 2015.
|
| 449 |
+
[29] Matthias Fey and Jan Eric Lenssen. Fast graph representation learning with pytorch geometric. CoRR, abs/1903.02428, 2019.
|
| 450 |
+
[30] Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(Nov):2579–2605, 2008.
|
| 451 |
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|
| 452 |
+
# Checklist
|
| 453 |
+
|
| 454 |
+
1. For all authors...
|
| 455 |
+
|
| 456 |
+
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
|
| 457 |
+
(b) Did you describe the limitations of your work? [Yes]
|
| 458 |
+
(c) Did you discuss any potential negative societal impacts of your work? [No]
|
| 459 |
+
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
|
| 460 |
+
|
| 461 |
+
2. If you are including theoretical results...
|
| 462 |
+
|
| 463 |
+
(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
|
| 464 |
+
|
| 465 |
+
3. If you ran experiments...
|
| 466 |
+
|
| 467 |
+
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The codes are included in the supplemental material and the datasets can be downlowded by the codes automaticly.
|
| 468 |
+
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] They are included in the supplemental material.
|
| 469 |
+
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Section 4.1.
|
| 470 |
+
|
| 471 |
+
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 4.1.
|
| 472 |
+
|
| 473 |
+
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
|
| 474 |
+
|
| 475 |
+
(a) If your work uses existing assets, did you cite the creators? [Yes] See Section 4.1.
|
| 476 |
+
(b) Did you mention the license of the assets? [No]
|
| 477 |
+
(c) Did you include any new assets either in the supplemental material or as a URL? [No]
|
| 478 |
+
(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No]
|
| 479 |
+
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No]
|
| 480 |
+
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| 481 |
+
5. If you used crowdsourcing or conducted research with human subjects...
|
| 482 |
+
|
| 483 |
+
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
|
| 484 |
+
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
|
| 485 |
+
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
|
parse/train/9QffERDO_rJ/9QffERDO_rJ_content_list.json
ADDED
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[
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{
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"type": "text",
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"text": "Auxiliary Learning Induced Graph Convolutional Networks ",
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"text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ",
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"bbox": [
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"text": "Abstract ",
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"text": "1 Graph convolutional networks (GCNs) have recently achieved great success in \n2 many applications. However they suffer from an incomplete annotation problem \n3 for complex graph-structured data. In this paper, we introduce a novel auxiliary \n4 learning method for GCNs in a multi-task fashion, which can efficiently enrich the \n5 data annotations. Specifically, both link prediction and label generation are used as \n6 two auxiliary tasks to complement the primary task of node classification. These \n7 two auxiliary tasks are jointly trained with the primary node classification task \n8 via a graph meta-learning strategy. The experimental results demonstrate that the \n9 proposed method consistently and significantly outperforms existing methods and \n10 achieves state-of-the-art results on several benchmark citation network datasets. ",
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"type": "text",
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"text": "11 1 Introduction ",
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| 51 |
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"text_level": 1,
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"text": "12 Graph-structured data is ubiquitous in real-world applications. However, general deep learning \n13 methods, such as convolutional neural networks (CNNs), cannot adapt to graph-structured data \n14 directly, because the nodes in a graph have different numbers of neighbors, which often lose the \n15 ranking information. To handle the graph data effectively, graph convolutional networks (GCNs) \n16 have recently been proposed and used in many applications such as biomolecular prediction [1] and \n17 recommendation systems [2]. \n18 Previous methods focus on designing models that can extract information from both the graph \n19 topology and node features. Specifically, existing GCN methods typically design different propagation \n20 strategies [3, 4, 5] for each network layer and stack more network layers [6, 7, 8] to derive larger \n21 receptive fields. However, the neighborhood aggregation is essentially a type of Laplacian smoothing \n22 and stacking too many layers may result in over-smoothing [9]. These drawbacks of existing methods \n23 limit further performance enhancement. \n24 In this paper, we try to explore the bottleneck of node classification from another point of view, i.e., \n25 from the training data itself. As shown in Fig. 1, graph-structured data has different properties from \n26 grid-like data such as an image. The most obvious difference is that the nodes in a graph are connected \n27 by edges. This causes two main issues whien it comes to annotating graph-structured data, resulting in \n28 that existing methods cannot fully leverage the graph-structured information. First, the edges in most \n29 graph data for semi-supervised node classification are unweighted. This arbitrary edge indication \n30 setting cannot effectively reflect the detailed graph structures. Besides, graph-structured data may be \n31 contaminated with noisy edges. These noisy edges cannot represent the true pairwise relationships \n32 between nodes. Second, using one-hot labels to train a graph-based model is inappropriate. One-hot \n33 labels are widely used in various machine learning tasks, assigning a training sample to a single class. \n34 However, nodes in a graph are connected; even nodes with different classes may have relations. In \n35 this scenario, it is more suitable to use soft labels to assign a node to multiple classes, with different \n36 probabilities indicating which class the node possibly belongs to. \n37 To fully leverage the graph-structured \n38 information for enhancing the node \n39 classification performance of GCNs, \n40 we introduce an auxiliary learning \n41 scheme to the GCN framework in \n42 a multi-task fashion. To this end, \n43 we add two auxiliary tasks to enrich \n44 the topology information of a graph \n45 by softening the node labels and re \n46 weighting the edges. Experimental \n47 results show that our model achieves \n48 state-of-the-art node classification per \n49 formance on several benchmark cita \n50 tion network datasets. Our contribu \n51 tions are as follows: \n52 (1) We propose two auxiliary tasks to capture more accurate graph information and enhance the \n53 model performance. The auxiliary link prediction task ensures that the model captures more graph \n54 topology information and generates probabilistic edges. The auxiliary label generation task softens \n55 the one-hot labels and generates pseudo-labels for unlabeled nodes. \n56 (2) The reconstructed edges and pseudo sudo labels derived via the two auxiliary tasks are iteratively \n57 updated with a node classifier of the primary task based on a meta auxiliary learning strategy, resulting \n8 in state-of-the-art node classification performance. ",
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"image_caption": [
|
| 108 |
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"Figure 1: The difference between image data and graphstructured data. The nodes in graph-structured data are connected. Further, in most graph data for the node classification task, edges are unweighted and noisy. "
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"type": "text",
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"text": "59 2 Related Work ",
|
| 144 |
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"type": "text",
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| 155 |
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"text": "Over the past few years, GCNs have achieved significant breakthroughs in graph data representation. \nGenerally, existing GCNs can be divided into spectral-based methods and spatial-based methods. ",
|
| 156 |
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|
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"text": "62 The spectral-based methods use graph spectral theory to define the graph convolutional operation \n63 in a graph Fourier domain. Spectral CNN [10] follows these mathematical foundations, assuming \n64 that a convolutional filter is a set of learnable parameters. To reduce computational complexity, \n65 ChebNet [11] approximates a graph convolutional filter as Chebyshev polynomials of the eigenvalues. \n66 GCN [12] introduces a first-order approximation of ChebNet and proposes a renormalization trick to \n67 alleviate numerical instabilities and exploding/vanishing gradients. DualGCN [13] introduces a dual \n68 GCN architecture with two graph convolutional layers in parallel to encode both local and global \n69 structural information. \n70 The spatial-based methods define feature aggregation in the spatial domain directly, which is more \n71 efficient, general, and flexible [14]. The key challenge for these spatial-based methods is to apply \n72 the convolution operation for different-sized neighborhoods, while at the same time maintaining the \n73 weight sharing property. Neural network for graphs (NN4G) [15] is the first spatial-based method, \n74 applying the graph convolutional operation in the spatial space. Diffusion convolutional neural \n75 networks (DCNN) [16] consider graph convolutions as diffusion processes to efficiently learn features \n76 that are invariant under isomorphism. Message passing neural networks (MPNN) [17] model graph \n77 convolution as a message passing process among the nodes. The graph attention network (GAT) \n78 [3] introduce masked self-attentional layers to assign different weights to adjacent nodes, leading to \n79 learnable filter weights. The mixture model network (MoNet) [18] introduces pseudo-coordinates to \n80 assign different weights to the neighbors of each node. To achieve weight sharing across different \n81 nodes, some spatial-based models attempt to rank a node’s neighbors via certain criteria or metrics, \n82 which transforms the graph-structured data into grid data for further processing. The large-scale \n83 graph convolutional network (LGCN) [19] ranks a node’s neighbors via the node feature values. Then, \n84 multiple 1D convolutional layers are stacked for feature aggregation. Approximate personalized \n85 propagation of neural predictions (APPNP) [4] takes the personalized PageRank algorithm as the \n86 model propagation method to avoid over-smoothing when stacking more layers or increasing the size \n87 of the neighborhood. \n88 Multi-task learning is designed to simultaneously learn a set of related but different tasks for ensuring \n89 that a learning model can derive the best performance across all tasks. Different from multi-task \n90 learning, auxiliary learning is only concerned with model performance on the primary task. For \n91 instance, Deepstereo [20] leverages auxiliary learning to predict the relative poses of multiple cameras \n92 for unsupervised monocular depth estimation. To improve the performance of conversational speech \n93 recognition, auxiliary learning [21] is applied to low-level representations. Compared to the common \n94 learning scheme, meta auxiliary learning can enhance learning performance. For instance, MAXL \n95 [22] adopts meta-learning to automatically generate the auxiliary task labels. Pseudo Label [23] is \n96 a semi-supervised learning method, where a deep neural network is trained using both labeled and \n97 unlabeled data. For unlabeled data, the model picks up the class that has the maximum predicted \n98 probability as the true label to train itself. MPL [24] extends the Pseudo Label [23] via a meta-learning \n99 strategy, where the pseudo-labels are not generated by itself, but by a teacher network. \n100 The existing GCNs are designed for a single task, where the properties of graph-structured data \n101 are not fully explored. We introduce the auxiliary learning scheme to leverage more detailed graph \n102 topology information for enhancing node classification performance. ",
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"Figure 2: Network architecture of the vanilla GCN model. The vanilla GCN contains multiple GCN layers. Each layer captures the graph structure to generate the hidden embeddings from the previous layer (For the first layer, it is the original feature of the node) as input, and obtains the output through the message calculation, aggregation and update step. The last layer uses a softmax function to generate classification probabilities for each node. "
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"text": "3 Method ",
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"text": "3.1 Preliminaries ",
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"text": "05 Given a graph $\\mathcal { G } = \\{ \\nu , \\varepsilon , \\mathbf { X } \\}$ , $\\nu$ is a set of nodes and $\\mathcal { E }$ is a set of the edges connecting the related nodes. 06 $\\mathbf { X } \\in \\mathbb { R } ^ { N \\times d }$ represents the features matrix of the nodes, where $d$ is the dimension of the node 107 features and $N = | \\nu |$ is the number of nodes. ",
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"text": "108 The proposed method adopts the vanilla GCN [12] as the backbone network, taking the graph \n109 adjacency matrix A, labeled training set $\\mathbf { Y } _ { \\mathrm { t r a i n } }$ , and original features $\\mathbf { X }$ of the nodes as inputs to \n110 perform the semi-supervised node classification task. Based on the MPNN framework [17], each \n111 layer of the vanilla GCN is defined in three parts: \n112 (1) Message computation: The message of node $v _ { i }$ and its neighbor node $v _ { j }$ is calculated, where \n113 $j \\in \\mathcal { N } ( i )$ , as: ",
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| 272 |
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"text": "$$\n\\mathbf { m } _ { j } ^ { ( k ) } = \\frac { 1 } { \\sqrt { \\deg ( i ) \\deg ( j ) } } \\mathbf { h } _ { j } ^ { ( k - 1 ) } \\mathbf { W } ^ { ( k ) } , \\ j \\in \\{ i \\} \\bigcup \\mathcal { N } ( i ) .\n$$",
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"text": "Here, 114 $\\deg ( i )$ is the degree of node $v _ { i }$ and $\\mathbf { W } ^ { ( k ) } \\in \\mathbb { R } ^ { c _ { k - 1 } \\times c _ { k } }$ are the learnable parameters of the 115 $k ^ { \\mathrm { t h } }$ -layer, where $c _ { k }$ is the size of the hidden embedding. ",
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"text": "116 (2) Aggregation: The messages of node $v _ { i }$ and its neighbors are aggregated by summing them up: ",
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"text": "$$\n\\mathbf { m } _ { \\mathcal { N } ( i ) } ^ { ( k ) } = \\sum _ { j \\in \\{ i \\} \\cup \\mathcal { N } ( i ) } \\mathbf { m } _ { j } ^ { ( k ) } ,\n$$",
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"img_path": "images/5bb1ad25699143eabc00afa5d26076d610a1c489fe13046a73dacd313172fc38.jpg",
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"image_caption": [
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"Figure 3: The overall network architecture. Our method consists of three networks: a backbone network for the primary task and two auxiliary task networks. (1) The backbone network is a vanilla GCN, which predicts the classification results of each node. (2) The first auxiliary task network is a link predictor, which focuses on the link prediction task and generates a probabilistic edge structure as the input of the backbone network. (3) The second auxiliary task network is a label generator, which employs the label generation task to generate the pseudo soft labels for supervising the node classifier. The parts connected by the dashed arrow share the same parameters, and the red slash on the arrow indicates a stop-gradient (detach) operation. "
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| 344 |
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],
|
| 345 |
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"image_footnote": [],
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| 346 |
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"type": "text",
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"text": "where m(k)(i)117 denotes the aggregated message. ",
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"text": "18 (3) Feature updating: Finally, the hidden representation is updated with the aggregated message. For \n19 the vanilla GCN, the update function can be considered as applying a non-linear operation to the \n120 aggregated message: ",
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"text": "$$\n\\mathbf { h } _ { i } ^ { k } = \\left\\{ \\begin{array} { l l } { \\mathrm { s o f t m a x } ( \\mathbf { m } _ { \\mathcal { N } ( i ) } ^ { ( k ) } ) , } & { \\mathrm { i f ~ } k = K } \\\\ { \\mathrm { R e L U } ( \\mathbf { m } _ { \\mathcal { N } ( i ) } ^ { ( k ) } ) , } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.\n$$",
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"text": "121 Here, $K$ is the number of model layers. The last layer of the vanilla GCN should output the \n122 classification probability via a softmax function. Otherwise, a ReLU operation is used. ",
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"text": "3.2 Multi-Task Network Architecture ",
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"text": "In this paper, we propose an auxiliary learning induced GCN for semi-supervised node classification (see Fig. 3). To enhance the node classification performance of the backbone vanilla GCN, we design two auxiliary tasks: link prediction and pseudo-label generation. ",
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"text": "127 The first $K - 1$ layers of a $K$ -layer vanilla GCN model can be considered as a feature extractor, while \n128 the last layer can be considered as a classifier. Denoting the feature extractor and node classifier of \n129 the backbone model as $h _ { \\theta _ { 2 } }$ and $f _ { \\theta _ { 1 } }$ , respectively, where $\\theta _ { 1 }$ and $\\theta _ { 2 }$ are the learnable parameters, the \n130 proposed two auxiliary task networks are defined as follows. ",
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"text": "3.2.1 Link Predictor ",
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"text": "132 To enrich the edge information, we design an auxiliary link prediction task to infer the missing edges \n133 and present the probability of edge existence. The link predictor we propose contains a decoder $\\bar { R } ( \\cdot )$ \n134 and the feature extractor $h _ { \\theta _ { 2 } }$ of the backbone model. ",
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"text": "The feature extractor 135 $h _ { \\theta _ { 2 } } ( \\cdot )$ takes reduced adjacency matrix ${ \\bf A } _ { s }$ which corresponds to the sampled 136 edge set $\\mathcal { E } _ { s } \\subset \\mathcal { E }$ , and node features $\\mathbf { X }$ as inputs to generate the hidden embedding: ",
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"text": "$$\n\\begin{array} { r } { { \\bf H } _ { s } = h _ { \\theta _ { 2 } } ( { \\bf X } , { \\bf A } _ { s } ) . } \\end{array}\n$$",
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"text": "137 Then, the decoder computes the similarity between each node based on $\\mathbf { H } _ { s }$ to predict the edge \n138 existence probabilities. We simply use an inner-product calculator with a sigmoid function as the \n139 implementation of decoder $R ( \\cdot )$ . The similarity of two hidden embeddings can be computed as ",
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"text": "$$\nr _ { i j } = R ( \\mathbf { h } _ { i } , \\mathbf { h } _ { j } ) = \\sigma ( \\mathbf { h } _ { i } \\mathbf { h } _ { j } ^ { T } ) ,\n$$",
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"text": "140 where $\\mathbf { h } _ { i }$ and $\\mathbf { h } _ { j }$ denote the hidden embeddings of node $v _ { i }$ and node $v _ { j }$ in $\\mathbf { H } _ { s }$ , respectively. ",
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"type": "text",
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"text": "141 3.2.2 Label Generator ",
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"text": "Although using the one-hot label is inappropriate, it is difficult to obtain soft labels with manual annotations for real graph-structured data. To tackle this issue, we introduce an auxiliary label generation task to generate soft labels that reflect the tendency of different classes each node belongs to. Label generator $g _ { \\varphi } ( { \\bf X } , { \\bf A } )$ is a vanilla GCN, where $\\varphi$ are the learnable parameters and A is the adjacency matrix. The label generation network predicts the label distribution of each node based on the graph structure $\\mathbf { A }$ and the raw features $\\mathbf { X }$ of the nodes, as: ",
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"text": "$$\n\\hat { \\mathbf { Y } } ^ { g } = g _ { \\varphi } ( \\mathbf { X } , \\mathbf { A } ) ,\n$$",
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"text": "where 48 $\\mathbf { Y } ^ { g }$ are the predicted pseudo labels for guiding the training of the backbone network and the 49 label generator. ",
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"text": "3.2.3 Node classifier ",
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"text": "151 The node classifier carries out the primary semi-supervised node classification task in our model. \n152 Compared to the vanilla GCN model, we add a graph reconstruction step at the beginning. The graph \n153 adjacency matrix reconstructed with the hidden embeddings contains richer edge information since \n154 the proposed auxiliary link prediction task can enhance the graph topology capturing ability of the \n155 feature extractor $h _ { \\theta _ { 2 } } ( \\cdot )$ . However, the hidden embeddings derived by the feature extractor change \n156 rapidly in the first few iterations, resulting in a changing reconstructed adjacency matrix. Directly \n157 applying the reconstructed adjacency matrix to the entire backbone network increases the training \n158 instability. Thus, we only apply it as the input of the feature extractor $h _ { \\theta _ { 2 } } ( \\cdot )$ , while the classifier \n159 $f _ { \\theta _ { 1 } } ( \\cdot )$ still adopts the original adjacency matrix as input. \n160 In each training iteration, the feature extractor $h _ { \\theta _ { 2 } } ( \\cdot )$ first generates the hidden embeddings of the \n161 nodes $\\mathbf { H } = h _ { \\theta _ { 2 } } ( \\mathbf { X } , \\mathbf { A } )$ . The decoder uses $\\mathbf { H }$ to reconstruct the adjacency matrix $\\mathbf { A } _ { \\mathrm { r e c o n } } = R ( \\mathbf { H } , \\mathbf { H } )$ . \n162 Then, the feature extractor takes the reconstructed adjacency matrix $\\mathbf { A } _ { \\mathrm { r e c o n } }$ to compute the hidden \n163 embeddings $\\mathbf { H } _ { \\mathrm { r e c o n } } = h _ { \\theta _ { 2 } } ( \\mathbf { X } , \\mathbf { A } _ { \\mathrm { r e c o n } } )$ . \n64 Finally, the computed hidden embeddings $\\mathbf { H } _ { \\mathrm { r e c o n } }$ and the original graph adjacency matrix A are fed \n65 to the classifier to obtain the final classification results: ",
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"text": "$$\n\\hat { \\mathbf { Y } } ^ { f } = f _ { \\theta _ { 1 } } ( \\mathbf { H } _ { \\mathrm { r e c o n } } , \\mathbf { A } ) .\n$$",
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},
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{
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"type": "text",
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"text": "166 3.3 Auxiliary Training Phases ",
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"text": "67 In addition to the backbone network, the proposed method contains two auxiliary task networks. In \n68 the following, we will introduce the objectives for the node classifier $f _ { \\theta _ { 1 } }$ , link predictor $h _ { \\theta _ { 2 } }$ , and \n69 label generator $g _ { \\varphi }$ in order, and then leverage a meta auxiliary learning scheme to train the proposed \n70 multi-task network. ",
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"text": "3.3.1 Training the Node Classifier $f _ { \\theta _ { 1 } }$ ",
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"text": "172 The purpose of the node classifier $f _ { \\theta _ { 1 } } ( \\cdot )$ is to carry out the graph-based semi-supervised node \n173 classification task, which yields the final prediction results. Naturally, it is necessary to use the \n174 classification loss between the predicted result and the real categories of nodes to supervise the \n175 training process. At the same time, to reflect the tendency of the class a node belongs to, the training \n176 should make the prediction labels of the classifier be close to the pseudo soft labels generated by \n177 label generator $g _ { \\varphi } ( \\cdot )$ . \n178 In the $t ^ { \\mathrm { t h } }$ iteration, denoting the pseudo soft labels as $\\hat { \\mathbf { Y } } ^ { g ( t ) } = g _ { \\varphi ^ { ( t ) } } ( \\mathbf { X } , \\mathbf { A } )$ , the real labels used in \n179 training as $\\mathbf { Y } _ { \\mathrm { t r a i n } }$ , and the prediction results of the node classifier of the backbone network $f _ { \\theta _ { 1 } }$ as \n180 $\\hat { \\mathbf { Y } } ^ { f ( t ) } = f _ { \\theta _ { 1 } ^ { ( t ) } } \\big ( h _ { \\theta _ { 2 } ^ { ( t ) } } \\big ( \\mathbf { X } , \\mathbf { A } _ { \\mathrm { r e c o n } } ^ { ( t ) } \\big ) , \\mathbf { A } \\big )$ , the objective for the node classifier $f _ { \\theta _ { 1 } } ( \\cdot )$ is defined as ",
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"text": "",
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"img_path": "images/ae646b9e6510c7d0bfc2b61e4dcfa759719c99634e9793cffaaefd1a9700cac8.jpg",
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"text": "$$\n\\mathcal { L } _ { \\theta _ { 1 } } ^ { ( t ) } = \\mathcal { L } _ { \\mathrm { C E } } ( \\hat { \\mathbf { Y } } _ { \\mathrm { t r a i n } } ^ { f ( t ) } , \\mathbf { Y } _ { \\mathrm { t r a i n } } ) + \\mathcal { L } _ { \\mathrm { M S E } } ( \\hat { \\mathbf { Y } } ^ { f ( t ) } , \\hat { \\mathbf { Y } } ^ { g ( t ) } ) .\n$$",
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"text": "181 The objective contains two parts: the loss on the real training labels, and the loss on the generated \n182 pseudo soft labels. $\\mathcal { L } _ { \\mathrm { C E } }$ denotes the cross-entropy loss and $\\mathcal { L } _ { \\mathrm { M S E } }$ denotes the mean squared loss. \n183 Although this objective can be used to update the learnable parameters $\\theta _ { 2 }$ of the feature extractor \n184 $h _ { \\theta _ { 2 } } ( \\cdot )$ , as it generates the hidden embeddings used for node classification, we only employ it to \n185 supervise the learning of the node parameters $\\theta _ { 1 }$ of the classifier $f _ { \\theta _ { 1 } } ( \\cdot )$ . To avoid adding unnecessary \n186 supervision for generating the reconstructed graph adjacency matrix $\\mathbf { A } _ { \\mathrm { r e c o n } }$ , we add a stop-gradient \n187 operation (detach) which is formulated as follows: ",
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"text": "$$\n{ \\bf A } _ { \\mathrm { r e c o n } } ^ { ( t ) } = { \\tt d e t a c h } ( R ( { \\bf H } , { \\bf H } ) ) .\n$$",
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"text": "3.3.2 Auxiliary Training for the Feature Extractor $h _ { \\theta _ { 2 } }$ ",
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"text": "189 The feature extractor $h _ { \\theta _ { 2 } }$ is shared by the link prediction module and the backbone network. In each \n190 iteration, we first randomly sample a certain percentage of edges from the real edge set $\\mathcal { E }$ to form a \n191 sampled edge set $\\mathcal { E } _ { s } \\subset \\mathcal { E }$ . Denoting the reduced graph adjacency matrix as ${ \\bf A } _ { s }$ , which corresponds to \n192 the sampled edge set $\\mathcal { E } _ { s }$ , and applying a message passing operation with ${ \\bf A } _ { s }$ as ",
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"img_path": "images/5608085ffea398b89c9b164b2c93b5b574e90e579dec55ca848c921466284d19.jpg",
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"text": "$$\n\\begin{array} { r } { \\mathbf { H } _ { s } ^ { ( t ) } = h _ { \\theta _ { 2 } ^ { ( t ) } } ( \\mathbf { X } , \\mathbf { A } _ { s } ^ { ( t ) } ) , } \\end{array}\n$$",
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"text": "193 then the objective for the feature extractor is defined as ",
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"text": "$$\n\\mathcal { L } _ { \\theta _ { 2 } } ^ { ( t ) } = \\mathcal { L } _ { \\mathrm { C E } } ( R ( \\mathbf { H } _ { s } ^ { ( t ) } , \\mathbf { H } _ { s } ^ { ( t ) } ) , \\mathbf { A } ) .\n$$",
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"text": "Here $R ( \\mathbf { H } _ { s } ^ { ( t ) } , \\mathbf { H } _ { s } ^ { ( t ) } )$ represents the correlation between the hidden embeddings of each pair of nodes. ",
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"text": "3.3.3 Auxiliary Training for the Label Generator $g _ { \\varphi }$ ",
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"text": "The label generator $g _ { \\varphi }$ is a vanilla GCN model used to predict the label distribution for each node, which naturally needs to be supervised by a classification loss. In the $t ^ { \\mathrm { t h } }$ iteration, a label generator first generates the prediction results using the original graph adjacency matrix A and the node features $\\mathbf { X }$ , which is formulated as $\\hat { \\mathbf { Y } } ^ { g ( t ) } = g _ { \\varphi ^ { ( t ) } } ( \\mathbf { X } , \\mathbf { A } )$ . Denoting the training labels as $\\mathbf { Y } _ { \\mathrm { t r a i n } }$ , the objective of the label generator $g _ { \\varphi }$ can be formulated as ",
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"text": "$$\n\\mathcal { L } _ { \\varphi } ^ { ( t ) } = \\mathcal { L } _ { \\mathrm { C E } } ( \\hat { \\mathbf { Y } } _ { \\mathrm { t r a i n } } ^ { g ( t ) } , \\mathbf { Y } _ { \\mathrm { t r a i n } } ) .\n$$",
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"text": "201 3.3.4 Meta-Learning Based Training Strategy ",
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"text": "The final node classification results only depend on the primary task, while the performance of the other modules, including the label generator and link predictor are not our ultimate concern. Simply using the classification loss to train the label generator $g _ { \\varphi }$ or using the link prediction loss to train the feature extractor $h _ { \\theta _ { 2 } }$ cannot provide the effects of auxiliary learning. Thus, it is crucial to make the node classifier $f _ { \\theta _ { 1 } }$ perform better after it is trained with the pseudo soft labels while taking the hidden embeddings derived by the feature extractor $h _ { \\theta _ { 2 } }$ as inputs. To this end, we use meta auxiliary learning to update the model parameters. ",
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"text": "209 For the auxiliary label generation task, we consider not only the classification performance of the \n210 label generator $g _ { \\varphi }$ , but also the auxiliary effect on the node classifier. We assume that the classifier \n211 parameters $\\theta _ { 1 }$ are updated with gradient descent based on the pseudo soft labels in the $t ^ { \\mathrm { t h } }$ iteration, ",
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"text": "$$\n\\boldsymbol { \\theta } _ { 1 } ^ { \\prime } = \\boldsymbol { \\theta } _ { 1 } ^ { ( t ) } - \\eta \\nabla _ { \\boldsymbol { \\theta } _ { 1 } } \\mathcal { L } _ { \\mathrm { M S E } } ( \\hat { \\mathbf { Y } } ^ { f ( t ) } , \\hat { \\mathbf { Y } } ^ { g ( t ) } ) .\n$$",
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"text": "212 A direct way to evaluate the auxiliary effect of the pseudo soft labels is to compute the classification loss of the prediction given by the node classifier using the updated parameters 213 $\\theta _ { 1 } ^ { \\prime }$ , as ",
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"text": "$$\n\\begin{array} { r } { \\mathcal { L } ^ { f \\prime } = \\mathcal { L } _ { \\mathrm { C E } } \\big ( f _ { \\theta _ { 1 } ^ { \\prime } } \\big ( h _ { \\theta _ { 2 } ^ { ( t ) } } \\big ( \\mathbf { X } , \\mathbf { A } _ { \\mathrm { r e c o n } } \\big ) , \\mathbf { A } \\big ) _ { \\mathrm { t r a i n } } , \\mathbf { Y } _ { \\mathrm { t r a i n } } \\big ) . } \\end{array}\n$$",
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"text": "214 This loss can quantify how much performance improvement the classifier gains from the two auxiliary \n215 tasks. Because $\\theta _ { 1 } ^ { \\prime }$ is updated with the pseudo soft labels generated by $g _ { \\varphi } ( \\cdot )$ , the loss $\\mathcal { L } ^ { f \\prime }$ is also a \n216 function of $\\varphi$ . This means that the objective could be used to supervise the learning of $\\varphi$ . Note \n217 that $\\nabla _ { \\varphi } \\mathcal { L } ^ { f \\prime }$ requires the gradient of the gradient to be computed [25], which can be considered as a \n218 meta-learning strategy. The final objective of the label generator $g _ { \\varphi }$ with meta-learning is formulated \n219 as ",
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"text": "$$\n\\mathcal { L } _ { \\varphi - \\mathrm { { m e t a } } } ^ { ( t ) } = \\mathcal { L } _ { \\varphi } ^ { ( t ) } + \\mathcal { L } ^ { f \\prime } .\n$$",
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"type": "text",
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"text": "220 Similar to the label generator $g _ { \\varphi } ( \\cdot )$ , the link predictor $h _ { \\theta _ { 2 } } ^ { b }$ should derive the effective node embed \n221 dings to enhance the node classification performance. Since the hidden embeddings used for node ",
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"text": "Algorithm 1 AL-GCN ",
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"text": "Input: Graph adjacency matrix A, the node features $\\mathbf { X }$ , the data labels $\\mathbf { Y } _ { \\mathrm { t r a i n } }$ of a training set. Output: A feature extractor $h _ { \\theta _ { 2 } }$ , a node classifier $f _ { \\theta _ { 1 } }$ ",
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"text": "1: Initialize learnable parameters $\\theta _ { 1 } , \\theta _ { 2 }$ , $\\varphi$ \n2: while not converged do \n3: # node classifier training phase \n4: $\\begin{array} { r l } & { \\hat { \\mathbf { Y } } ^ { g } \\gets g _ { \\varphi } ( \\mathbf { X } , \\mathbf { A } ) } \\\\ & { \\mathbf { H } \\gets h _ { \\theta _ { 2 } } ( \\mathbf { X } , \\mathbf { A } ) } \\\\ & { \\mathbf { A } _ { \\mathrm { r e c o n } } = \\mathsf { d e t a c h } ( R ( \\mathbf { H } , \\mathbf { H } ) ) } \\\\ & { \\mathbf { H } _ { \\mathrm { r e c o n } } \\gets h _ { \\theta _ { 2 } } ( \\mathbf { X } , \\mathbf { A } _ { \\mathrm { r e c o n } } ) } \\\\ & { \\hat { \\mathbf { Y } } ^ { f } \\gets f _ { \\theta _ { 1 } } ( \\mathbf { H } _ { \\mathrm { r e c o n } } , \\mathbf { A } ) } \\\\ & { \\mathcal { L } _ { \\theta _ { 1 } } \\gets \\mathcal { L } _ { \\mathrm { C E } } ( \\hat { \\mathbf { Y } } _ { \\mathrm { t r a i n } } ^ { f } , \\mathbf { Y } _ { \\mathrm { t r a i n } } ) + \\mathcal { L } _ { \\mathrm { M S E } } ( \\hat { \\mathbf { Y } } ^ { f } , \\hat { \\mathbf { Y } } ^ { g } ) } \\\\ & { \\mathbf { \\Delta } _ { \\mathrm { r e } } ^ { \\mathsf { x } } , \\quad \\mathbf { \\Delta } _ { \\mathrm { L } } ^ { \\mathsf { x } } , } \\end{array}$ \n5: \n6: \n7: \n8: \n9: \n10: Update: $\\theta _ { 1 } \\gets \\mathrm { A d a m } ( \\mathcal { L } _ { \\theta _ { 1 } } , \\theta _ { 1 } )$ \n11: # meta-learning preparation \n12: Compute: $-$ \n13: $\\begin{array} { r l } { } & { { } \\mathcal { L } _ { \\mathrm { C E } } ( f ^ { b } ( h _ { \\theta _ { 2 } } ^ { b } ( \\mathbf { X } , \\mathbf { A } _ { \\mathrm { r e c o n } } ) , \\mathbf { A } ) _ { \\mathrm { t r a i n } } , \\mathbf { Y } _ { \\mathrm { t r a i n } } ) } \\end{array}$ \n14: # label generator training phase \n15: $\\begin{array} { r l } & { \\hat { \\mathbf Y } ^ { g } \\gets g _ { \\varphi } ( \\mathbf X , \\mathbf A ) } \\\\ & { \\mathcal { L } _ { \\varphi } \\gets \\mathcal { L } _ { \\mathrm { C E } } ( \\hat { \\mathbf Y } _ { \\mathrm { t r a i n } } ^ { g } , \\mathbf Y _ { \\mathrm { t r a i n } } ) _ { , } + } \\end{array}$ \n16: \n17: Update: $\\varphi \\gets \\mathrm { A d a m } ( \\mathcal { L } _ { \\varphi } , \\varphi )$ \n18: # feature extractor training phase \n19: ${ \\bf A } _ { s } \\gets$ RandomSample(A) \n20: Hs ← hθ (X, As) \n21: $\\mathcal { L } _ { \\boldsymbol { \\theta } _ { 2 } } \\gets \\mathcal { L } _ { \\mathrm { C E } } ( R ( \\mathbf { H } _ { s } , \\mathbf { H } _ { s } ) , \\mathbf { A } ) + $ \n22: Update: $\\theta _ { 2 } \\gets \\mathrm { A d a m } ( \\mathcal { L } _ { \\theta _ { 2 } } , \\theta _ { 2 } )$ \n23: end while \n222 classification are derived from the feature extractor $h _ { \\theta _ { 2 } }$ , the objective defined in Eq. 14 can also be \n223 considered as a function of $\\theta _ { 2 }$ . Thus, the objective of the feature extractor $h _ { \\theta _ { 2 } }$ with meta-learning is \n224 defined as ",
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"text": "$$\n\\mathcal { L } _ { \\theta _ { 2 } \\mathrm { - m e t a } } ^ { ( t ) } = \\mathcal { L } _ { \\theta _ { 2 } } ^ { ( t ) } + \\mathcal { L } ^ { f ^ { \\prime } } .\n$$",
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"text": "225 This objective means that, except for the link prediction task, the feature extractor $h _ { \\theta _ { 2 } }$ should ensure \n226 that the generated hidden embedding enhance the classification accuracy of the node classifier $f _ { \\theta _ { 1 } }$ \n227 updated with the pseudo soft labels. The overall training process is shown in Alg. 1. ",
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"text": "4 Experiments ",
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"text": "4.1 Experimental Settings and Compared Methods ",
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"text": "We demonstrate the classification performance of our method via semi-supervised document classification on three citation network datasets, including Cora, Citeseer, and Pubmed [26], where nodes represent the documents and edges are citation links. Dataset statistics are summarized in Table 1. ",
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"text": "233 The proposed method is a novel GCN, which is leveraged to carry out a node classification task. \n234 We compare our method with several popular graph-based node classification methods including \n235 GCN [12], GAT [3], DualGCN [13], SGC [27], and APPNP [4]. As we use a three-layer GCN as \n236 the backbone in our proposed model, we compare both two-layer and three-layer GCNs, denoted as \n237 GCN2 and GCN3, respectively. ",
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"text": "For the semi-supervised node classification, we use all node features but only 20 labels per class for training and 500 nodes as the validation set. We train the proposed method for a maximum of 400 epochs using Adam [28] with a learning rate of 0.01. We use the well-trained parameters, which achieve the best performance on the validation set during the training phases, to evaluate classification accuracy on a test set of 1,000 labeled examples. We run each method 100 times and compute the average classification accuracy on a single NVIDIA GTX 1080Ti GPU. To implement all the compared methods more conveniently, we use PyG [29] as the graph-based learning framework. ",
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"table_caption": [
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"Table 1: Dataset statistics. "
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"table_body": "<table><tr><td>Dataset</td><td>Nodes</td><td>Edges</td><td>Features</td><td>Classes</td></tr><tr><td>Cora</td><td>2,708</td><td>5,278</td><td>1,433</td><td>7</td></tr><tr><td>CiteSeer</td><td>3,327</td><td>4,552</td><td>3,703</td><td>6</td></tr><tr><td>Cora</td><td>19,717</td><td>44,324</td><td>500</td><td>3</td></tr></table>",
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"table_caption": [
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"Table 2: Classification results on the datasets (bold: best, underline: runner-up). "
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"table_body": "<table><tr><td>Method</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td></tr><tr><td>GAT [3] DualGCN[13]</td><td>82.5 ±0.8%</td><td>71.4±0.7%</td><td>78.4± 0.4%</td></tr><tr><td rowspan=\"3\">SGC [27] APPNP [4]</td><td>83.4 ± 0.5%</td><td>72.6 ± 0.6%</td><td>79.9 ± 0.3%</td></tr><tr><td>81.3 ± 0.7%</td><td>70.9 ± 0.6%</td><td>78.2 ± 0.5%</td></tr><tr><td>83.2 ± 0.4%</td><td>71.6 ± 0.5%</td><td>79.8 ± 0.3%</td></tr><tr><td rowspan=\"2\">GCN2 [12] AL-GCN2 (ours)</td><td>81.5±0.7%</td><td>71.5 ± 0.5%</td><td>79.2 ± 0.4%</td></tr><tr><td>82.3 ± 0.4%</td><td>72.6 ± 0.5%</td><td>79.6± 0.5%</td></tr><tr><td rowspan=\"2\">GCN3[12] AL-GCN3 (ours)</td><td>80.7±1.2%</td><td>68.0±1.4%</td><td>77.7 ± 0.5%</td></tr><tr><td>84.7 ± 0.4%</td><td>72.3± 0.5%</td><td>81.4± 0.6%</td></tr></table>",
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"text": "245 4.2 Experimental Results ",
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"text": "46 In this section, we provide the experimental results of the node classification, an ablation study, and \n47 the visualization of hidden embeddings. More experimental results, such as parameter and model \n48 robustness studies can be found in the supplementary materials. ",
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"text": "We conduct the node classification task on three citation network datasets. As shown in Table 2, our method consistently and significantly enhances the learning performance compared to the other methods. In particular, for the Cora dataset, the proposed method is superior to GCN by $4 . 9 \\%$ Compared to the other methods, our model considers more graph-structured information via auxiliary learning. Thus, it is consistently and significantly superior to the compared methods, achieving state-of-the-art results. ",
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"text": "4.3 Ablation Studies ",
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"text": "4.3.1 On the Auxiliary Learning Modules ",
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"text": "To determine how the link predictor $( P )$ and the pseudo label generator $( G )$ affect the node classification performance, we apply the following two ablation models: (1) Vanilla GCN with the link predictor, termed $\\mathrm { G C N } { + } P$ . (2) Vanilla GCN with the lable generator, termed ${ \\mathrm { G C N } } { + } G$ . We compare ${ \\mathrm { G C N } } { \\mathrm { + } } G$ and $\\mathrm { G C N } { + } P$ with the proposed method, AL-GCN $( \\mathbf { G } \\mathbf { C } \\mathbf { N } { + } \\mathbf { } \\mathbf { } P { + } G )$ , and the original GCN. Table 3 lists the results. As can be seen, the proposed method consistently outperforms $\\mathrm { G C N } { + } P$ and ${ \\mathrm { G C N } } { + } G$ . Specifically, compared to the link predictor, the label generator has more effect on the Citeseer dataset. However, for the Pubmed dataset, the link predictor has much more effect on the learning performance. ",
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"text": "4.3.2 On the Reconstructed Graph Adjacency Matrix ",
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"text": "For the backbone network, a reconstructed graph adjacency matrix via a link prediction task is used as input. To determine how the reconstructed graph adjacency matrix affects the node classification performance, we compare the proposed model with the following two models: (1) A model that takes the original graph adjacency matrix without the reconstructed one as the input of the backbone network, termed w/o-recG; (2) A model that takes only the reconstructed graph adjacency matrix as the input of the backbone network, termed w/o-oriG. ",
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"text": "272 As shown in Table 4, our method consistently outperforms w/o-recG and w/o-oriG. This can be \n273 attributed to the fact that the reconstructed graph adjacency matrix via the link predictor can capture \n274 the detailed topology information of a graph and the fixed original graph adjacency matrix increases \n275 the training stability. ",
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"table_caption": [
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"Table 3: The ablation experiment results. "
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"table_body": "<table><tr><td>Method</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td></tr><tr><td>GCN</td><td>80.7 ± 1.2%</td><td>68.0 ± 1.4%</td><td>77.7 ± 0.5%</td></tr><tr><td>GCN+P</td><td>83.0 ± 0.7%</td><td>70.6 ± 0.8%</td><td>81.3 ± 0.6%</td></tr><tr><td>GCN+G</td><td>83.0 ± 0.6%</td><td>71.7 ± 0.8%</td><td>78.8± 0.6%</td></tr><tr><td>GCN+P+G (AL-GCN)</td><td>84.7 ± 0.4%</td><td>72.3± 0.5%</td><td>81.4 ± 0.6%</td></tr></table>",
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"table_body": "<table><tr><td>Method</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td></tr><tr><td>GCN</td><td>80.7±1.2%</td><td>68.0±1.4%</td><td>77.7 ± 0.5%</td></tr><tr><td>w/o-recG</td><td>84.5 ± 0.5%</td><td>71.9 ± 0.6%</td><td>80.1 ± 0.5%</td></tr><tr><td>W/0-oriG</td><td>84.1 ±0.6%</td><td>71.4 ± 0.9%</td><td>80.6 ± 1.5%</td></tr><tr><td>AL-GCN</td><td>84.7 ±0.4%</td><td>72.3 ± 0.5%</td><td>81.4± 0.6%</td></tr></table>",
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"text": "To determine how the hidden embeddings affect the learning performance, we use the visualization tool t-SNE [30] to observe their distribution. As shown in Fig. 4, the embedding results of GCN and GAT are denser, and the separation of different clusters is not obvious. In contrast, the node distributions learned by our proposed method are more separate, with most of the nodes from the same classes being close to each other, resulting in obvious cluster structures. These experimental results demonstrate that the proposed method can capture more detailed structure information of a graph, including the nodes and edges, resulting in more effective hidden embeddings. ",
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"text": "5 Conclusion ",
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"text": "We have proposed a novel graph convolutional network for semi-supervised node classification. Different from existing methods, the proposed model focuses on enriching the graph data and adopts meta auxiliary learning to enhance the representations of nodes and edges in a graph. To enrich node label information, an auxiliary label generator is used to generate pseudo probabilistic labels. Meanwhile, an auxiliary link predictor is used to generate probabilistic edges to enrich the graph structure information. The enriched node and edge information can iteratively enhance the performance of the node classification task. Experimental results on several benchmark citation datasets show that the proposed model is superior to the existing methods. For future work, we note that real-world data is usually contaminated by noise, which results in a robustness problem for graph learning methods. We plan to extend our model to handle noisy data by designing a more robust learning method for graph-structured data. ",
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"text": "References ",
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"text": "[1] David Duvenaud, Dougal Maclaurin, Jorge Aguilera-Iparraguirre, Rafael Gómez-Bombarelli, Timothy Hirzel, Alán Aspuru-Guzik, and Ryan P. Adams. Convolutional networks on graphs for learning molecular fingerprints. In Advances in Neural Information Processing Systems, pages 2224–2232, 2015. \n[2] Rex Ying, Ruining He, Kaifeng Chen, Pong Eksombatchai, William L. Hamilton, and Jure Leskovec. Graph convolutional neural networks for web-scale recommender systems. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 974–983, 2018. [3] Petar Velickovic, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph attention networks. In 6th International Conference on Learning Representations, 2018. [4] Johannes Klicpera, Aleksandar Bojchevski, and Stephan Günnemann. Predict then propagate: Graph neural networks meet personalized pagerank. In 7th International Conference on Learning Representations, 2019. [5] Xiao Wang, Meiqi Zhu, Deyu Bo, Peng Cui, Chuan Shi, and Jian Pei. AM-GCN: adaptive multichannel graph convolutional networks. In Proceedings of the 26t h ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 1243–1253, 2020. [6] Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Stefanie Jegelka. Representation learning on graphs with jumping knowledge networks. In Proceedings of the 35th International Conference on Machine Learning, volume 80, pages 5449–5458, 2018. [7] Lingxiao Zhao and Leman Akoglu. Pairnorm: Tackling oversmoothing in gnns. In 8th International Conference on Learning Representations, 2020. \n[8] Ming Chen, Zhewei Wei, Zengfeng Huang, Bolin Ding, and Yaliang Li. Simple and deep graph convolutional networks. In Proceedings of the 37th International Conference on Machine Learning, volume 119, pages 1725–1735, 2020. [9] Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into graph convolutional networks for semi-supervised learning. In Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, pages 3538–3545, 2018. \n[10] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. In 2nd International Conference on Learning Representations, 2014. \n[11] Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In Advances in Neural Information Processing Systems, pages 3837–3845, 2016. \n[12] Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In 5th International Conference on Learning Representations, 2017. \n[13] Chenyi Zhuang and Qiang Ma. Dual graph convolutional networks for graph-based semisupervised classification. In Proceedings of the 2018 World Wide Web Conference on World Wide Web, pages 499–508, 2018. \n[14] Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S. Yu. A comprehensive survey on graph neural networks. CoRR, abs/1901.00596, 2019. \n[15] Alessio Micheli. Neural network for graphs: A contextual constructive approach. IEEE Trans. Neural Networks, 20(3):498–511, 2009. \n[16] James Atwood and Don Towsley. Diffusion-convolutional neural networks. In Advances in Neural Information Processing Systems, pages 1993–2001, 2016. \n[17] Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural message passing for quantum chemistry. In Proceedings of the 34th International Conference on Machine Learning, volume 70, pages 1263–1272, 2017. \n[18] Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodolà, Jan Svoboda, and Michael M. Bronstein. Geometric deep learning on graphs and manifolds using mixture model cnns. In 2017 IEEE Conference on Computer Vision and Pattern Recognition, pages 5425–5434, 2017. \n[19] Hongyang Gao, Zhengyang Wang, and Shuiwang Ji. Large-scale learnable graph convolutional networks. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 1416–1424, 2018. \n[20] John Flynn, Ivan Neulander, James Philbin, and Noah Snavely. Deepstereo: Learning to predict new views from the world’s imagery. CoRR, abs/1506.06825, 2015. \n[21] Shubham Toshniwal, Hao Tang, Liang Lu, and Karen Livescu. Multitask learning with low-level auxiliary tasks for encoder-decoder based speech recognition. In 18th Annual Conference of the International Speech Communication Association, pages 3532–3536, 2017. \n[22] Timothy M. Hospedales, Antreas Antoniou, Paul Micaelli, and Amos J. Storkey. Meta-learning in neural networks: A survey. CoRR, abs/2004.05439, 2020. \n[23] Dong-Hyun Lee. Pseudo-label: The simple and efficient semi-supervised learning method for deep neural networks. In Workshop on Challenges in Representation Learning, volume 3, 2013. \n[24] Hieu Pham, Qizhe Xie, Zihang Dai, and Quoc V. Le. Meta pseudo labels. CoRR, abs/2003.10580, 2020. \n[25] Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In Proceedings of the 34th International Conference on Machine Learning, volume 70, pages 1126–1135, 2017. \n[26] Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina EliassiRad. Collective classification in network data. AI Magazine, 29(3):93, September 2008. \n[27] Felix Wu, Amauri H. Souza Jr., Tianyi Zhang, Christopher Fifty, Tao Yu, and Kilian Q. Weinberger. Simplifying graph convolutional networks. In Proceedings of the 36th International Conference on Machine Learning, volume 97, pages 6861–6871, 2019. \n[28] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In 3rd International Conference on Learning Representations, 2015. \n[29] Matthias Fey and Jan Eric Lenssen. Fast graph representation learning with pytorch geometric. CoRR, abs/1903.02428, 2019. \n[30] Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(Nov):2579–2605, 2008. ",
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"text": "1. For all authors... ",
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"text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [Yes] \n(c) Did you discuss any potential negative societal impacts of your work? [No] \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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"text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ",
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"text": "3. If you ran experiments... ",
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| 1 |
+
# ANALYTICAL MOMENT REGULARIZER FOR TRAINING ROBUST NETWORKS
|
| 2 |
+
|
| 3 |
+
Anonymous authors Paper under double-blind review
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
Despite the impressive performance of deep neural networks (DNNs) on numerous learning tasks, they still exhibit uncouth behaviours. One puzzling behaviour is the subtle sensitive reaction of DNNs to various noise attacks. Such a nuisance has strengthened the line of research around developing and training noise-robust networks. In this work, we propose a new training regularizer that aims to minimize the probabilistic expected training loss of a DNN subject to a generic Gaussian input. We provide an efficient and simple approach to approximate such a regularizer for arbitrarily deep networks. This is done by leveraging the analytic expression of the output mean of a shallow neural network, avoiding the need for memory and computation expensive data augmentation. We conduct extensive experiments on LeNet and AlexNet on various datasets including MNIST, CIFAR10, and CIFAR100 to demonstrate the effectiveness of our proposed regularizer. In particular, we show that networks that are trained with the proposed regularizer benefit from a boost in robustness against Gaussian noise to an equivalent amount of performing 3-21 folds of noisy data augmentation. Moreover, we empirically show on several architectures and datasets that improving robustness against Gaussian noise, by using the new regularizer, can improve the overall robustness against 6 other types of attacks by two orders of magnitude.
|
| 8 |
+
|
| 9 |
+
# 1 INTRODUCTION
|
| 10 |
+
|
| 11 |
+
Deep neural networks (DNNs) have emerged as generic models that can be trained to perform impressively well in a variety of learning tasks ranging from object recognition (He et al., 2016) and semantic segmentation (Long et al., 2015) to speech recognition (Hinton et al., 2012) and bioinformatics (Angermueller et al., 2016). Despite their increasing popularity, flexibility, generality, and performance, DNNs have been recently shown to be quite susceptible to small imperceptible input noise (Szegedy et al., 2014; Moosavi-Dezfooli et al., 2016; Goodfellow et al., 2015). Such analysis gives a clear indication that even state-of-the-art DNNs may lack robustness. Consequently, there has been an ever-growing interest in the machine learning community to study this uncanny behaviour. In particular, the work of (Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016) demonstrates that there are systematic approaches to constructing adversarial attacks that result in misclassification errors with high probability. Even more peculiarly, some noise perturbations seem to be doubly agnostic (Moosavi-Dezfooli et al., 2017), i.e. there exist deterministic perturbations that can result in misclassification errors with high probability when applied to different networks, irrespective of the input (denoted network and input agnostic).
|
| 12 |
+
|
| 13 |
+
Understanding this degradation in performance under adversarial attacks is of tremendous importance, especially for real-world DNN deployment, e.g. self-driving cars/drones and equipment for the visually impaired. A standard and popular means to alleviate this nuisance is noisy data augmentation in training, i.e. a DNN is exposed to noisy input images during training so as to bolster its robustness during inference. Several works have demonstrated that DNNs can in fact benefit from such augmentation (Moosavi-Dezfooli et al., 2016; Goodfellow et al., 2015). However, data augmentation in general might not be sufficient for two reasons. (1) Particularly with high-dimensional input noise, the amount of data augmentation necessary to sufficiently capture the noise space will be very large, which will increase training time. (2) Data augmentation with high energy noise can negatively impact the performance on noise-free test examples. This can be explained by the fundamental trade-off between accuracy and robustness (Tsipras et al., 2018; Boopathy et al., 2019). It can also arise from the fact that augmentation forces the DNN to have the same prediction for two vastly different versions of the same input, noise-free and a substantially corrupted version.
|
| 14 |
+
|
| 15 |
+
Therefore, in this paper, we propose a new regularizer for noise-robust networks to circumvent the aforementioned setbacks of data augmentation.
|
| 16 |
+
|
| 17 |
+
A natural objective for training against attacks sampled from a distribution $\mathcal { D }$ , that bypasses the need for data augmentation, is the expected loss under this distribution. Since a closed-form expression is generally difficult to obtain or an approximate surrogate is expensive to evaluate (Monte Carlo estimates), we propose instead a closely related objective that is the loss of the expected predictions of the network under $\mathcal { D }$ -distributed adversarial noise. Since it has been shown that Gaussian noise can be adversarial (Bibi et al., 2018) and that such noise is widely studied in applications such as image processing, we restrict the focus in this paper to the case where $\mathcal { D }$ is Gaussian. While this may seem to be too restrictive, we later show that improving the robustness of networks against Gaussian attacks also improves the robustness against a family of other types of attacks. However, even under such an assumption, only a memory and computationally expensive (expensive due to two-stage network linearization), closed-form approximate surrogate for network expected predictions exists (Bibi et al., 2018).
|
| 18 |
+
|
| 19 |
+
Contributions. (i) We formalize a new regularizer that is a function of the probabilistic first moment of the output of a DNN to train robust DNNs against noise sampled from distribution $\mathcal { D }$ . (ii) Under the special choice of Gaussian attacks, i.e. $\mathcal { D }$ is Gaussian, we show how the first moment expression can be evaluated very efficiently during training for an arbitrary deep DNN by bypassing the need to perform memory and computationally expensive two-stage linearization. (iii) Extensive experiments using LeNet (LeCun et al., 1999) and AlexNet (Krizhevsky et al., 2012) architectures on MNIST (LeCun, 1998), CIFAR10, and CIFAR100 (Krizhevsky & Hinton, 2009) datasets demonstrate that a substantial enhancement in robustness can be achieved when using our regularizer in training. In fact, in the majority of the experiments, the improvement is better than training on the same dataset, augmented with 3 to 21 times Gaussian noisy data. Interestingly, the results suggest an excellent trade-off between accuracy and robustness. Moreover, we show that networks that are trained to be robust against Gaussian attacks using our proposed regularizer enjoy orders of magnitude boost in robustness against a family of other types of attacks.
|
| 20 |
+
|
| 21 |
+
# 2 RELATED WORK
|
| 22 |
+
|
| 23 |
+
Despite the impressive performance of DNNs on various tasks, they have been shown to be very sensitive to certain types of noise, commonly referred to as adversarial examples, particularly in the recognition task (Moosavi-Dezfooli et al., 2016; Goodfellow et al., 2015). Adversarial examples can be viewed as small imperceptible noise that, once added to the input of a DNN, its performance is severely degraded. This finding has incited interest in studying/measuring the robustness of DNNs.
|
| 24 |
+
|
| 25 |
+
The literature is rich with work that aims to unify and understand the notion of network robustness. For instance, Szegedy et al. (2014) suggested a spectral stability analysis for a wide class of DNNs by measuring the Lipschitz constant of the affine transformation describing a fully-connected or a convolutional layer. This result was extended to compute an upper bound for a composition of layers, i.e. a DNN. However, this measure sets an upper bound on the robustness over the entire input domain and does not take into account the noise distribution. Later, Fawzi et al. (2017a) defined robustness as the mean support of the minimum adversarial perturbation, which is now the most common definition for robustness. Not only was robustness studied against adversarial perturbations but also against geometric transformations to the input. Fawzi et al. (2018) emphasized the independence of the robustness measure to the ground truth class labels and that it should only depend on the classifier and the dataset distribution. Subsequently, two different metrics to measure DNN robustness were proposed: one for general adversarial attacks and another for noise sampled from uniform distribution. Recently, Gilmer et al. (2018) showed the trade-off between robustness and test error from a theoretical point of view on a simple classification problem with hyperspheres.
|
| 26 |
+
|
| 27 |
+
On the other hand, and based on various robustness analyses, several works proposed various approaches in building networks that are robust against noise sampled from well known distributions and against generic adversarial attacks. For instance, Grosse et al. (2017) proposed a model that was trained to classify adversarial examples with statistical hypothesis testing on the distribution of the dataset. Another approach is to perform statistical analysis on the latent feature space instead (Li & Li, 2017; Feinman et al., 2017), or train a DNN that rejects adversarial attacks (Lu et al., 2017). Moreover, the geometry of the decision boundaries of DNN classifiers was studied by Fawzi et al. (2017b) to infer a simple curvature test for this purpose. Using this method, one can restore the original label and classify the input correctly. Restoring the original input using defense mechanisms, which can only detect adversarial examples, can be done by denoising (ridding it from its adversarial nature) so long as the noise perturbation is well-known and modeled apriori (Zhu et al., 2016). A fresh approach to robustness was proposed by Zantedeschi et al. (2017), where they showed that using bounded ReLUs (if augmented with Gaussian noise) to limit the output range can improve robustness. A different work proposed to distill the learned knowledge from a deep model to retrain a similar model architecture as a means to improving robustness (Papernot et al., 2016). This training approach is one of many adversarial training strategies for robustness Makhzani et al. (2016). More closely to our work is (Cisse et al., 2017), where a new training regularizer was proposed for a large family of DNNs. The proposed regularizer softly enforces that the upper bound of the Lipshitz constant of the output of the network to be less than or equal to one. Moreover and very recently, the work of Bibi et al. (2018) has derived analytic expressions for the output mean and covariance of networks in the form of (Affine, ReLU, Affine) under a generic Gaussian input. This work also demonstrates how a (memory and computation expensive) two-stage linearization can be employed to locally approximate a deep network with a two layer one, thus enabling the application of the derived expressions on the approximated shallower network.
|
| 28 |
+
|
| 29 |
+
|
| 30 |
+
Figure 1: Overview of the proposed graph for training Gaussian robust networks. The yellow block corresponds to an arbitrary network $\Phi ( . , \theta )$ viewed as the composition of two subnetworks separated by a ReLU. The stream on the bottom computes the output mean $\mu _ { 4 }$ of the network $\Phi ( : , \theta )$ assuming that (i) the noise input distribution is independent Gaussian with variances $\sigma _ { x } ^ { 2 }$ , and (ii) $\Omega ( . ~ : ~ \theta _ { 2 } )$ is approximated by a linear function. This evaluation for the output mean is efficient as it only requires an extra forward pass (bottom stream), as opposed to other methods that employ computationally and memory intensive network linearizations or data augmentation.
|
| 31 |
+
|
| 32 |
+
Most prior work requires data augmentation, training new architectures that distill knowledge, or detect adversaries a priori, resulting in expensive training routines that may be ineffective in the presence of several input noise types. To this end, we address these limitations through our new regularizer that aims to fundamentally tackle Gaussian input noise without data augmentation and, as a consequence, improves overall robustness against other types of attacks.
|
| 33 |
+
|
| 34 |
+
# 3 METHODOLOGY
|
| 35 |
+
|
| 36 |
+
Background on Network Moments. Networks with a single hidden layer of the form (Affine, ReLU, Affine) can be written in the functional form $\mathbf { g } ( \mathbf { x } ) = \bar { \mathbf { B } } \mathrm { m a x } \left( \mathbf { A } \mathbf { x } + \mathbf { \bar { c } } _ { 1 } , \mathbf { 0 } _ { p } \right) + \mathbf { c } _ { 2 }$ . The max(.) is an element-wise operator, $\textbf { A } \in \mathbb { R } ^ { p \times n }$ , and $\textbf { B } \in \mathbb { R } ^ { d \times p }$ . Thus, $\mathbf { g } : \mathbb { R } ^ { n } \mathbb { R } ^ { d }$ . Given that $\mathbf { x } \sim \mathcal { N } \left( \mu _ { x } , \Sigma _ { x } \right)$ , Bibi et al. (2018) showed that:
|
| 37 |
+
|
| 38 |
+
$$
|
| 39 |
+
\begin{array} { r } { \mathbf { T h e o r e m 1 textit { T h e f i r s t m o m e n t } } o f g ( \mathbf { x } ) \ i s \mathbb { E } [ \mathbf { g } ( \mathbf { x } ) ] = \mathbf { B } \left( \mu _ { 2 } \odot \Phi \left( \frac { \mu _ { 2 } } { \sigma _ { 2 } } \right) + \sigma _ { 2 } \odot \varphi \left( \frac { \mu _ { 2 } } { \sigma _ { 2 } } \right) \right) + \mathbf { c } _ { 2 } . } \end{array}
|
| 40 |
+
$$
|
| 41 |
+
|
| 42 |
+
Note that $\mu _ { 2 } = \mathbf { A } \mu _ { x } + \mathbf { c } _ { 1 }$ , $\sigma _ { 2 } = \sqrt { \mathrm { d i a g } \left( \Sigma _ { 2 } \right) } .$ , $\Sigma _ { 2 } = \mathbf { A } \Sigma _ { x } \mathbf { A } ^ { \top }$ , $\Phi$ and $\varphi$ are the standard Gaussian cumulative (CDF) and density (PDF) functions, respectively. The vector multiplication and division are element-wise operations. Lastly, $\mathrm { d i a g ( . ) }$ extracts the diagonal elements of a matrix into a vector. For ease of notation, we let $\begin{array} { r } { \mu _ { 3 } = \dot { T } ( \mu _ { 2 } , \dot { \sigma } _ { 2 } ) = ( \mu _ { 2 } \odot \Phi ( \frac { \mu _ { 2 } } { \sigma _ { 2 } } \bar { ) } + \sigma _ { 2 } \odot \varphi ( \frac { \mu _ { 2 } } { \sigma _ { 2 } } ) ) } \end{array}$ .
|
| 43 |
+
|
| 44 |
+
To extend the results of Theorem (1) to deeper models, a two-stage linearization was proposed in Bibi et al. (2018), where $( \mathbf { A } , \mathbf { B } )$ and $( \mathbf { c } _ { 1 } , \mathbf { c } _ { 2 } )$ are taken to be the Jacobians and biases of the first order Taylor approximation to the two network functions around a ReLU layer in a DNN. Refer to Bibi et al. (2018) for more details about this expression and the proposed linearization.
|
| 45 |
+
|
| 46 |
+
Proposed Robust Training Regularizer. To propose an alternative to noisy data augmentation to address its drawbacks, one has to realize that this augmentation strategy aims to minimize the expected training loss of a DNN when subjected to noisy input distribution $\mathcal { D }$ through sampling. In fact, it minimizes an empirical loss that approximates this expected loss when enough samples are present during training. When sampling is insufficient (a drawback of data augmentation in highdimensions), this approximation is too loose and robustness can suffer. However, if we have access to an analytic expression for the expected loss, expensive data augmentation can be averted. This is the key motivation of the paper. Mathematically, the training loss can be modeled as
|
| 47 |
+
|
| 48 |
+
$$
|
| 49 |
+
\operatorname* { m i n } _ { \theta } \ \sum _ { i = 1 } ^ { N } \Big ( \ell \left( \Phi ( \mathbf { x } _ { i } ; \theta ) , y _ { i } \right) + \alpha \mathbb { E } _ { \mathbf { n } \sim \mathcal { D } } \left[ \ell \left( \Phi ( \mathbf { x } _ { i } + \mathbf { n } ; \theta ) , y _ { i } \right) \right] \Big ) .
|
| 50 |
+
$$
|
| 51 |
+
|
| 52 |
+
Here, $\Phi : \mathbb { R } ^ { n } \mathbb { R } ^ { d }$ is any arbitrary network with parameters $\theta , \ell$ is the loss function, $\{ ( \mathbf { x } _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ are the noise-free data-label training pairs, and $\alpha \geq 0$ is a trade off parameter. While the first term in Equation 1 is the standard empirical loss commonly used for training, the second term is often replaced with its Monte Carlo estimate through data augmentation. That is, for each training example $\mathbf { x } _ { i }$ , the second term is approximated with an empirical average of $\tilde { N }$ noisy examples of $\mathbf { x } _ { i }$ such that $\begin{array} { r } { \mathbb { E } _ { \mathbf { n } \sim \mathcal { D } } [ \ell \left( \boldsymbol { \Phi } ( \mathbf { x } _ { i } + \mathbf { n } ; \boldsymbol { \theta } ) , y _ { i } \right) ] \approx \frac { 1 } { \tilde { N } } \sum _ { j = 1 } ^ { \tilde { N } } \ell \left( \boldsymbol { \Phi } ( \mathbf { x } _ { i } + \mathbf { n } _ { j } ; \boldsymbol { \theta } ) , y _ { i } \right) . } \end{array}$ . This will increase the size of the dataset by a factor of $\tilde { N }$ , which will in turn increase training complexity. As discussed earlier, network performance on the noise-free examples can also be negatively impacted. Note that obtaining a closed form expression for the second term in Equation 1 for some of the popularly used losses $\ell$ is more complicated than deriving expressions for the output mean of the network $\Phi$ itself, e.g. in Theorem (1). Therefore, we propose to replace this loss with the following surrogate
|
| 53 |
+
|
| 54 |
+
$$
|
| 55 |
+
\operatorname* { m i n } _ { \theta } \ \sum _ { i = 1 } ^ { N } \Big ( \ell \left( \Phi ( \mathbf { x } _ { i } ; \theta ) , y _ { i } \right) + \alpha \ell \left( \mathbb { E } _ { \mathbf { n } \sim \mathcal { D } } \left[ \Phi ( \mathbf { x } _ { i } + \mathbf { n } ; \theta ) \right] , y _ { i } \right) . \Big )
|
| 56 |
+
$$
|
| 57 |
+
|
| 58 |
+
Because of Jensen’s inequality, Equation 2 is a lower bound to Eq Equation 1 when $\ell$ is convex, which is the case for most popular losses including $\ell _ { 2 }$ -loss and cross-entropy loss. The proposed second term in Equation 2 encourages that the output mean of the network $\Phi$ of every noisy example $( \mathbf { x } _ { i } + \mathbf { n } )$ matches the correct class label $y _ { i }$ . This regularizer will stimulate a separation among the output mean of the classes if the training data is subjected to noise sampled from $\mathcal { D }$ . Having access to an analytic expression for these means will prompt a simple inexpensive training, where the actual size of the training set is unaffected and augmentation is avoided. This form of regularization is proposed to replace data augmentation.
|
| 59 |
+
|
| 60 |
+
While a closed-form expression for the second term of Equation 2 might be infeasible for a general network $\Phi ( . )$ , an expensive approximation can be attained. In particular, Theorem Equation 1 provides an analytic expression to evaluate the second term in Equation 2, for when $\mathcal { D }$ is Gaussian and when the network is approximated by a two-stage linearization procedure as $\Phi ( \mathbf { x } ) \approx \mathbf { B } \mathrm { m a x } \left( \mathbf { A } \mathbf { x } + \mathbf { c } _ { 1 } , \mathbf { 0 } _ { p } \right) + \mathbf { c } _ { 2 }$ . However, it is not clear how to utilize such a result to regularize networks during training with Equation 2 as a loss. This is primarily due to the computationally expensive and memory intensive network linearization proposed in Bibi et al. (2018). Specifically, the linearization parameters $\left( \mathbf { A } , \mathbf { B } , \mathbf { c } _ { 1 } , \mathbf { c } _ { 2 } \right)$ are a function of the network parameters, $\theta$ , which are updated with every gradient descent step on Equation 2; thus, two-stage linearization has to be performed in every $\theta$ update step, which is infeasible.
|
| 61 |
+
|
| 62 |
+
On an Efficient Approximation to Equation 2. The loss in Equation 2 proposes a generic approach to train robust arbitrary networks against noise sampled from an arbitrary distribution $\mathcal { D }$ . Since the problem in its general setting is too broad for detailed analysis, we restrict the scope of this work to the class of networks, which are most popularly used and parameterized by $\theta$ , $\bar { \Phi ( . ; \theta ) } : \mathbb { R } ^ { n } \to \mathbb { R } ^ { d }$ with ReLUs as nonlinear activations. Moreover, since random Gaussian noise was shown to exhibit an adversarial nature Bibi et al. (2018); Rauber et al. (2017); Franceschi et al. (2018), and it is one of the most well studied noise models for the useful properties it exhibits, we restrict $\mathcal { D }$ to the case of Gaussian noise. In particular, $\mathcal { D }$ is independent zero-mean Gaussian noise at the input, i.e. $\mathbf { n } \sim \mathcal { D } = \mathcal { N } \left( \mathbf { 0 } , \Sigma _ { x } = \mathrm { D i a g } \left( \sigma _ { x } ^ { 2 } \right) \right)$ , where $\sigma _ { x } ^ { 2 } \in \mathbb { R } ^ { n }$ is a vector of variances and $\mathrm { D i a g ( . ) }$ reshapes the vector elements into a diagonal matrix. Generally, it is still difficult to compute the second term in Equation 2 under Gaussian noise for arbitrary networks. However, if we have access to an inexpensive approximation of the network, avoiding the computationally and memory expensive network linearization in Bibi et al. (2018), an approximation to the second term in Equation 2 can be used for efficient robust training directly on $\theta$ .
|
| 63 |
+
|
| 64 |
+
Consider the $l ^ { \mathrm { t h } }$ ReLU layer in $\Phi ( . ; \theta )$ . the network can be expressed as $\begin{array} { r l } { \Phi ( . ; \theta ) } & { { } = } \end{array}$ $\Omega ( \mathrm { R e L U } _ { l } ( \Upsilon ( . , \theta _ { 1 } ) ) ; \theta _ { 2 } )$ . Note that the parameters of the overall network $\Phi ( . ; \theta )$ is the union of the parameters of the two subnetworks $\Upsilon ( . ; \theta _ { 1 } )$ and $\Omega ( . ; \theta _ { 2 } )$ , i.e. $\theta = \theta _ { 1 } \cup \theta _ { 2 }$ . Throughout this work and to simplify the analysis, we set $l = 1$ . With such a choice of $l$ , the first subnetwork $\Upsilon ( . , \theta _ { 1 } )$ is affine with $\bar { \theta _ { 1 } } = \{ \mathbf { A } , \mathbf { c } _ { 1 } \}$ . However, the second subnetwork $\Omega ( . , \theta _ { 2 } )$ is not linear in general, and thus, one can linearize $\Omega ( . , \theta _ { 2 } )$ at $\mathbb { E } _ { \mathbf { n } \sim \mathcal { N } ( \mathbf { 0 } , \Sigma _ { x } ) } \left[ \mathrm { R e L U } _ { 1 } \left( \Upsilon \left( \mathbf { x } _ { i } + \mathbf { n } ; \theta _ { 1 } \right) \right) \right] = T ( \mu _ { 2 } , \sigma _ { 2 } ) = \mu _ { 3 }$ . Note that $\mu _ { 3 }$ is the output mean after the ReLU and $\mu _ { 2 } = \mathbf { A } \mathbf { x } _ { i } + \mathbf { c } _ { 1 }$ , since $\Upsilon ( \mathbf { x } _ { i } + \mathbf { n } ; \theta _ { 1 } ) = \mathbf { A } \left( \mathbf { x } _ { i } + \mathbf { n } \right) + \mathbf { c } _ { 1 }$ . Both $T ( . , . )$ and $\sigma _ { 2 }$ are defined in Equation 1. Thus, linearizing $\Omega$ at $\mu _ { 3 }$ with linearization parameters $\mathbf { ( B , c _ { 2 } ) }$ being the Jacobian of $\Omega$ and ${ \bf c } _ { 2 } = \Omega ( \mu _ { 3 } , \theta _ { 2 } ) - { \bf B } \mu _ { 3 }$ , we have that, for any point $\mathbf { v } _ { i }$ close to $\mu _ { 3 }$ : $\Omega ( \mathbf { v } _ { i } , \mathbf { \bar { \theta } } _ { 2 } ) \approx \mathbf { B } \mathbf { v } _ { i } + \mathbf { c } _ { 2 }$ . While computing $\mathbf { ( B , c _ { 2 } ) }$ through linearization is generally very expensive, computing the approximation to Equation 2 requires explicit access to neither $\mathbf { B }$ nor $\mathbf { c } _ { 2 }$ . Note that this second term for $l = 1$ is given as:
|
| 65 |
+
|
| 66 |
+
$$
|
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+
\begin{array} { r l } & { \quad \ell \left( \mathbb { E } _ { \mathbf { n } \sim \mathcal { N } ( \mathbf { 0 } , \Sigma _ { x } ) } [ \Phi ( \mathbf { x } _ { i } + \mathbf { n } ; \boldsymbol { \theta } ) ] , y _ { i } \right) = \ell \left( \mathbb { E } _ { \mathbf { z } _ { i } \sim \mathcal { N } ( \mathbf { x } _ { i } , \Sigma _ { x } ) } [ \Omega \left( \mathrm { R e L U } _ { 1 } \left( \Upsilon ( \mathbf { z } _ { i } ; \boldsymbol { \theta } _ { 1 } ) \right) ; \boldsymbol { \theta } _ { 2 } \right) ] , y _ { i } \right) } \\ & { = \ell \left( \mathbb { E } _ { \mathbf { z } _ { i } \sim \mathcal { N } ( \mathbf { x } _ { i } , \Sigma _ { x } ) } [ \Omega \left( \mathrm { R e L U } _ { 1 } \left( \mathbf { A } \mathbf { z } _ { i } + \mathbf { c } _ { 1 } \right) ; \boldsymbol { \theta } _ { 2 } \right) ] , y _ { i } \right) } \\ & { \approx \ell \left( \mathbb { E } _ { \mathbf { z } _ { i } \sim \mathcal { N } ( \mathbf { x } _ { i } , \Sigma _ { x } ) } [ \mathbf { B } \left( \mathrm { R e L U } _ { 1 } \left( \mathbf { A } \mathbf { z } _ { i } + \mathbf { c } _ { 1 } \right) \right) + \mathbf { c } _ { 2 } ] , y _ { i } \right) } \\ & { = \ell \left( \mathbf { B } \mu _ { 3 } + \mathbf { c } _ { 2 } , y _ { i } \right) = \ell \left( \Omega ( \mu _ { 3 } , \boldsymbol { \theta } _ { 2 } ) , y _ { i } \right) . } \end{array}
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$$
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The approximation follows from the assumption that the input to the second subnetwork $\Omega ( . ; \theta _ { 2 } )$ , i.e. $\mathbf { v } _ { i } = \mathrm { R e L U } _ { 1 } \left( \mathbf { A } \mathbf { z } _ { i } + \mathbf { c } _ { 1 } \right) )$ , is close to the point of linearization $\mu _ { 3 }$ such that $\Omega ( \mathbf { v } _ { i } ; \theta _ { 2 } ) \approx \mathbf { B } \mathbf { v } _ { i } + \mathbf { c } _ { 2 }$ . Or simply, that the input to $\Omega$ is close to the mean inputs, i.e. $\mu _ { 3 }$ , to $\Omega$ under Gaussian noise. The penultimate equality follows from the linearity of the expectation. As for the last equality, $\mathbf { ( B , c _ { 2 } ) }$ are the linearization parameters of $\Omega$ at $\mu _ { 3 }$ , where $\mathbf { c } _ { 2 } = \mathbf { \bar { \Omega } } ( \mu _ { 3 } , \theta _ { 2 } ) - \mathbf { B } \mu _ { 3 }$ by the first order Taylor approximation. Thus, computing the second term of Equation 2 according to Equation 3 can be simply approximated by a forward pass of $\mu _ { 3 }$ through the second network $\Omega$ . As for computing $\mu _ { 3 } = T ( \mu _ { 2 } , \sigma _ { 2 } )$ , note that $\mu _ { 2 } = \mathbf { A } \mathbf { x } _ { i } + \mathbf { c } _ { 1 }$ in Equation 3, which is equivalent to a forward pass of $\mathbf { x } _ { i }$ through the first subnetwork because $\Upsilon ( . , \theta _ { 1 } )$ is linear with $\theta _ { 1 } = \{ \bar { \bf A } , { \bf c } _ { 1 } \}$ . Moreover, since $\sigma _ { 2 } =$ $\sqrt { \mathrm { d i a g } \left( \mathbf { A } \Sigma _ { x } \mathbf { A } ^ { \top } \right) }$ , we have: $\sigma _ { 2 } = \sqrt { \operatorname { d i a g } \left( \mathbf { A D i a g } \left( \sigma _ { x } ^ { 2 } \right) \mathbf { A } ^ { \top } \right) } = \sqrt { \left( \mathbf { A } \odot \mathbf { A } \right) \sigma _ { x } ^ { 2 } }$ . The expression for $\sigma _ { 2 }$ can be efficiently computed by simply squaring the linear parameters in the first subnetwork and performing a forward pass of the input noise variance $\sigma _ { x } ^ { 2 }$ through $\Upsilon$ without the bias $\mathbf { c } _ { 1 }$ and taking the element-wise square root. Lastly, it is straightforward to compute $T ( \mu _ { 2 } , \sigma _ { 2 } )$ as it is an elementwise function in Equation 1. The overall computational graph in Figure 1 shows a summary of the computation needed to evaluate the loss in Equation 2 using only forward passes through the two subnetworks $\Upsilon$ and $\Omega$ . It is now possible with the proposed efficient approximation of our proposed regularizer in Equation 2 to efficiently train networks on noisy training examples that are corrupted with noise $\bar { \mathcal { N } } ( \mathbf { 0 } , \bar { \Sigma } _ { x } )$ without any form of prohibitive data augmentation.
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# 4 EXPERIMENTS
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In this section, we conduct experiments on multiple network architectures and datasets to demonstrate the effectiveness of our proposed regularizer in training more robust networks, especially in comparison with data augmentation. To standardize robustness evaluation, we first propose a new unified robustness metric against additive noise from a general distribution $\mathcal { D }$ and later specialize it when $\mathcal { D }$ is Gaussian. Lastly, we show that networks trained with our proposed regularizer not only outperform in robustness networks trained with Gaussian augmented data. Moreover, we show that such networks are also much more magnitudes times robust against other types of attacks.
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On the Robustness Evaluation Metric. While there is a consensus on the definition of robustness in the presence of adversarial attacks, as the smallest perturbation required to fool a network, i.e. to change its prediction, it is not straightforward to extend such a definition to additive noise sampled from a distribution $\mathcal { D }$ . In particular, the work of Fawzi et al. (2018) tried to address this difficulty by defining the robustness of a classifier around an example x as the distance between x and the closest decision boundary. However, this definition is difficult to compute in practice and is not scalable, as it requires solving a generally nonconvex optimization problem for every testing example $\mathbf { x }$ that may also suffer from poor local minima. To remedy these drawbacks, we present a new robustness metric for generic additive noise.
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Figure 2: General trade-off between accuracy and robustness on LeNet. We see, in all plots, that the accuracy tends to be negatively correlated with robustness over varying noise levels and amount of augmentation. Baseline refers to training with neither data augmentation nor our regularizer. However, it is hard to compare the performance of our method against data augmentation from these plots as we can only compare the robustness of models with similar noise-free testing accuracy.
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Robustness Against Additive Noise. Consider a classifier $\Psi ( . )$ with $\psi ( \mathbf { x } ) = \arg \operatorname* { m a x } _ { i } \Psi _ { i } ( \mathbf { x } )$ as the predicted class label for the example $\mathbf { x }$ regardless of the correct class label $y _ { i }$ . We define the robustness on a sample $\mathbf { x }$ against a generic additive noise sampled from a distribution $\mathcal { D }$ as
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$$
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\Re _ { \mathcal { D } } ( \mathbf { x } ) = \mathbb { P } _ { \mathbf { n } \sim \mathcal { D } } \{ \psi ( \mathbf { x } + \mathbf { n } ) = \psi ( \mathbf { x } ) \} .
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$$
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Here, the proposed robustness metric $\Re _ { \mathcal { D } } ( \mathbf { x } )$ measures the probability of the classifier to preserve the original prediction of the noise-free example $\psi ( \mathbf { x } )$ after adding noise, $\psi ( \mathbf { x } + \mathbf { n } )$ , from distribution $\mathcal { D }$ . Therefore, the robustness over a testing dataset $\tau$ can be defined as the expected robustness over the test dataset: $\Re _ { \mathcal { D } } ( \mathcal { T } ) = \mathbb { E } _ { \mathbf { x } \sim \mathcal { T } } \left[ \Re _ { \mathcal { D } } ( \mathbf { x } ) \right]$ . Inspired by Franceschi et al. (2018), for ease, we relax Equation 4 from the probability of preserving the prediction score to a 0/1 robustness over $m$ - randomly sampled examples from $\mathcal { D }$ . That is, $\Re _ { \mathcal { D } } ( { \bf x } ) = 1$ means that, among $m$ randomly sampled noise from $\mathcal { D }$ added to $\mathbf { x }$ , none changed the prediction from $\psi ( \mathbf { x } )$ . However, if a single example of these $m$ samples changed the prediction from $\psi ( \mathbf { x } )$ , we set $\Re _ { \mathcal { D } } ( { \bf x } ) = 0$ . Thus, the robustness score is the average of this measure over the testing dataset $\tau$ .
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Robustness Against Gaussian Noise. For additive Gaussian noise, i.e. $\mathcal { D } ~ = ~ \mathcal { N } ( \mathbf { 0 } , \Sigma _ { x } ~ =$ Diag $\left( \sigma _ { x } ^ { 2 } \right)$ ), robustness is averaged over a range of testing variances $\sigma _ { x } ^ { 2 }$ . We restrict $\sigma _ { x }$ to 30 evenly sampled values in $[ 0 , 0 . 5 ]$ , where this set is denoted as $\mathcal { A } ^ { 1 }$ . In practice, this is equivalent to sampling $m$ Gaussian examples for each $\sigma _ { x } \in { \mathcal { A } }$ , and if none of the $m$ samples changes the prediction of the classifier $\psi$ from the original noise-free example, the robustness for that sample at that $\sigma _ { x }$ noise level is set to 1 and then averaged over the complete testing set. Then, the robustness is the average over multiple $\sigma _ { x } \in { \mathcal { A } }$ . To make the computation even more efficient, instead of sampling a large number of Gaussian noise samples $( m )$ , we only sample a single noise sample with the average√ energy over $\mathcal { D }$ . That is, we sample a single $\mathbf { n }$ of norm $\mathbf { \bar { \mathbf { \rho } } } _ { \| \mathbf { n } \| _ { 2 } } = \mathbf { \bar { \sigma } } \sigma _ { x } \sqrt { n }$ . This is due to the fact that
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Experimental Setup. In this section, we demonstrate the effectiveness of the proposed regularizer in improving robustness. Several experiments are performed with our objective Equation 2, where we strike a comparison with data augmentation approaches.
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Architecture Details. The input images in MNIST (gray-scale) and CIFAR (colored) are squares with sides equal to 28 and 32, respectively. Since AlexNet was originally trained on ImageNet of sides equal to 224, we will marginally alter the implementation of AlexNet in TorchVision Marcel & Rodriguez (2010) to accommodate for this difference. First, we change the number of hidden units in the first fully-connected layer (in LeNet to 4096, AlexNet to 256, LeNet on MNIST to 3136). For AlexNet, we changed all pooling kernel sizes from 3 to 2 and the padding size of conv1 from 2 to 5. Second, we swapped each maxpool with the preceding ReLU, which makes training and inference more efficient. Third, we enforce that the first layer in all the models is a convolution followed by ReLU as discussed earlier. Lastly, to simplify analysis, we removed all dropout layers. We leave the details of the optimization hyper-parameters to the appendix.
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Figure 3: Fair robustness comparison of LeNet with data augmentation and our regularizer. We only report results for models with a test accuracy that is at least as good as the accuracy of the baseline with a tolerance: $0 \%$ , $0 . 3 9 \%$ , and $0 . 7 5 \%$ for MNIST, CIFAR10, CIFAR100, respectively. Only the models with the highest robustness are presented. Training with our regularizer can attain similar/better robustness than 21-fold noisy data augmentation on MNIST and CIFAR100, while maintaining a high noise-free test accuracy.
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Results. For each model and dataset, we compare baseline models, i.e. models trained with noisefree data and without our regularization, with two others: one using data augmentation and another using our proposed regularizer. Each of the latter has two configurable variables: the level of noise controlled by $\textstyle { \mathcal { \sigma } } _ { x } ^ { 2 }$ during training, and the amount of noise controlled by the trade-off coefficient $\alpha$ in Equation 2 or $\tilde { N }$ (number of added noisy training examples) in the case of augmentation.
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Accuracy vs. Robustness. We start by demonstrating that data augmentation tends to improve the robustness, as captured by $\Re ( \mathcal { T } )$ over the test set, at the expense of decreasing the testing accuracy on the noise-free examples. Realizing this is essential for a fair comparison, as one would need to compare the robustness of networks that only have similar noise-free testing accuracies. To this end, we ran 60 training experiments with data augmentation on LeNet with three datasets (MNIST, CIFAR10, and CIFAR100), four augmentation levels $( \tilde { N } \in \{ 2 , 6 , 1 1 , 2 1 \} )$ , and five noise levels $( \sigma _ { x } \in \mathcal { A } = \{ 0 . 1 2 5 , 0 . 2 5 , 0 . 3 2 5 , 0 . 5 , 1 . 0 \} )$ . In contrast, we ran robust training experiments using Equation 2 with the trade-off coefficient $\overset { \cdot } { \alpha } \in \{ 0 . 5 , 1 , 1 . 5 , 2 , 5 , 1 0 , 2 0 \}$ on the same datasets, but we extended the noise levels $\sigma _ { x }$ to include the extreme noise regime of $\sigma _ { x } \in \{ 2 , 5 , 1 0 , 2 0 \}$ . These noise levels are too large to be used for data augmentation, especially since $\mathbf { x } \in [ 0 , 1 ] ^ { n }$ ; however, as we will see, they are still beneficial for our proposed regularizer. Figure 2 shows both the testing accuracy and robustness as measured by $\Re ( \mathcal { T } )$ over a varying range of training $\sigma _ { x }$ for the data augmentation approach of LeNet on MNIST, CIFAR-10 and CIFAR-100. It is important to note here that the main goal of these plots is not to compare the robustness score, but rather, to demonstrate a very important trend. In particular, increasing the training $\sigma _ { x }$ for each approach degrades testing accuracy on noise-free data. However, the degradation in our approach is much more graceful since the trained LeNet model was never directly exposed to individually corrupted examples during training as opposed to the data augmentation approach. Note that our regularizer enforces the separation between the expected output prediction analytically. Moreover, the robustness of both methods consistently improves as the training $\sigma _ { x }$ increases. This trend holds even on the easiest dataset (MNIST). Interestingly, models trained with our regularizer enjoy an improvement in testing accuracy over the baseline model. Such behaviour only emerges with a large factor of augmentation, $\tilde { N } = 2 \bar { 1 }$ , and a small enough training $\sigma _ { x }$ on MNIST. This indicates that models can benefit from better accuracy with a good approximation of Equation 1 through our proposed objective or through extensive Monte Carlo estimation. However, as $\underset { \cdots } { \sigma } { _ { x } }$ increases, Monte Carlo estimates of the second term in Equation 1 via data augmentation (with $\ddot { N } = 2 1$ ) is no longer enough to capture the noise.
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Robustness Comparison. For fair comparison, it is essential to only compare the robustness of networks that achieve similar testing accuracy, since perfect robustness is attainable with a deterministic classifier that assigns the same class label regardless of the input. In fact, we proposed a unified robustness metric for the reason that most commonly used metrics are disassociated from
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<table><tr><td></td><td>g</td><td>PGD</td><td>LBFGS</td><td>FGSM</td><td>DF2</td><td>GNR</td><td>ACC</td></tr><tr><td rowspan="6">EE nenen JSINW</td><td>0</td><td>1.02 × 10-03</td><td>5.50×10-04</td><td>2.24×10-02</td><td>5.91×10-04</td><td>77.45</td><td>98.50</td></tr><tr><td>0.125</td><td>3.36×10-01</td><td>3.43×10-01</td><td>8.19 ×10-01</td><td>2.05×10-01</td><td>93.14</td><td>97.50</td></tr><tr><td>0.250</td><td>4.58×10-01</td><td>4.31×10-01</td><td>1.21</td><td>2.63×10-01</td><td>95.64</td><td>98.75</td></tr><tr><td>0.325</td><td>4.21×10-01</td><td>4.51 ×10-01</td><td>1.17</td><td>2.33×10-01</td><td>96.75</td><td>97.50</td></tr><tr><td>1.0</td><td>5.44×10-01</td><td>5.22×10-01</td><td>1.34</td><td>2.95×10-01</td><td>97.32</td><td>99.00</td></tr><tr><td></td><td></td><td></td><td>-05</td><td></td><td></td><td></td></tr><tr><td rowspan="5">A neeae CEITIIIO</td><td>0</td><td>2.30×10-05</td><td>2.50 × 10-05</td><td>1.50 ×10</td><td>2.10×10-05</td><td>29.69</td><td>34.75</td></tr><tr><td>0.12</td><td>3.64×10-04</td><td>2.83×10-04</td><td>5.06×10-04</td><td>2.16×10-04</td><td>31.65</td><td>33.50</td></tr><tr><td>0.250</td><td>4.37 ×10-04</td><td>3.86×10-04</td><td>6.50×10-04</td><td>2.47×10-04</td><td>32.85</td><td>32.25</td></tr><tr><td>0.325</td><td>5.37×10-04</td><td>4.04×10-04</td><td>7.29 ×10-04</td><td>3.18×10-04</td><td>33.84</td><td>34.25</td></tr><tr><td>1.0</td><td>4.92×10-04</td><td>3.26×10-04</td><td>6.50×10-04</td><td>2.85×10-04</td><td>34.65</td><td>35.50</td></tr></table>
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Table 1: Gaussian robustness improves overall robustness. We report the robustness metrics corresponding to various attacks (PGD, LBFGS, FGSM, and DF2), our proposed GNR metric, and the test accuracy ACC for LeNet and AlexNet networks trained on MNIST and CIFAR100 using our proposed regularizer with noise variance $\sigma$ in training. Note that $\sigma = 0$ corresponds to baseline models trained without our regularizer. We observe that training networks with our proposed regularizer (designed for additive Gaussian attacks) not only improves the robustness against Gaussian attacks but also against 6 other types of attacks which 4 of them listed here and the others are left for appendix.
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the ground-truth labels and only consider model predictions. Therefore, we filtered out the results from Figure 2 by removing all the experiments that achieved lower test accuracy than the baseline model. Figure 3 summarizes these results for LeNet. Now, we can clearly see the difference between training with data augmentation and our approach. For MNIST (Figure 3a), we achieved the same robustness as 21-fold data augmentation without feeding the network with any noisy examples during training and while preserving the same baseline accuracy. Interestingly, for CIFAR10 (Figure 3b), our method is twice as robust as the best robustness achieved via data augmentation. Moreover, for CIFAR100 (Figure 3c), we are able to outperform data augmentation by around $5 \%$ . Finally, for extra validation, we also conducted the same experiments with AlexNet on CIFAR10 and CIFAR100 which can be found in the appendix. We can see that our proposed regularizer can improve robustness by $1 5 \%$ on CIFAR10 and around $2 5 \%$ on CIFAR100. It is interesting to note that for CIFAR10, data augmentation could not improve the robustness of the trained models without drastically degrading the testing accuracy on the noise-free examples. Moreover, it is interesting to observe that the best robustness achieved through data augmentation is even worse than the baseline. This could be due to the trade-off coefficient $\alpha$ in Equation 1.
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Towards General Robustness via Gaussian Robustness. Here, we investigate whether improving robustness to Gaussian input noise can improve robustness against other types of attacks. Specifically, we compare the robustness of models trained using our proposed regularizer (robust again Gaussian attacks) with baseline models subject to different types of attacks: Projected Gradient Descent (PGD) and LBFGS attacks Szegedy et al. (2014), Fast Sign Gradient Method (FGSM) Goodfellow et al. (2015), and DeepFool L2Attack (DF2) Moosavi-Dezfooli et al. (2016) as provide by Rauber et al. (2017). For all these attacks, we report the minimum energy perturbation that can change the network prediction. We also report our Gaussian Network Robustness (GNR) metric, which is the Gaussian version of Equation 4 along with the testing accuracy (ACC). We perform experiments on LeNet on MNIST, CIFAR10 and CIFAR100 datasets and on AlexNet on both CIFAR10 and CIFAR100. Due to space constraints, we show the robustness results for only LeNet on MNIST and AlexNet of CIFAR100 and leave the rest along with two other types of attacks for the appendix. Table 1 shows that improving GNR through our data augmentation free regularizer can significantly improve all robustness metrics. For instance, comparing LeNet trained with our proposed regularizer against LeNet trained without any regularization, i.e. $\sigma = 0$ , we see that robustness against all types of attacks improves by almost two orders of magnitude, while maintaining a similar testing accuracy. A similar improvement in performance is consistently present for AlexNet on CIFAR100.
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# 5 CONCLUSION
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Addressing the sensitivity problem of deep neural networks to adversarial perturbation is of great importance to the machine learning community. However, building robust classifiers against this noises is computationally expensive, as it is generally done through the means of data augmentation. We propose a generic lightweight analytic regularizer, which can be applied to any deep neural network with a ReLU activation after the first affine layer. It is designed to increase the robustness of the trained models under additive Gaussian noise. We demonstrate this with multiple architectures and datasets and show that it outperforms data augmentation without observing any noisy examples.
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Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. In Conference on Neural Information Processing Systems Workshops (NeurIPSW17), 2017.
|
| 175 |
+
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| 176 |
+
Jonas Rauber, Wieland Brendel, and Matthias Bethge. Foolbox v0.8.0: A python toolbox to benchmark the robustness of machine learning models. CoRR, 2017.
|
| 177 |
+
|
| 178 |
+
Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. In International Conference on Learning Representations (ICLR14), 2014.
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| 179 |
+
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| 180 |
+
Dimitris Tsipras, Shibani Santurkar, Logan Engstrom, Alexander Turner, and Aleksander Madry. Robustness may be at odds with accuracy. stat, 2018.
|
| 181 |
+
|
| 182 |
+
Valentina Zantedeschi, Maria-Irina Nicolae, and Ambrish Rawat. Efficient defenses against adversarial attacks. In ACM Workshop on AI and Security, 2017.
|
| 183 |
+
|
| 184 |
+
Fengyuan Zhu, Guangyong Chen, and Pheng-Ann Heng. From noise modeling to blind image denoising. In Computer Vision and Patter Recognition Conference (CVPR16), 2016.
|
| 185 |
+
|
| 186 |
+
# A EXPERIMENTAL SETUP AND DETAILS.
|
| 187 |
+
|
| 188 |
+
All experiments, are conducted using PyTorch version 0.4.1 Paszke et al. (2017). All hyperparameters are fixed and Table 2 we report the setup for the two optimizers. In particular, we use the Adam optimizaer Kingma & Ba (2015) with $\beta _ { 1 } = 0 . 9 , \beta _ { 2 } = 0 . 9 9 9$ , $\epsilon = 1 0 ^ { \div 8 }$ with amsgrad set to False. The second optimizer is SGD Loshchilov & Hutter (2017) with momentum $\scriptstyle 1 = 0 . 9$ , dampening $= 0$ , with Nesterov acceleration. In each experiment, we randomly split the training dataset into $10 \%$ validation and $90 \%$ training and monitor the validation loss after each epoch. If validation loss did not improve for lr patience epochs, we reduce the learning rate by multiplying it by lr factor. We start with an initial learning rate of lr initial. The training is terminated only if the validation loss did not improve for loss patience number of epochs or if the training reached 100 epochs. We report the results of the model with the best validation loss.
|
| 189 |
+
|
| 190 |
+
Table 2: Lists the training optimization hyper-parameters.
|
| 191 |
+
|
| 192 |
+
<table><tr><td rowspan=1 colspan=6>Hyper-parameter</td><td rowspan=1 colspan=1>LeNet</td><td rowspan=1 colspan=1>AlexNet</td></tr><tr><td rowspan=7 colspan=6>optimizerminibatch_sizelr_initiallr_patiencelr_factorloss-patienceweight_decay</td><td rowspan=1 colspan=1>er</td><td rowspan=1 colspan=1>Adam</td></tr><tr><td rowspan=1 colspan=2>oatch_si2</td><td rowspan=1 colspan=1>e</td><td rowspan=1 colspan=1>1000</td><td rowspan=1 colspan=1>128</td></tr><tr><td rowspan=1 colspan=1>0.0001</td><td rowspan=1 colspan=1>0.1</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.5</td></tr><tr><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>20</td></tr><tr><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0.0005</td></tr></table>
|
| 193 |
+
|
| 194 |
+
# B EXAMPLES ON NOISE LEVELS
|
| 195 |
+
|
| 196 |
+
Figure 4 provides examples of the different levels of noise on a given digit 8.
|
| 197 |
+
|
| 198 |
+
|
| 199 |
+
Figure 4: This Figure shows an example of the noise level over varying level of input $\sigma$ on the digit 8. In particular, one can observe that with $\sigma$ large than 0.7 the among of noise is severe even for the human level. Training on such extreme noise levels will deem data augmentation to be difficult.
|
| 200 |
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| 201 |
+
# C A COMMENT ON THE ROBUSTNESS METRIC
|
| 202 |
+
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| 203 |
+
We measure the robustness against Gaussian noise by averaging over a range of input noise levels, where at each level for each image, we consider it misclassified if the probability of it being misclassified is greater than a certain threshold. The final robustness is the average over multiple testing $\sigma _ { x }$ . This is special case of the more general case in Equation (4). We then report the area under the curve of the robustness with varying testing $\sigma _ { x }$ as shown in Figure 6. The area under this curve thus represents the overall robustness of a given model under several varying input noise standard deviation $\sigma _ { x }$ .
|
| 204 |
+
|
| 205 |
+
# D OTHER ROBUSTNESS METRICS
|
| 206 |
+
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| 207 |
+
We report the robustness of several architectures over several datasets with and without our trained regularizer. We show that our proposed efficient regularizer not only improves the robustness against Gaussin noise attacks but againts several other types of attacks. Table 3 summarizes the types of attacks used for robustness evaluation.
|
| 208 |
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| 209 |
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|
| 210 |
+
Figure 5: The robustness is a function of the ratio of the orange area to the blue area in the white circle.
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| 211 |
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| 212 |
+
|
| 213 |
+
Figure 6: The robustness is thus measured as the area under the curve of testing accuracy versus input noise level (standard deviation).
|
| 214 |
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| 215 |
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|
| 216 |
+
Figure 7: Fair robustness comparison of AlexNet with data augmentation and our regularizer. The reported models trained with our regularizer on CIFAR10 and CIFAR100 on all training $\sigma _ { x }$ are within $1 . 6 8 \%$ and $4 . 8 3 \%$ of the baseline accuracy, respectively. The models trained with the proposed regularizer achieve better robustness than 11-fold and 6-fold noisy data augmentation on CIFAR10 and CIFAR100, respectively.
|
| 217 |
+
|
| 218 |
+
Table 3: The table lists all the attacks performed.
|
| 219 |
+
|
| 220 |
+
<table><tr><td>Attack Abbreviation</td><td>AttackName</td></tr><tr><td>PGD LBF</td><td>Projected Gradient Descent</td></tr><tr><td>GSM</td><td>LBFGS Attack FGSM</td></tr><tr><td>AGA</td><td>Additive Gaussian Noise Attack</td></tr><tr><td>AUA</td><td>AdditiveUniformNoiseAttack</td></tr><tr><td>DF2</td><td>DeepFool l2 Attack</td></tr></table>
|
| 221 |
+
|
| 222 |
+
<table><tr><td rowspan=1 colspan=1>ACC</td><td rowspan=1 colspan=1>09'86598609:2600'66</td><td rowspan=1 colspan=1>30.99122255</td><td rowspan=1 colspan=1>78.285850358398986860</td><td rowspan=1 colspan=1>730:2900'2930.5506'99</td><td rowspan=1 colspan=1>3530322533550</td></tr><tr><td rowspan=1 colspan=1>GNN</td><td rowspan=1 colspan=1>24225317599657326</td><td rowspan=1 colspan=1>2077800039.2031405</td><td rowspan=1 colspan=1>E87242706:00</td><td rowspan=1 colspan=1>353105098.29</td><td rowspan=1 colspan=1>696753368505820953</td></tr><tr><td rowspan=1 colspan=1>P</td><td rowspan=1 colspan=1>10-01X169 10-01X89710-01038720-01 X067</td><td rowspan=1 colspan=1>£0-013£0-0011X591 11×590×86[</td><td rowspan=1 colspan=1>×1333 ×81I15</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>0 10-0I X 81820-0131801 X917</td></tr><tr><td rowspan=1 colspan=1>AAA</td><td rowspan=1 colspan=1>20-01 X 65780%68333</td><td rowspan=1 colspan=1>10-0I X 66'910-0I X08°2 1</td><td rowspan=1 colspan=1>10-01 × 00'910-0I X 9694443</td><td rowspan=1 colspan=1>10-0I X86*210-01 X278</td><td rowspan=1 colspan=1>10-01 X39110-01X107</td></tr><tr><td rowspan=1 colspan=1>AAA</td><td rowspan=1 colspan=1>20-01 X 8975737</td><td rowspan=1 colspan=1>£0-01X29710-01 X82'910-0I X 26210-0I × ∠9'990'[</td><td rowspan=1 colspan=1>10-01 X 90'9I0-0IX80'9TO-OI×1</td><td rowspan=1 colspan=1>10-0I X892</td><td rowspan=1 colspan=1>20-013335 10-0I X 66T</td></tr><tr><td rowspan=1 colspan=1>SSS</td><td rowspan=1 colspan=1>20-012710-0I X 61'81215184</td><td rowspan=1 colspan=1>£0-0I X291 20-012755</td><td rowspan=1 colspan=1>25-21225035-213250XI67</td><td rowspan=1 colspan=1>0- 20-0IX58%2133508</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>ERT</td><td rowspan=1 colspan=1>10-011000 10-01X 11110-01X 111</td><td rowspan=1 colspan=1>90-013830 20-01100030--0[ × 99'[</td><td rowspan=1 colspan=1>£0-01 X 79.011-111150 £0-11 3 39:51</td><td rowspan=1 colspan=1>10-05[1 70</td><td rowspan=1 colspan=1>10- 50-013 950233 330</td></tr><tr><td rowspan=1 colspan=1>PPG</td><td rowspan=1 colspan=1>£0-01X70110-0I X89510-01X110-01X1</td><td rowspan=1 colspan=1>90-01X218 20-11270 20-113330</td><td rowspan=1 colspan=1>90-01X2£0-01 X 22720-010580£0-01 X 9555</td><td rowspan=1 colspan=1>10-111105</td><td rowspan=1 colspan=1>20-01X005 10-0132210-111 2304-11 1 76.5</td></tr><tr><td rowspan=1 colspan=1>b</td><td rowspan=1 colspan=1>10500000976001</td><td rowspan=1 colspan=1>10110000970001</td><td rowspan=1 colspan=1>0051297600</td><td rowspan=1 colspan=1>00509760</td><td rowspan=1 colspan=1>10000801</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>LSINNuoGNe</td><td rowspan=1 colspan=1>01TIITuo</td><td rowspan=1 colspan=1>00ICTIAITuoGNe</td><td rowspan=1 colspan=1>CIIIIIIuoJEere</td><td rowspan=1 colspan=1>001CTIAITuo</td></tr></table>
|
| 223 |
+
|
| 224 |
+
corresponds to baseline models trained without our regularizer. We observe that training networks with our proposed reg
|
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|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "ANALYTICAL MOMENT REGULARIZER FOR TRAINING ROBUST NETWORKS ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
176,
|
| 8 |
+
98,
|
| 9 |
+
823,
|
| 10 |
+
146
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Anonymous authors Paper under double-blind review ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
183,
|
| 19 |
+
170,
|
| 20 |
+
398,
|
| 21 |
+
198
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "ABSTRACT ",
|
| 28 |
+
"text_level": 1,
|
| 29 |
+
"bbox": [
|
| 30 |
+
454,
|
| 31 |
+
234,
|
| 32 |
+
544,
|
| 33 |
+
250
|
| 34 |
+
],
|
| 35 |
+
"page_idx": 0
|
| 36 |
+
},
|
| 37 |
+
{
|
| 38 |
+
"type": "text",
|
| 39 |
+
"text": "Despite the impressive performance of deep neural networks (DNNs) on numerous learning tasks, they still exhibit uncouth behaviours. One puzzling behaviour is the subtle sensitive reaction of DNNs to various noise attacks. Such a nuisance has strengthened the line of research around developing and training noise-robust networks. In this work, we propose a new training regularizer that aims to minimize the probabilistic expected training loss of a DNN subject to a generic Gaussian input. We provide an efficient and simple approach to approximate such a regularizer for arbitrarily deep networks. This is done by leveraging the analytic expression of the output mean of a shallow neural network, avoiding the need for memory and computation expensive data augmentation. We conduct extensive experiments on LeNet and AlexNet on various datasets including MNIST, CIFAR10, and CIFAR100 to demonstrate the effectiveness of our proposed regularizer. In particular, we show that networks that are trained with the proposed regularizer benefit from a boost in robustness against Gaussian noise to an equivalent amount of performing 3-21 folds of noisy data augmentation. Moreover, we empirically show on several architectures and datasets that improving robustness against Gaussian noise, by using the new regularizer, can improve the overall robustness against 6 other types of attacks by two orders of magnitude. ",
|
| 40 |
+
"bbox": [
|
| 41 |
+
233,
|
| 42 |
+
262,
|
| 43 |
+
764,
|
| 44 |
+
511
|
| 45 |
+
],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "1 INTRODUCTION ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
+
176,
|
| 54 |
+
516,
|
| 55 |
+
336,
|
| 56 |
+
531
|
| 57 |
+
],
|
| 58 |
+
"page_idx": 0
|
| 59 |
+
},
|
| 60 |
+
{
|
| 61 |
+
"type": "text",
|
| 62 |
+
"text": "Deep neural networks (DNNs) have emerged as generic models that can be trained to perform impressively well in a variety of learning tasks ranging from object recognition (He et al., 2016) and semantic segmentation (Long et al., 2015) to speech recognition (Hinton et al., 2012) and bioinformatics (Angermueller et al., 2016). Despite their increasing popularity, flexibility, generality, and performance, DNNs have been recently shown to be quite susceptible to small imperceptible input noise (Szegedy et al., 2014; Moosavi-Dezfooli et al., 2016; Goodfellow et al., 2015). Such analysis gives a clear indication that even state-of-the-art DNNs may lack robustness. Consequently, there has been an ever-growing interest in the machine learning community to study this uncanny behaviour. In particular, the work of (Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016) demonstrates that there are systematic approaches to constructing adversarial attacks that result in misclassification errors with high probability. Even more peculiarly, some noise perturbations seem to be doubly agnostic (Moosavi-Dezfooli et al., 2017), i.e. there exist deterministic perturbations that can result in misclassification errors with high probability when applied to different networks, irrespective of the input (denoted network and input agnostic). ",
|
| 63 |
+
"bbox": [
|
| 64 |
+
174,
|
| 65 |
+
545,
|
| 66 |
+
825,
|
| 67 |
+
738
|
| 68 |
+
],
|
| 69 |
+
"page_idx": 0
|
| 70 |
+
},
|
| 71 |
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| 72 |
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"type": "text",
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| 73 |
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"text": "Understanding this degradation in performance under adversarial attacks is of tremendous importance, especially for real-world DNN deployment, e.g. self-driving cars/drones and equipment for the visually impaired. A standard and popular means to alleviate this nuisance is noisy data augmentation in training, i.e. a DNN is exposed to noisy input images during training so as to bolster its robustness during inference. Several works have demonstrated that DNNs can in fact benefit from such augmentation (Moosavi-Dezfooli et al., 2016; Goodfellow et al., 2015). However, data augmentation in general might not be sufficient for two reasons. (1) Particularly with high-dimensional input noise, the amount of data augmentation necessary to sufficiently capture the noise space will be very large, which will increase training time. (2) Data augmentation with high energy noise can negatively impact the performance on noise-free test examples. This can be explained by the fundamental trade-off between accuracy and robustness (Tsipras et al., 2018; Boopathy et al., 2019). It can also arise from the fact that augmentation forces the DNN to have the same prediction for two vastly different versions of the same input, noise-free and a substantially corrupted version. ",
|
| 74 |
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"text": "Therefore, in this paper, we propose a new regularizer for noise-robust networks to circumvent the aforementioned setbacks of data augmentation. ",
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| 85 |
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"text": "A natural objective for training against attacks sampled from a distribution $\\mathcal { D }$ , that bypasses the need for data augmentation, is the expected loss under this distribution. Since a closed-form expression is generally difficult to obtain or an approximate surrogate is expensive to evaluate (Monte Carlo estimates), we propose instead a closely related objective that is the loss of the expected predictions of the network under $\\mathcal { D }$ -distributed adversarial noise. Since it has been shown that Gaussian noise can be adversarial (Bibi et al., 2018) and that such noise is widely studied in applications such as image processing, we restrict the focus in this paper to the case where $\\mathcal { D }$ is Gaussian. While this may seem to be too restrictive, we later show that improving the robustness of networks against Gaussian attacks also improves the robustness against a family of other types of attacks. However, even under such an assumption, only a memory and computationally expensive (expensive due to two-stage network linearization), closed-form approximate surrogate for network expected predictions exists (Bibi et al., 2018). ",
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"text": "Contributions. (i) We formalize a new regularizer that is a function of the probabilistic first moment of the output of a DNN to train robust DNNs against noise sampled from distribution $\\mathcal { D }$ . (ii) Under the special choice of Gaussian attacks, i.e. $\\mathcal { D }$ is Gaussian, we show how the first moment expression can be evaluated very efficiently during training for an arbitrary deep DNN by bypassing the need to perform memory and computationally expensive two-stage linearization. (iii) Extensive experiments using LeNet (LeCun et al., 1999) and AlexNet (Krizhevsky et al., 2012) architectures on MNIST (LeCun, 1998), CIFAR10, and CIFAR100 (Krizhevsky & Hinton, 2009) datasets demonstrate that a substantial enhancement in robustness can be achieved when using our regularizer in training. In fact, in the majority of the experiments, the improvement is better than training on the same dataset, augmented with 3 to 21 times Gaussian noisy data. Interestingly, the results suggest an excellent trade-off between accuracy and robustness. Moreover, we show that networks that are trained to be robust against Gaussian attacks using our proposed regularizer enjoy orders of magnitude boost in robustness against a family of other types of attacks. ",
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"type": "text",
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"text": "2 RELATED WORK ",
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| 118 |
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"type": "text",
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"text": "Despite the impressive performance of DNNs on various tasks, they have been shown to be very sensitive to certain types of noise, commonly referred to as adversarial examples, particularly in the recognition task (Moosavi-Dezfooli et al., 2016; Goodfellow et al., 2015). Adversarial examples can be viewed as small imperceptible noise that, once added to the input of a DNN, its performance is severely degraded. This finding has incited interest in studying/measuring the robustness of DNNs. ",
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"type": "text",
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"text": "The literature is rich with work that aims to unify and understand the notion of network robustness. For instance, Szegedy et al. (2014) suggested a spectral stability analysis for a wide class of DNNs by measuring the Lipschitz constant of the affine transformation describing a fully-connected or a convolutional layer. This result was extended to compute an upper bound for a composition of layers, i.e. a DNN. However, this measure sets an upper bound on the robustness over the entire input domain and does not take into account the noise distribution. Later, Fawzi et al. (2017a) defined robustness as the mean support of the minimum adversarial perturbation, which is now the most common definition for robustness. Not only was robustness studied against adversarial perturbations but also against geometric transformations to the input. Fawzi et al. (2018) emphasized the independence of the robustness measure to the ground truth class labels and that it should only depend on the classifier and the dataset distribution. Subsequently, two different metrics to measure DNN robustness were proposed: one for general adversarial attacks and another for noise sampled from uniform distribution. Recently, Gilmer et al. (2018) showed the trade-off between robustness and test error from a theoretical point of view on a simple classification problem with hyperspheres. ",
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"type": "text",
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"text": "On the other hand, and based on various robustness analyses, several works proposed various approaches in building networks that are robust against noise sampled from well known distributions and against generic adversarial attacks. For instance, Grosse et al. (2017) proposed a model that was trained to classify adversarial examples with statistical hypothesis testing on the distribution of the dataset. Another approach is to perform statistical analysis on the latent feature space instead (Li & Li, 2017; Feinman et al., 2017), or train a DNN that rejects adversarial attacks (Lu et al., 2017). Moreover, the geometry of the decision boundaries of DNN classifiers was studied by Fawzi et al. (2017b) to infer a simple curvature test for this purpose. Using this method, one can restore the original label and classify the input correctly. Restoring the original input using defense mechanisms, which can only detect adversarial examples, can be done by denoising (ridding it from its adversarial nature) so long as the noise perturbation is well-known and modeled apriori (Zhu et al., 2016). A fresh approach to robustness was proposed by Zantedeschi et al. (2017), where they showed that using bounded ReLUs (if augmented with Gaussian noise) to limit the output range can improve robustness. A different work proposed to distill the learned knowledge from a deep model to retrain a similar model architecture as a means to improving robustness (Papernot et al., 2016). This training approach is one of many adversarial training strategies for robustness Makhzani et al. (2016). More closely to our work is (Cisse et al., 2017), where a new training regularizer was proposed for a large family of DNNs. The proposed regularizer softly enforces that the upper bound of the Lipshitz constant of the output of the network to be less than or equal to one. Moreover and very recently, the work of Bibi et al. (2018) has derived analytic expressions for the output mean and covariance of networks in the form of (Affine, ReLU, Affine) under a generic Gaussian input. This work also demonstrates how a (memory and computation expensive) two-stage linearization can be employed to locally approximate a deep network with a two layer one, thus enabling the application of the derived expressions on the approximated shallower network. ",
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| 152 |
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| 159 |
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|
| 160 |
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{
|
| 161 |
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"type": "image",
|
| 162 |
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"img_path": "images/6a8b44ab9cc4a92b240046d156abb707fc39b6013fdf553b6bae86b47048dd38.jpg",
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"image_caption": [
|
| 164 |
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"Figure 1: Overview of the proposed graph for training Gaussian robust networks. The yellow block corresponds to an arbitrary network $\\Phi ( . , \\theta )$ viewed as the composition of two subnetworks separated by a ReLU. The stream on the bottom computes the output mean $\\mu _ { 4 }$ of the network $\\Phi ( : , \\theta )$ assuming that (i) the noise input distribution is independent Gaussian with variances $\\sigma _ { x } ^ { 2 }$ , and (ii) $\\Omega ( . ~ : ~ \\theta _ { 2 } )$ is approximated by a linear function. This evaluation for the output mean is efficient as it only requires an extra forward pass (bottom stream), as opposed to other methods that employ computationally and memory intensive network linearizations or data augmentation. "
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| 177 |
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"text": "",
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| 178 |
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"text": "Most prior work requires data augmentation, training new architectures that distill knowledge, or detect adversaries a priori, resulting in expensive training routines that may be ineffective in the presence of several input noise types. To this end, we address these limitations through our new regularizer that aims to fundamentally tackle Gaussian input noise without data augmentation and, as a consequence, improves overall robustness against other types of attacks. ",
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| 189 |
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"text": "3 METHODOLOGY ",
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| 200 |
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| 201 |
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"type": "text",
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| 211 |
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"text": "Background on Network Moments. Networks with a single hidden layer of the form (Affine, ReLU, Affine) can be written in the functional form $\\mathbf { g } ( \\mathbf { x } ) = \\bar { \\mathbf { B } } \\mathrm { m a x } \\left( \\mathbf { A } \\mathbf { x } + \\mathbf { \\bar { c } } _ { 1 } , \\mathbf { 0 } _ { p } \\right) + \\mathbf { c } _ { 2 }$ . The max(.) is an element-wise operator, $\\textbf { A } \\in \\mathbb { R } ^ { p \\times n }$ , and $\\textbf { B } \\in \\mathbb { R } ^ { d \\times p }$ . Thus, $\\mathbf { g } : \\mathbb { R } ^ { n } \\mathbb { R } ^ { d }$ . Given that $\\mathbf { x } \\sim \\mathcal { N } \\left( \\mu _ { x } , \\Sigma _ { x } \\right)$ , Bibi et al. (2018) showed that: ",
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| 219 |
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| 220 |
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| 221 |
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"type": "equation",
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| 222 |
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"img_path": "images/09a201e92599746e0952633750c2dadd793b0e340db0db58ff014b5a0921f087.jpg",
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| 223 |
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"text": "$$\n\\begin{array} { r } { \\mathbf { T h e o r e m 1 textit { T h e f i r s t m o m e n t } } o f g ( \\mathbf { x } ) \\ i s \\mathbb { E } [ \\mathbf { g } ( \\mathbf { x } ) ] = \\mathbf { B } \\left( \\mu _ { 2 } \\odot \\Phi \\left( \\frac { \\mu _ { 2 } } { \\sigma _ { 2 } } \\right) + \\sigma _ { 2 } \\odot \\varphi \\left( \\frac { \\mu _ { 2 } } { \\sigma _ { 2 } } \\right) \\right) + \\mathbf { c } _ { 2 } . } \\end{array}\n$$",
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| 224 |
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| 225 |
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"type": "text",
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| 235 |
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"text": "Note that $\\mu _ { 2 } = \\mathbf { A } \\mu _ { x } + \\mathbf { c } _ { 1 }$ , $\\sigma _ { 2 } = \\sqrt { \\mathrm { d i a g } \\left( \\Sigma _ { 2 } \\right) } .$ , $\\Sigma _ { 2 } = \\mathbf { A } \\Sigma _ { x } \\mathbf { A } ^ { \\top }$ , $\\Phi$ and $\\varphi$ are the standard Gaussian cumulative (CDF) and density (PDF) functions, respectively. The vector multiplication and division are element-wise operations. Lastly, $\\mathrm { d i a g ( . ) }$ extracts the diagonal elements of a matrix into a vector. For ease of notation, we let $\\begin{array} { r } { \\mu _ { 3 } = \\dot { T } ( \\mu _ { 2 } , \\dot { \\sigma } _ { 2 } ) = ( \\mu _ { 2 } \\odot \\Phi ( \\frac { \\mu _ { 2 } } { \\sigma _ { 2 } } \\bar { ) } + \\sigma _ { 2 } \\odot \\varphi ( \\frac { \\mu _ { 2 } } { \\sigma _ { 2 } } ) ) } \\end{array}$ . ",
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| 236 |
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| 243 |
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"text": "To extend the results of Theorem (1) to deeper models, a two-stage linearization was proposed in Bibi et al. (2018), where $( \\mathbf { A } , \\mathbf { B } )$ and $( \\mathbf { c } _ { 1 } , \\mathbf { c } _ { 2 } )$ are taken to be the Jacobians and biases of the first order Taylor approximation to the two network functions around a ReLU layer in a DNN. Refer to Bibi et al. (2018) for more details about this expression and the proposed linearization. ",
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| 258 |
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"type": "text",
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| 268 |
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"text": "Proposed Robust Training Regularizer. To propose an alternative to noisy data augmentation to address its drawbacks, one has to realize that this augmentation strategy aims to minimize the expected training loss of a DNN when subjected to noisy input distribution $\\mathcal { D }$ through sampling. In fact, it minimizes an empirical loss that approximates this expected loss when enough samples are present during training. When sampling is insufficient (a drawback of data augmentation in highdimensions), this approximation is too loose and robustness can suffer. However, if we have access to an analytic expression for the expected loss, expensive data augmentation can be averted. This is the key motivation of the paper. Mathematically, the training loss can be modeled as ",
|
| 269 |
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| 277 |
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|
| 278 |
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"type": "equation",
|
| 279 |
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| 280 |
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"text": "$$\n\\operatorname* { m i n } _ { \\theta } \\ \\sum _ { i = 1 } ^ { N } \\Big ( \\ell \\left( \\Phi ( \\mathbf { x } _ { i } ; \\theta ) , y _ { i } \\right) + \\alpha \\mathbb { E } _ { \\mathbf { n } \\sim \\mathcal { D } } \\left[ \\ell \\left( \\Phi ( \\mathbf { x } _ { i } + \\mathbf { n } ; \\theta ) , y _ { i } \\right) \\right] \\Big ) .\n$$",
|
| 281 |
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| 282 |
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"type": "text",
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"text": "Here, $\\Phi : \\mathbb { R } ^ { n } \\mathbb { R } ^ { d }$ is any arbitrary network with parameters $\\theta , \\ell$ is the loss function, $\\{ ( \\mathbf { x } _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { N }$ are the noise-free data-label training pairs, and $\\alpha \\geq 0$ is a trade off parameter. While the first term in Equation 1 is the standard empirical loss commonly used for training, the second term is often replaced with its Monte Carlo estimate through data augmentation. That is, for each training example $\\mathbf { x } _ { i }$ , the second term is approximated with an empirical average of $\\tilde { N }$ noisy examples of $\\mathbf { x } _ { i }$ such that $\\begin{array} { r } { \\mathbb { E } _ { \\mathbf { n } \\sim \\mathcal { D } } [ \\ell \\left( \\boldsymbol { \\Phi } ( \\mathbf { x } _ { i } + \\mathbf { n } ; \\boldsymbol { \\theta } ) , y _ { i } \\right) ] \\approx \\frac { 1 } { \\tilde { N } } \\sum _ { j = 1 } ^ { \\tilde { N } } \\ell \\left( \\boldsymbol { \\Phi } ( \\mathbf { x } _ { i } + \\mathbf { n } _ { j } ; \\boldsymbol { \\theta } ) , y _ { i } \\right) . } \\end{array}$ . This will increase the size of the dataset by a factor of $\\tilde { N }$ , which will in turn increase training complexity. As discussed earlier, network performance on the noise-free examples can also be negatively impacted. Note that obtaining a closed form expression for the second term in Equation 1 for some of the popularly used losses $\\ell$ is more complicated than deriving expressions for the output mean of the network $\\Phi$ itself, e.g. in Theorem (1). Therefore, we propose to replace this loss with the following surrogate ",
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| 293 |
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|
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"text": "$$\n\\operatorname* { m i n } _ { \\theta } \\ \\sum _ { i = 1 } ^ { N } \\Big ( \\ell \\left( \\Phi ( \\mathbf { x } _ { i } ; \\theta ) , y _ { i } \\right) + \\alpha \\ell \\left( \\mathbb { E } _ { \\mathbf { n } \\sim \\mathcal { D } } \\left[ \\Phi ( \\mathbf { x } _ { i } + \\mathbf { n } ; \\theta ) \\right] , y _ { i } \\right) . \\Big )\n$$",
|
| 305 |
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| 306 |
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"text": "Because of Jensen’s inequality, Equation 2 is a lower bound to Eq Equation 1 when $\\ell$ is convex, which is the case for most popular losses including $\\ell _ { 2 }$ -loss and cross-entropy loss. The proposed second term in Equation 2 encourages that the output mean of the network $\\Phi$ of every noisy example $( \\mathbf { x } _ { i } + \\mathbf { n } )$ matches the correct class label $y _ { i }$ . This regularizer will stimulate a separation among the output mean of the classes if the training data is subjected to noise sampled from $\\mathcal { D }$ . Having access to an analytic expression for these means will prompt a simple inexpensive training, where the actual size of the training set is unaffected and augmentation is avoided. This form of regularization is proposed to replace data augmentation. ",
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"text": "While a closed-form expression for the second term of Equation 2 might be infeasible for a general network $\\Phi ( . )$ , an expensive approximation can be attained. In particular, Theorem Equation 1 provides an analytic expression to evaluate the second term in Equation 2, for when $\\mathcal { D }$ is Gaussian and when the network is approximated by a two-stage linearization procedure as $\\Phi ( \\mathbf { x } ) \\approx \\mathbf { B } \\mathrm { m a x } \\left( \\mathbf { A } \\mathbf { x } + \\mathbf { c } _ { 1 } , \\mathbf { 0 } _ { p } \\right) + \\mathbf { c } _ { 2 }$ . However, it is not clear how to utilize such a result to regularize networks during training with Equation 2 as a loss. This is primarily due to the computationally expensive and memory intensive network linearization proposed in Bibi et al. (2018). Specifically, the linearization parameters $\\left( \\mathbf { A } , \\mathbf { B } , \\mathbf { c } _ { 1 } , \\mathbf { c } _ { 2 } \\right)$ are a function of the network parameters, $\\theta$ , which are updated with every gradient descent step on Equation 2; thus, two-stage linearization has to be performed in every $\\theta$ update step, which is infeasible. ",
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"text": "On an Efficient Approximation to Equation 2. The loss in Equation 2 proposes a generic approach to train robust arbitrary networks against noise sampled from an arbitrary distribution $\\mathcal { D }$ . Since the problem in its general setting is too broad for detailed analysis, we restrict the scope of this work to the class of networks, which are most popularly used and parameterized by $\\theta$ , $\\bar { \\Phi ( . ; \\theta ) } : \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { d }$ with ReLUs as nonlinear activations. Moreover, since random Gaussian noise was shown to exhibit an adversarial nature Bibi et al. (2018); Rauber et al. (2017); Franceschi et al. (2018), and it is one of the most well studied noise models for the useful properties it exhibits, we restrict $\\mathcal { D }$ to the case of Gaussian noise. In particular, $\\mathcal { D }$ is independent zero-mean Gaussian noise at the input, i.e. $\\mathbf { n } \\sim \\mathcal { D } = \\mathcal { N } \\left( \\mathbf { 0 } , \\Sigma _ { x } = \\mathrm { D i a g } \\left( \\sigma _ { x } ^ { 2 } \\right) \\right)$ , where $\\sigma _ { x } ^ { 2 } \\in \\mathbb { R } ^ { n }$ is a vector of variances and $\\mathrm { D i a g ( . ) }$ reshapes the vector elements into a diagonal matrix. Generally, it is still difficult to compute the second term in Equation 2 under Gaussian noise for arbitrary networks. However, if we have access to an inexpensive approximation of the network, avoiding the computationally and memory expensive network linearization in Bibi et al. (2018), an approximation to the second term in Equation 2 can be used for efficient robust training directly on $\\theta$ . ",
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"text": "Consider the $l ^ { \\mathrm { t h } }$ ReLU layer in $\\Phi ( . ; \\theta )$ . the network can be expressed as $\\begin{array} { r l } { \\Phi ( . ; \\theta ) } & { { } = } \\end{array}$ $\\Omega ( \\mathrm { R e L U } _ { l } ( \\Upsilon ( . , \\theta _ { 1 } ) ) ; \\theta _ { 2 } )$ . Note that the parameters of the overall network $\\Phi ( . ; \\theta )$ is the union of the parameters of the two subnetworks $\\Upsilon ( . ; \\theta _ { 1 } )$ and $\\Omega ( . ; \\theta _ { 2 } )$ , i.e. $\\theta = \\theta _ { 1 } \\cup \\theta _ { 2 }$ . Throughout this work and to simplify the analysis, we set $l = 1$ . With such a choice of $l$ , the first subnetwork $\\Upsilon ( . , \\theta _ { 1 } )$ is affine with $\\bar { \\theta _ { 1 } } = \\{ \\mathbf { A } , \\mathbf { c } _ { 1 } \\}$ . However, the second subnetwork $\\Omega ( . , \\theta _ { 2 } )$ is not linear in general, and thus, one can linearize $\\Omega ( . , \\theta _ { 2 } )$ at $\\mathbb { E } _ { \\mathbf { n } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\Sigma _ { x } ) } \\left[ \\mathrm { R e L U } _ { 1 } \\left( \\Upsilon \\left( \\mathbf { x } _ { i } + \\mathbf { n } ; \\theta _ { 1 } \\right) \\right) \\right] = T ( \\mu _ { 2 } , \\sigma _ { 2 } ) = \\mu _ { 3 }$ . Note that $\\mu _ { 3 }$ is the output mean after the ReLU and $\\mu _ { 2 } = \\mathbf { A } \\mathbf { x } _ { i } + \\mathbf { c } _ { 1 }$ , since $\\Upsilon ( \\mathbf { x } _ { i } + \\mathbf { n } ; \\theta _ { 1 } ) = \\mathbf { A } \\left( \\mathbf { x } _ { i } + \\mathbf { n } \\right) + \\mathbf { c } _ { 1 }$ . Both $T ( . , . )$ and $\\sigma _ { 2 }$ are defined in Equation 1. Thus, linearizing $\\Omega$ at $\\mu _ { 3 }$ with linearization parameters $\\mathbf { ( B , c _ { 2 } ) }$ being the Jacobian of $\\Omega$ and ${ \\bf c } _ { 2 } = \\Omega ( \\mu _ { 3 } , \\theta _ { 2 } ) - { \\bf B } \\mu _ { 3 }$ , we have that, for any point $\\mathbf { v } _ { i }$ close to $\\mu _ { 3 }$ : $\\Omega ( \\mathbf { v } _ { i } , \\mathbf { \\bar { \\theta } } _ { 2 } ) \\approx \\mathbf { B } \\mathbf { v } _ { i } + \\mathbf { c } _ { 2 }$ . While computing $\\mathbf { ( B , c _ { 2 } ) }$ through linearization is generally very expensive, computing the approximation to Equation 2 requires explicit access to neither $\\mathbf { B }$ nor $\\mathbf { c } _ { 2 }$ . Note that this second term for $l = 1$ is given as: ",
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"text": "$$\n\\begin{array} { r l } & { \\quad \\ell \\left( \\mathbb { E } _ { \\mathbf { n } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\Sigma _ { x } ) } [ \\Phi ( \\mathbf { x } _ { i } + \\mathbf { n } ; \\boldsymbol { \\theta } ) ] , y _ { i } \\right) = \\ell \\left( \\mathbb { E } _ { \\mathbf { z } _ { i } \\sim \\mathcal { N } ( \\mathbf { x } _ { i } , \\Sigma _ { x } ) } [ \\Omega \\left( \\mathrm { R e L U } _ { 1 } \\left( \\Upsilon ( \\mathbf { z } _ { i } ; \\boldsymbol { \\theta } _ { 1 } ) \\right) ; \\boldsymbol { \\theta } _ { 2 } \\right) ] , y _ { i } \\right) } \\\\ & { = \\ell \\left( \\mathbb { E } _ { \\mathbf { z } _ { i } \\sim \\mathcal { N } ( \\mathbf { x } _ { i } , \\Sigma _ { x } ) } [ \\Omega \\left( \\mathrm { R e L U } _ { 1 } \\left( \\mathbf { A } \\mathbf { z } _ { i } + \\mathbf { c } _ { 1 } \\right) ; \\boldsymbol { \\theta } _ { 2 } \\right) ] , y _ { i } \\right) } \\\\ & { \\approx \\ell \\left( \\mathbb { E } _ { \\mathbf { z } _ { i } \\sim \\mathcal { N } ( \\mathbf { x } _ { i } , \\Sigma _ { x } ) } [ \\mathbf { B } \\left( \\mathrm { R e L U } _ { 1 } \\left( \\mathbf { A } \\mathbf { z } _ { i } + \\mathbf { c } _ { 1 } \\right) \\right) + \\mathbf { c } _ { 2 } ] , y _ { i } \\right) } \\\\ & { = \\ell \\left( \\mathbf { B } \\mu _ { 3 } + \\mathbf { c } _ { 2 } , y _ { i } \\right) = \\ell \\left( \\Omega ( \\mu _ { 3 } , \\boldsymbol { \\theta } _ { 2 } ) , y _ { i } \\right) . } \\end{array}\n$$",
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"text": "The approximation follows from the assumption that the input to the second subnetwork $\\Omega ( . ; \\theta _ { 2 } )$ , i.e. $\\mathbf { v } _ { i } = \\mathrm { R e L U } _ { 1 } \\left( \\mathbf { A } \\mathbf { z } _ { i } + \\mathbf { c } _ { 1 } \\right) )$ , is close to the point of linearization $\\mu _ { 3 }$ such that $\\Omega ( \\mathbf { v } _ { i } ; \\theta _ { 2 } ) \\approx \\mathbf { B } \\mathbf { v } _ { i } + \\mathbf { c } _ { 2 }$ . Or simply, that the input to $\\Omega$ is close to the mean inputs, i.e. $\\mu _ { 3 }$ , to $\\Omega$ under Gaussian noise. The penultimate equality follows from the linearity of the expectation. As for the last equality, $\\mathbf { ( B , c _ { 2 } ) }$ are the linearization parameters of $\\Omega$ at $\\mu _ { 3 }$ , where $\\mathbf { c } _ { 2 } = \\mathbf { \\bar { \\Omega } } ( \\mu _ { 3 } , \\theta _ { 2 } ) - \\mathbf { B } \\mu _ { 3 }$ by the first order Taylor approximation. Thus, computing the second term of Equation 2 according to Equation 3 can be simply approximated by a forward pass of $\\mu _ { 3 }$ through the second network $\\Omega$ . As for computing $\\mu _ { 3 } = T ( \\mu _ { 2 } , \\sigma _ { 2 } )$ , note that $\\mu _ { 2 } = \\mathbf { A } \\mathbf { x } _ { i } + \\mathbf { c } _ { 1 }$ in Equation 3, which is equivalent to a forward pass of $\\mathbf { x } _ { i }$ through the first subnetwork because $\\Upsilon ( . , \\theta _ { 1 } )$ is linear with $\\theta _ { 1 } = \\{ \\bar { \\bf A } , { \\bf c } _ { 1 } \\}$ . Moreover, since $\\sigma _ { 2 } =$ $\\sqrt { \\mathrm { d i a g } \\left( \\mathbf { A } \\Sigma _ { x } \\mathbf { A } ^ { \\top } \\right) }$ , we have: $\\sigma _ { 2 } = \\sqrt { \\operatorname { d i a g } \\left( \\mathbf { A D i a g } \\left( \\sigma _ { x } ^ { 2 } \\right) \\mathbf { A } ^ { \\top } \\right) } = \\sqrt { \\left( \\mathbf { A } \\odot \\mathbf { A } \\right) \\sigma _ { x } ^ { 2 } }$ . The expression for $\\sigma _ { 2 }$ can be efficiently computed by simply squaring the linear parameters in the first subnetwork and performing a forward pass of the input noise variance $\\sigma _ { x } ^ { 2 }$ through $\\Upsilon$ without the bias $\\mathbf { c } _ { 1 }$ and taking the element-wise square root. Lastly, it is straightforward to compute $T ( \\mu _ { 2 } , \\sigma _ { 2 } )$ as it is an elementwise function in Equation 1. The overall computational graph in Figure 1 shows a summary of the computation needed to evaluate the loss in Equation 2 using only forward passes through the two subnetworks $\\Upsilon$ and $\\Omega$ . It is now possible with the proposed efficient approximation of our proposed regularizer in Equation 2 to efficiently train networks on noisy training examples that are corrupted with noise $\\bar { \\mathcal { N } } ( \\mathbf { 0 } , \\bar { \\Sigma } _ { x } )$ without any form of prohibitive data augmentation. ",
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"text": "4 EXPERIMENTS ",
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"text": "In this section, we conduct experiments on multiple network architectures and datasets to demonstrate the effectiveness of our proposed regularizer in training more robust networks, especially in comparison with data augmentation. To standardize robustness evaluation, we first propose a new unified robustness metric against additive noise from a general distribution $\\mathcal { D }$ and later specialize it when $\\mathcal { D }$ is Gaussian. Lastly, we show that networks trained with our proposed regularizer not only outperform in robustness networks trained with Gaussian augmented data. Moreover, we show that such networks are also much more magnitudes times robust against other types of attacks. ",
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"text": "On the Robustness Evaluation Metric. While there is a consensus on the definition of robustness in the presence of adversarial attacks, as the smallest perturbation required to fool a network, i.e. to change its prediction, it is not straightforward to extend such a definition to additive noise sampled from a distribution $\\mathcal { D }$ . In particular, the work of Fawzi et al. (2018) tried to address this difficulty by defining the robustness of a classifier around an example x as the distance between x and the closest decision boundary. However, this definition is difficult to compute in practice and is not scalable, as it requires solving a generally nonconvex optimization problem for every testing example $\\mathbf { x }$ that may also suffer from poor local minima. To remedy these drawbacks, we present a new robustness metric for generic additive noise. ",
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"Figure 2: General trade-off between accuracy and robustness on LeNet. We see, in all plots, that the accuracy tends to be negatively correlated with robustness over varying noise levels and amount of augmentation. Baseline refers to training with neither data augmentation nor our regularizer. However, it is hard to compare the performance of our method against data augmentation from these plots as we can only compare the robustness of models with similar noise-free testing accuracy. "
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"text": "Robustness Against Additive Noise. Consider a classifier $\\Psi ( . )$ with $\\psi ( \\mathbf { x } ) = \\arg \\operatorname* { m a x } _ { i } \\Psi _ { i } ( \\mathbf { x } )$ as the predicted class label for the example $\\mathbf { x }$ regardless of the correct class label $y _ { i }$ . We define the robustness on a sample $\\mathbf { x }$ against a generic additive noise sampled from a distribution $\\mathcal { D }$ as ",
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"text": "$$\n\\Re _ { \\mathcal { D } } ( \\mathbf { x } ) = \\mathbb { P } _ { \\mathbf { n } \\sim \\mathcal { D } } \\{ \\psi ( \\mathbf { x } + \\mathbf { n } ) = \\psi ( \\mathbf { x } ) \\} .\n$$",
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"text": "Here, the proposed robustness metric $\\Re _ { \\mathcal { D } } ( \\mathbf { x } )$ measures the probability of the classifier to preserve the original prediction of the noise-free example $\\psi ( \\mathbf { x } )$ after adding noise, $\\psi ( \\mathbf { x } + \\mathbf { n } )$ , from distribution $\\mathcal { D }$ . Therefore, the robustness over a testing dataset $\\tau$ can be defined as the expected robustness over the test dataset: $\\Re _ { \\mathcal { D } } ( \\mathcal { T } ) = \\mathbb { E } _ { \\mathbf { x } \\sim \\mathcal { T } } \\left[ \\Re _ { \\mathcal { D } } ( \\mathbf { x } ) \\right]$ . Inspired by Franceschi et al. (2018), for ease, we relax Equation 4 from the probability of preserving the prediction score to a 0/1 robustness over $m$ - randomly sampled examples from $\\mathcal { D }$ . That is, $\\Re _ { \\mathcal { D } } ( { \\bf x } ) = 1$ means that, among $m$ randomly sampled noise from $\\mathcal { D }$ added to $\\mathbf { x }$ , none changed the prediction from $\\psi ( \\mathbf { x } )$ . However, if a single example of these $m$ samples changed the prediction from $\\psi ( \\mathbf { x } )$ , we set $\\Re _ { \\mathcal { D } } ( { \\bf x } ) = 0$ . Thus, the robustness score is the average of this measure over the testing dataset $\\tau$ . ",
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"text": "Robustness Against Gaussian Noise. For additive Gaussian noise, i.e. $\\mathcal { D } ~ = ~ \\mathcal { N } ( \\mathbf { 0 } , \\Sigma _ { x } ~ =$ Diag $\\left( \\sigma _ { x } ^ { 2 } \\right)$ ), robustness is averaged over a range of testing variances $\\sigma _ { x } ^ { 2 }$ . We restrict $\\sigma _ { x }$ to 30 evenly sampled values in $[ 0 , 0 . 5 ]$ , where this set is denoted as $\\mathcal { A } ^ { 1 }$ . In practice, this is equivalent to sampling $m$ Gaussian examples for each $\\sigma _ { x } \\in { \\mathcal { A } }$ , and if none of the $m$ samples changes the prediction of the classifier $\\psi$ from the original noise-free example, the robustness for that sample at that $\\sigma _ { x }$ noise level is set to 1 and then averaged over the complete testing set. Then, the robustness is the average over multiple $\\sigma _ { x } \\in { \\mathcal { A } }$ . To make the computation even more efficient, instead of sampling a large number of Gaussian noise samples $( m )$ , we only sample a single noise sample with the average√ energy over $\\mathcal { D }$ . That is, we sample a single $\\mathbf { n }$ of norm $\\mathbf { \\bar { \\mathbf { \\rho } } } _ { \\| \\mathbf { n } \\| _ { 2 } } = \\mathbf { \\bar { \\sigma } } \\sigma _ { x } \\sqrt { n }$ . This is due to the fact that ",
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"text": "Experimental Setup. In this section, we demonstrate the effectiveness of the proposed regularizer in improving robustness. Several experiments are performed with our objective Equation 2, where we strike a comparison with data augmentation approaches. ",
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"text": "Architecture Details. The input images in MNIST (gray-scale) and CIFAR (colored) are squares with sides equal to 28 and 32, respectively. Since AlexNet was originally trained on ImageNet of sides equal to 224, we will marginally alter the implementation of AlexNet in TorchVision Marcel & Rodriguez (2010) to accommodate for this difference. First, we change the number of hidden units in the first fully-connected layer (in LeNet to 4096, AlexNet to 256, LeNet on MNIST to 3136). For AlexNet, we changed all pooling kernel sizes from 3 to 2 and the padding size of conv1 from 2 to 5. Second, we swapped each maxpool with the preceding ReLU, which makes training and inference more efficient. Third, we enforce that the first layer in all the models is a convolution followed by ReLU as discussed earlier. Lastly, to simplify analysis, we removed all dropout layers. We leave the details of the optimization hyper-parameters to the appendix. ",
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"image_caption": [
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"Figure 3: Fair robustness comparison of LeNet with data augmentation and our regularizer. We only report results for models with a test accuracy that is at least as good as the accuracy of the baseline with a tolerance: $0 \\%$ , $0 . 3 9 \\%$ , and $0 . 7 5 \\%$ for MNIST, CIFAR10, CIFAR100, respectively. Only the models with the highest robustness are presented. Training with our regularizer can attain similar/better robustness than 21-fold noisy data augmentation on MNIST and CIFAR100, while maintaining a high noise-free test accuracy. "
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"page_idx": 6
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{
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"type": "text",
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"text": "Results. For each model and dataset, we compare baseline models, i.e. models trained with noisefree data and without our regularization, with two others: one using data augmentation and another using our proposed regularizer. Each of the latter has two configurable variables: the level of noise controlled by $\\textstyle { \\mathcal { \\sigma } } _ { x } ^ { 2 }$ during training, and the amount of noise controlled by the trade-off coefficient $\\alpha$ in Equation 2 or $\\tilde { N }$ (number of added noisy training examples) in the case of augmentation. ",
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"bbox": [
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"page_idx": 6
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{
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"type": "text",
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"text": "Accuracy vs. Robustness. We start by demonstrating that data augmentation tends to improve the robustness, as captured by $\\Re ( \\mathcal { T } )$ over the test set, at the expense of decreasing the testing accuracy on the noise-free examples. Realizing this is essential for a fair comparison, as one would need to compare the robustness of networks that only have similar noise-free testing accuracies. To this end, we ran 60 training experiments with data augmentation on LeNet with three datasets (MNIST, CIFAR10, and CIFAR100), four augmentation levels $( \\tilde { N } \\in \\{ 2 , 6 , 1 1 , 2 1 \\} )$ , and five noise levels $( \\sigma _ { x } \\in \\mathcal { A } = \\{ 0 . 1 2 5 , 0 . 2 5 , 0 . 3 2 5 , 0 . 5 , 1 . 0 \\} )$ . In contrast, we ran robust training experiments using Equation 2 with the trade-off coefficient $\\overset { \\cdot } { \\alpha } \\in \\{ 0 . 5 , 1 , 1 . 5 , 2 , 5 , 1 0 , 2 0 \\}$ on the same datasets, but we extended the noise levels $\\sigma _ { x }$ to include the extreme noise regime of $\\sigma _ { x } \\in \\{ 2 , 5 , 1 0 , 2 0 \\}$ . These noise levels are too large to be used for data augmentation, especially since $\\mathbf { x } \\in [ 0 , 1 ] ^ { n }$ ; however, as we will see, they are still beneficial for our proposed regularizer. Figure 2 shows both the testing accuracy and robustness as measured by $\\Re ( \\mathcal { T } )$ over a varying range of training $\\sigma _ { x }$ for the data augmentation approach of LeNet on MNIST, CIFAR-10 and CIFAR-100. It is important to note here that the main goal of these plots is not to compare the robustness score, but rather, to demonstrate a very important trend. In particular, increasing the training $\\sigma _ { x }$ for each approach degrades testing accuracy on noise-free data. However, the degradation in our approach is much more graceful since the trained LeNet model was never directly exposed to individually corrupted examples during training as opposed to the data augmentation approach. Note that our regularizer enforces the separation between the expected output prediction analytically. Moreover, the robustness of both methods consistently improves as the training $\\sigma _ { x }$ increases. This trend holds even on the easiest dataset (MNIST). Interestingly, models trained with our regularizer enjoy an improvement in testing accuracy over the baseline model. Such behaviour only emerges with a large factor of augmentation, $\\tilde { N } = 2 \\bar { 1 }$ , and a small enough training $\\sigma _ { x }$ on MNIST. This indicates that models can benefit from better accuracy with a good approximation of Equation 1 through our proposed objective or through extensive Monte Carlo estimation. However, as $\\underset { \\cdots } { \\sigma } { _ { x } }$ increases, Monte Carlo estimates of the second term in Equation 1 via data augmentation (with $\\ddot { N } = 2 1$ ) is no longer enough to capture the noise. ",
|
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"bbox": [
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{
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"type": "text",
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| 571 |
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"text": "Robustness Comparison. For fair comparison, it is essential to only compare the robustness of networks that achieve similar testing accuracy, since perfect robustness is attainable with a deterministic classifier that assigns the same class label regardless of the input. In fact, we proposed a unified robustness metric for the reason that most commonly used metrics are disassociated from ",
|
| 572 |
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"bbox": [
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"page_idx": 6
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{
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"type": "table",
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"img_path": "images/0f9926d3d88c56381050e441973808a09f97d96d06512b2d0e21609720f6794b.jpg",
|
| 583 |
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"table_caption": [],
|
| 584 |
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"table_footnote": [],
|
| 585 |
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"table_body": "<table><tr><td></td><td>g</td><td>PGD</td><td>LBFGS</td><td>FGSM</td><td>DF2</td><td>GNR</td><td>ACC</td></tr><tr><td rowspan=\"6\">EE nenen JSINW</td><td>0</td><td>1.02 × 10-03</td><td>5.50×10-04</td><td>2.24×10-02</td><td>5.91×10-04</td><td>77.45</td><td>98.50</td></tr><tr><td>0.125</td><td>3.36×10-01</td><td>3.43×10-01</td><td>8.19 ×10-01</td><td>2.05×10-01</td><td>93.14</td><td>97.50</td></tr><tr><td>0.250</td><td>4.58×10-01</td><td>4.31×10-01</td><td>1.21</td><td>2.63×10-01</td><td>95.64</td><td>98.75</td></tr><tr><td>0.325</td><td>4.21×10-01</td><td>4.51 ×10-01</td><td>1.17</td><td>2.33×10-01</td><td>96.75</td><td>97.50</td></tr><tr><td>1.0</td><td>5.44×10-01</td><td>5.22×10-01</td><td>1.34</td><td>2.95×10-01</td><td>97.32</td><td>99.00</td></tr><tr><td></td><td></td><td></td><td>-05</td><td></td><td></td><td></td></tr><tr><td rowspan=\"5\">A neeae CEITIIIO</td><td>0</td><td>2.30×10-05</td><td>2.50 × 10-05</td><td>1.50 ×10</td><td>2.10×10-05</td><td>29.69</td><td>34.75</td></tr><tr><td>0.12</td><td>3.64×10-04</td><td>2.83×10-04</td><td>5.06×10-04</td><td>2.16×10-04</td><td>31.65</td><td>33.50</td></tr><tr><td>0.250</td><td>4.37 ×10-04</td><td>3.86×10-04</td><td>6.50×10-04</td><td>2.47×10-04</td><td>32.85</td><td>32.25</td></tr><tr><td>0.325</td><td>5.37×10-04</td><td>4.04×10-04</td><td>7.29 ×10-04</td><td>3.18×10-04</td><td>33.84</td><td>34.25</td></tr><tr><td>1.0</td><td>4.92×10-04</td><td>3.26×10-04</td><td>6.50×10-04</td><td>2.85×10-04</td><td>34.65</td><td>35.50</td></tr></table>",
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| 586 |
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"page_idx": 7
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},
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| 594 |
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{
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| 595 |
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"type": "text",
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| 596 |
+
"text": "Table 1: Gaussian robustness improves overall robustness. We report the robustness metrics corresponding to various attacks (PGD, LBFGS, FGSM, and DF2), our proposed GNR metric, and the test accuracy ACC for LeNet and AlexNet networks trained on MNIST and CIFAR100 using our proposed regularizer with noise variance $\\sigma$ in training. Note that $\\sigma = 0$ corresponds to baseline models trained without our regularizer. We observe that training networks with our proposed regularizer (designed for additive Gaussian attacks) not only improves the robustness against Gaussian attacks but also against 6 other types of attacks which 4 of them listed here and the others are left for appendix. ",
|
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"bbox": [
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"page_idx": 7
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{
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"type": "text",
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| 607 |
+
"text": "the ground-truth labels and only consider model predictions. Therefore, we filtered out the results from Figure 2 by removing all the experiments that achieved lower test accuracy than the baseline model. Figure 3 summarizes these results for LeNet. Now, we can clearly see the difference between training with data augmentation and our approach. For MNIST (Figure 3a), we achieved the same robustness as 21-fold data augmentation without feeding the network with any noisy examples during training and while preserving the same baseline accuracy. Interestingly, for CIFAR10 (Figure 3b), our method is twice as robust as the best robustness achieved via data augmentation. Moreover, for CIFAR100 (Figure 3c), we are able to outperform data augmentation by around $5 \\%$ . Finally, for extra validation, we also conducted the same experiments with AlexNet on CIFAR10 and CIFAR100 which can be found in the appendix. We can see that our proposed regularizer can improve robustness by $1 5 \\%$ on CIFAR10 and around $2 5 \\%$ on CIFAR100. It is interesting to note that for CIFAR10, data augmentation could not improve the robustness of the trained models without drastically degrading the testing accuracy on the noise-free examples. Moreover, it is interesting to observe that the best robustness achieved through data augmentation is even worse than the baseline. This could be due to the trade-off coefficient $\\alpha$ in Equation 1. ",
|
| 608 |
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"bbox": [
|
| 609 |
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| 611 |
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],
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"page_idx": 7
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},
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{
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| 617 |
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"type": "text",
|
| 618 |
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"text": "Towards General Robustness via Gaussian Robustness. Here, we investigate whether improving robustness to Gaussian input noise can improve robustness against other types of attacks. Specifically, we compare the robustness of models trained using our proposed regularizer (robust again Gaussian attacks) with baseline models subject to different types of attacks: Projected Gradient Descent (PGD) and LBFGS attacks Szegedy et al. (2014), Fast Sign Gradient Method (FGSM) Goodfellow et al. (2015), and DeepFool L2Attack (DF2) Moosavi-Dezfooli et al. (2016) as provide by Rauber et al. (2017). For all these attacks, we report the minimum energy perturbation that can change the network prediction. We also report our Gaussian Network Robustness (GNR) metric, which is the Gaussian version of Equation 4 along with the testing accuracy (ACC). We perform experiments on LeNet on MNIST, CIFAR10 and CIFAR100 datasets and on AlexNet on both CIFAR10 and CIFAR100. Due to space constraints, we show the robustness results for only LeNet on MNIST and AlexNet of CIFAR100 and leave the rest along with two other types of attacks for the appendix. Table 1 shows that improving GNR through our data augmentation free regularizer can significantly improve all robustness metrics. For instance, comparing LeNet trained with our proposed regularizer against LeNet trained without any regularization, i.e. $\\sigma = 0$ , we see that robustness against all types of attacks improves by almost two orders of magnitude, while maintaining a similar testing accuracy. A similar improvement in performance is consistently present for AlexNet on CIFAR100. ",
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},
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{
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"type": "text",
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"text": "5 CONCLUSION ",
|
| 630 |
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"text_level": 1,
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"bbox": [
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"page_idx": 7
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{
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"type": "text",
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"text": "Addressing the sensitivity problem of deep neural networks to adversarial perturbation is of great importance to the machine learning community. However, building robust classifiers against this noises is computationally expensive, as it is generally done through the means of data augmentation. We propose a generic lightweight analytic regularizer, which can be applied to any deep neural network with a ReLU activation after the first affine layer. It is designed to increase the robustness of the trained models under additive Gaussian noise. We demonstrate this with multiple architectures and datasets and show that it outperforms data augmentation without observing any noisy examples. ",
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"type": "text",
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"text": "REFERENCES ",
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"text": "A EXPERIMENTAL SETUP AND DETAILS. ",
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"text": "All experiments, are conducted using PyTorch version 0.4.1 Paszke et al. (2017). All hyperparameters are fixed and Table 2 we report the setup for the two optimizers. In particular, we use the Adam optimizaer Kingma & Ba (2015) with $\\beta _ { 1 } = 0 . 9 , \\beta _ { 2 } = 0 . 9 9 9$ , $\\epsilon = 1 0 ^ { \\div 8 }$ with amsgrad set to False. The second optimizer is SGD Loshchilov & Hutter (2017) with momentum $\\scriptstyle 1 = 0 . 9$ , dampening $= 0$ , with Nesterov acceleration. In each experiment, we randomly split the training dataset into $10 \\%$ validation and $90 \\%$ training and monitor the validation loss after each epoch. If validation loss did not improve for lr patience epochs, we reduce the learning rate by multiplying it by lr factor. We start with an initial learning rate of lr initial. The training is terminated only if the validation loss did not improve for loss patience number of epochs or if the training reached 100 epochs. We report the results of the model with the best validation loss. ",
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"table_caption": [
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"Table 2: Lists the training optimization hyper-parameters. "
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"table_body": "<table><tr><td rowspan=1 colspan=6>Hyper-parameter</td><td rowspan=1 colspan=1>LeNet</td><td rowspan=1 colspan=1>AlexNet</td></tr><tr><td rowspan=7 colspan=6>optimizerminibatch_sizelr_initiallr_patiencelr_factorloss-patienceweight_decay</td><td rowspan=1 colspan=1>er</td><td rowspan=1 colspan=1>Adam</td></tr><tr><td rowspan=1 colspan=2>oatch_si2</td><td rowspan=1 colspan=1>e</td><td rowspan=1 colspan=1>1000</td><td rowspan=1 colspan=1>128</td></tr><tr><td rowspan=1 colspan=1>0.0001</td><td rowspan=1 colspan=1>0.1</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.5</td></tr><tr><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>20</td></tr><tr><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0.0005</td></tr></table>",
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"text": "B EXAMPLES ON NOISE LEVELS ",
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"text": "Figure 4 provides examples of the different levels of noise on a given digit 8. ",
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"image_caption": [
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"Figure 4: This Figure shows an example of the noise level over varying level of input $\\sigma$ on the digit 8. In particular, one can observe that with $\\sigma$ large than 0.7 the among of noise is severe even for the human level. Training on such extreme noise levels will deem data augmentation to be difficult. "
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"text": "C A COMMENT ON THE ROBUSTNESS METRIC ",
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"text": "We measure the robustness against Gaussian noise by averaging over a range of input noise levels, where at each level for each image, we consider it misclassified if the probability of it being misclassified is greater than a certain threshold. The final robustness is the average over multiple testing $\\sigma _ { x }$ . This is special case of the more general case in Equation (4). We then report the area under the curve of the robustness with varying testing $\\sigma _ { x }$ as shown in Figure 6. The area under this curve thus represents the overall robustness of a given model under several varying input noise standard deviation $\\sigma _ { x }$ . ",
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"text": "D OTHER ROBUSTNESS METRICS ",
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"text": "We report the robustness of several architectures over several datasets with and without our trained regularizer. We show that our proposed efficient regularizer not only improves the robustness against Gaussin noise attacks but againts several other types of attacks. Table 3 summarizes the types of attacks used for robustness evaluation. ",
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"image_caption": [
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"Figure 5: The robustness is a function of the ratio of the orange area to the blue area in the white circle. "
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"image_caption": [
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"Figure 6: The robustness is thus measured as the area under the curve of testing accuracy versus input noise level (standard deviation). "
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{
|
| 1190 |
+
"type": "image",
|
| 1191 |
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"img_path": "images/a2d167a08a6826f73d37181210759353dbe7f0df93f0aed505e01aa2f76f517c.jpg",
|
| 1192 |
+
"image_caption": [
|
| 1193 |
+
"Figure 7: Fair robustness comparison of AlexNet with data augmentation and our regularizer. The reported models trained with our regularizer on CIFAR10 and CIFAR100 on all training $\\sigma _ { x }$ are within $1 . 6 8 \\%$ and $4 . 8 3 \\%$ of the baseline accuracy, respectively. The models trained with the proposed regularizer achieve better robustness than 11-fold and 6-fold noisy data augmentation on CIFAR10 and CIFAR100, respectively. "
|
| 1194 |
+
],
|
| 1195 |
+
"image_footnote": [],
|
| 1196 |
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"bbox": [
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| 1197 |
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|
| 1201 |
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|
| 1202 |
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"page_idx": 11
|
| 1203 |
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},
|
| 1204 |
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{
|
| 1205 |
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"type": "table",
|
| 1206 |
+
"img_path": "images/d718f186b7c0959fc34400f6ac47535eacd7c38254d7e5a5d3bb51498a3230f2.jpg",
|
| 1207 |
+
"table_caption": [
|
| 1208 |
+
"Table 3: The table lists all the attacks performed. "
|
| 1209 |
+
],
|
| 1210 |
+
"table_footnote": [],
|
| 1211 |
+
"table_body": "<table><tr><td>Attack Abbreviation</td><td>AttackName</td></tr><tr><td>PGD LBF</td><td>Projected Gradient Descent</td></tr><tr><td>GSM</td><td>LBFGS Attack FGSM</td></tr><tr><td>AGA</td><td>Additive Gaussian Noise Attack</td></tr><tr><td>AUA</td><td>AdditiveUniformNoiseAttack</td></tr><tr><td>DF2</td><td>DeepFool l2 Attack</td></tr></table>",
|
| 1212 |
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"bbox": [
|
| 1213 |
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|
| 1214 |
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|
| 1215 |
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| 1216 |
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|
| 1217 |
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],
|
| 1218 |
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"page_idx": 11
|
| 1219 |
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},
|
| 1220 |
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{
|
| 1221 |
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"type": "table",
|
| 1222 |
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"img_path": "images/2e6f596bb0055e34c5ecc5cc3891221907aeb22fde72bac7190aa8e5978ce36c.jpg",
|
| 1223 |
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"table_caption": [],
|
| 1224 |
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"table_footnote": [
|
| 1225 |
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"corresponds to baseline models trained without our regularizer. We observe that training networks with our proposed reg "
|
| 1226 |
+
],
|
| 1227 |
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"table_body": "<table><tr><td rowspan=1 colspan=1>ACC</td><td rowspan=1 colspan=1>09'86598609:2600'66</td><td rowspan=1 colspan=1>30.99122255</td><td rowspan=1 colspan=1>78.285850358398986860</td><td rowspan=1 colspan=1>730:2900'2930.5506'99</td><td rowspan=1 colspan=1>3530322533550</td></tr><tr><td rowspan=1 colspan=1>GNN</td><td rowspan=1 colspan=1>24225317599657326</td><td rowspan=1 colspan=1>2077800039.2031405</td><td rowspan=1 colspan=1>E87242706:00</td><td rowspan=1 colspan=1>353105098.29</td><td rowspan=1 colspan=1>696753368505820953</td></tr><tr><td rowspan=1 colspan=1>P</td><td rowspan=1 colspan=1>10-01X169 10-01X89710-01038720-01 X067</td><td rowspan=1 colspan=1>£0-013£0-0011X591 11×590×86[</td><td rowspan=1 colspan=1>×1333 ×81I15</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>0 10-0I X 81820-0131801 X917</td></tr><tr><td rowspan=1 colspan=1>AAA</td><td rowspan=1 colspan=1>20-01 X 65780%68333</td><td rowspan=1 colspan=1>10-0I X 66'910-0I X08°2 1</td><td rowspan=1 colspan=1>10-01 × 00'910-0I X 9694443</td><td rowspan=1 colspan=1>10-0I X86*210-01 X278</td><td rowspan=1 colspan=1>10-01 X39110-01X107</td></tr><tr><td rowspan=1 colspan=1>AAA</td><td rowspan=1 colspan=1>20-01 X 8975737</td><td rowspan=1 colspan=1>£0-01X29710-01 X82'910-0I X 26210-0I × ∠9'990'[</td><td rowspan=1 colspan=1>10-01 X 90'9I0-0IX80'9TO-OI×1</td><td rowspan=1 colspan=1>10-0I X892</td><td rowspan=1 colspan=1>20-013335 10-0I X 66T</td></tr><tr><td rowspan=1 colspan=1>SSS</td><td rowspan=1 colspan=1>20-012710-0I X 61'81215184</td><td rowspan=1 colspan=1>£0-0I X291 20-012755</td><td rowspan=1 colspan=1>25-21225035-213250XI67</td><td rowspan=1 colspan=1>0- 20-0IX58%2133508</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>ERT</td><td rowspan=1 colspan=1>10-011000 10-01X 11110-01X 111</td><td rowspan=1 colspan=1>90-013830 20-01100030--0[ × 99'[</td><td rowspan=1 colspan=1>£0-01 X 79.011-111150 £0-11 3 39:51</td><td rowspan=1 colspan=1>10-05[1 70</td><td rowspan=1 colspan=1>10- 50-013 950233 330</td></tr><tr><td rowspan=1 colspan=1>PPG</td><td rowspan=1 colspan=1>£0-01X70110-0I X89510-01X110-01X1</td><td rowspan=1 colspan=1>90-01X218 20-11270 20-113330</td><td rowspan=1 colspan=1>90-01X2£0-01 X 22720-010580£0-01 X 9555</td><td rowspan=1 colspan=1>10-111105</td><td rowspan=1 colspan=1>20-01X005 10-0132210-111 2304-11 1 76.5</td></tr><tr><td rowspan=1 colspan=1>b</td><td rowspan=1 colspan=1>10500000976001</td><td rowspan=1 colspan=1>10110000970001</td><td rowspan=1 colspan=1>0051297600</td><td rowspan=1 colspan=1>00509760</td><td rowspan=1 colspan=1>10000801</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>LSINNuoGNe</td><td rowspan=1 colspan=1>01TIITuo</td><td rowspan=1 colspan=1>00ICTIAITuoGNe</td><td rowspan=1 colspan=1>CIIIIIIuoJEere</td><td rowspan=1 colspan=1>001CTIAITuo</td></tr></table>",
|
| 1228 |
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"bbox": [
|
| 1229 |
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| 1230 |
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|
| 1231 |
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| 1232 |
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|
| 1233 |
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],
|
| 1234 |
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"page_idx": 12
|
| 1235 |
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}
|
| 1236 |
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]
|
parse/train/B1xDq2EFDH/B1xDq2EFDH_middle.json
ADDED
|
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See raw diff
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|
parse/train/B1xDq2EFDH/B1xDq2EFDH_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
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|
parse/train/BJxhijAcY7/BJxhijAcY7.md
ADDED
|
@@ -0,0 +1,543 @@
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|
| 1 |
+
# SIGNSGD WITH MAJORITY VOTE IS COMMUNICATION EFFICIENT AND FAULT TOLERANT
|
| 2 |
+
|
| 3 |
+
Jeremy Bernstein1∗, Jiawei Zhao12∗, Kamyar Azizzadenesheli3, Anima Anandkumar1 1Caltech, 2Nanjing University of Aeronautics and Astronautics, 3UC Irvine bernstein@caltech.edu, jiaweizhao@nuaa.edu.cn, kazizzad@uci.edu, anima@caltech.edu
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
Training neural networks on large datasets can be accelerated by distributing the workload over a network of machines. As datasets grow ever larger, networks of hundreds or thousands of machines become economically viable. The time cost of communicating gradients limits the effectiveness of using such large machine counts, as may the increased chance of network faults. We explore a particularly simple algorithm for robust, communication-efficient learning—SIGNSGD. Workers transmit only the sign of their gradient vector to a server, and the overall update is decided by a majority vote. This algorithm uses $3 2 \times$ less communication per iteration than full-precision, distributed SGD. Under natural conditions verified by experiment, we prove that SIGNSGD converges in the large and mini-batch settings, establishing convergence for a parameter regime of ADAM as a byproduct. Aggregating sign gradients by majority vote means that no individual worker has too much power. We prove that unlike SGD, majority vote is robust when up to $50 \%$ of workers behave adversarially. The class of adversaries we consider includes as special cases those that invert or randomise their gradient estimate. On the practical side, we built our distributed training system in Pytorch. Benchmarking against the state of the art collective communications library (NCCL), our framework—with the parameter server housed entirely on one machine—led to a $2 5 \%$ reduction in time for training resnet50 on Imagenet when using 15 AWS $\mathtt { p 3 . 2 x 1 }$ arge machines.
|
| 8 |
+
|
| 9 |
+
# 1 INTRODUCTION
|
| 10 |
+
|
| 11 |
+
The most powerful supercomputer in the world is currently a cluster of over 27,000 GPUs at Oak Ridge National Labs (TOP500, 2018). Distributed algorithms designed for such large-scale systems typically involve both computation and communication: worker nodes compute intermediate results locally, before sharing them with their peers. When devising new machine learning algorithms for distribution over networks of thousands of workers, we posit the following desiderata:
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D1 fast algorithmic convergence; D3 communication efficiency;
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D2 good generalisation performance; D4 robustness to network faults.
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When seeking an algorithm that satisfies all four desiderata D1–4, inevitably some tradeoff must be made. Stochastic gradient descent (SGD) naturally satisfies D1–2, and this has buoyed recent advances in deep learning. Yet when it comes to large neural network models with hundreds of millions of parameters, distributed SGD can suffer large communication overheads. To make matters worse, any faulty SGD worker can corrupt the entire model at any time by sending an infinite gradient, meaning that SGD without modification is not robust.
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A simple algorithm with aspirations towards all desiderata D1–4 is as follows: workers send the sign of their gradient up to the parameter server, which aggregates the signs and sends back only the majority decision. We refer to this algorithm as SIGNSGD with majority vote. All communication to and from the parameter server is compressed to one bit, so the algorithm certainly gives us D3. What’s more, in deep learning folklore sign based methods are known to perform well, indeed inspiring the popular RMSPROP and ADAM optimisers (Balles & Hennig, 2018), giving hope for D1. As far as robustness goes, aggregating gradients by a majority vote denies any individual worker too much power, suggesting it may be a natural way to achieve D4.
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Figure 1: Toy experiments. SIGNSGD with majority vote is run on a 1000-dimensional quadratic with $\mathcal { N } ( 0 , 1 )$ noise added to each gradient component. Adversarial experiments are run with 27 total workers. These plots may be reproduced in a web browser by running this Jupyter notebook.
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In this work, we make the above aspirations rigorous. Whilst D3 is immmediate, we provide the first convergence guarantees for SIGNSGD in the mini-batch setting, providing theoretical grounds for D1. We show how theoretically the behaviour of SIGNSGD changes as gradients move from high to low signal-to-noise ratio. We also extend the theory of majority vote to show that it achieves a notion of Byzantine fault tolerance. A distributed algorithm is Byzantine fault tolerant (Blanchard et al., 2017) if its convergence is robust when up to $50 \%$ of workers behave adversarially. The class of adversaries we consider contains interesting special cases, such as robustness to a corrupted worker sending random bits, or a worker that inverts their gradient estimate. Though our adversarial model is not the most general, it is interesting as a model of network faults, and so gives us D4.
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Next, we embark on a large-scale empirical validation of our theory. We implement majority vote in the Pytorch deep learning framework, using CUDA kernels to bit pack sign tensors down to one bit. Our results provide experimental evidence for D1–D4. Comparing our framework to NCCL (the state of the art communications library), we were able to speed up Imagenet training by $2 5 \%$ when distributing over 7 to 15 AWS p3.2xlarge machines, albeit at a slight loss in generalisation.
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Finally, in an interesting twist, the theoretical tools we develop may be brought to bear on a seemingly unrelated problem in the machine learning literature. Reddi et al. (2018) proved that the extremely popular ADAM optimiser in general does not converge in the mini-batch setting. This result belies the success of the algorithm in a wide variety of practical applications. SIGNSGD is equivalent to a special case of ADAM, and we establish the convergence rate of mini-batch SIGNSGD for a large class of practically realistic objectives. Therefore, we expect that these tools should carry over to help understand the success modes of ADAM. Our insight is that gradient noise distributions in practical problems are often unimodal and symmetric because of the Central Limit Theorem, yet Reddi et al. (2018)’s construction relies on bimodal noise distributions.
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# 2 RELATED WORK
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For decades, neural network researchers have adapted biologically inspired algorithms for efficient hardware implementation. Hopfield (1982), for example, considered taking the sign of the synaptic weights of his memory network for readier adaptation into integrated circuits. This past decade, neural network research has focused on training feedforward networks by gradient descent (LeCun et al., 2015). It is natural to ask what practical efficiency may accompany simply taking the sign of the backpropagated gradient. In this section, we explore related work pertaining to this question.
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Deep learning: whilst stochastic gradient descent (SGD) is the workhorse of machine learning (Robbins & Monro, 1951), algorithms like RMSPROP (Tieleman & Hinton, 2012) and ADAM (Kingma & Ba, 2015) are also extremely popular neural net optimisers. These algorithms have their roots in the RPROP optimiser (Riedmiller & Braun, 1993), which is a sign-based method similar to SIGNSGD except for a component-wise adaptive learning rate.
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<table><tr><td>Algorithm 1 SIGNUM with majority vote,the proposed algorithm for distributed optimisation. Good default setings for the tested machine learning problems are n = 0.Oool and β = O.9, though tuning is recommended.All operations on vectors are element-wise. Settng β = O yields SIGNSGD.</td></tr><tr><td>Require: learning rate n > O, momentum constant β ∈ [0,1),weight decay 入 ≥ 0, mini-batch size n,initial point x held by each of M workers,initial momentum Um ← O on mth worker</td></tr><tr><td>repeat on mth worker</td></tr><tr><td>gm← 1 stochasticGradient(x) >mini-batch gradient n i=1</td></tr><tr><td>Um↑ (1-β)gm+βvm >update momentum push sign(Um) to server > send sign momentum</td></tr><tr><td>on server M</td></tr><tr><td>V← sign(Um) V aggregate sign momenta m push sign(V) to each worker 二 >broadcast majority vote</td></tr><tr><td>on every worker</td></tr><tr><td>x←x-n(sign(V)+入x) > update parameters</td></tr><tr><td>until convergence</td></tr><tr><td></td></tr></table>
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Non-convex optimisation: parallel to (and oftentimes in isolation from) advances in deep learning practice, a sophisticated optimisation literature has developed. Nesterov & Polyak (2006) proposed cubic regularisation as an algorithm that can escape saddle points and provide guaranteed convergence to local minima of non-convex functions. This has been followed up by more recent works such as NATASHA (Allen-Zhu, 2017) that use other theoretical tricks to escape saddle points. It is still unclear how relevant these works are to deep learning, since it is not clear to what extent saddle points are an obstacle in practical problems. We avoid this issue altogether and satisfy ourselves with establishing convergence to critical points.
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Gradient compression: prior work on gradient compression generally falls into two camps. In the first camp, algorithms like QSGD (Alistarh et al., 2017), TERNGRAD (Wen et al., 2017) and ATOMO (Wang et al., 2018) use stochastic quantisation schemes to ensure that the compressed stochastic gradient remains an unbiased approximation to the true gradient. These works are therefore able to bootstrap existing SGD convergence theory. In the second camp, more heuristic algorithms like 1BITSGD (Seide et al., 2014) and deep gradient compression (Lin et al., 2018) pay less attention to theoretical guarantees and focus more on practical performance. These algorithms track quantisation errors and feed them back into subsequent updates. The commonality between the two camps is an effort to, one way or another, correct for bias in the compression.
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SIGNSGD with majority vote takes a different approach to these two existing camps. In directly employing the sign of the stochastic gradient, the algorithm unabashedly uses a biased approximation of the stochastic gradient. Carlson et al. (2016) and Bernstein et al. (2018) provide theoretical and empirical evidence that signed gradient schemes can converge well in spite of their biased nature. Their theory only applies in the large batch setting, meaning the theoretical results are less relevant to deep learning practice. Still Bernstein et al. (2018) showed promising experimental results in the mini-batch setting. An appealing feature of majority vote is that it naturally leads to compression in both directions of communication between workers and parameter server. As far as we are aware, all existing gradient compression schemes lose compression before scattering results back to workers.
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Byzantine fault tolerant optimisation: the problem of modifying SGD to make it Byzantine fault tolerant has recently attracted interest in the literature (Yin et al., 2018). For example, Blanchard et al. (2017) proposed KRUM, which operates by detecting and excluding outliers in the gradient aggregation. Alistarh et al. (2018) propose BYZANTINESGD which instead focuses on detecting and eliminating adversaries. Clearly both these strategies incur overheads, and eliminating adversaries precludes the possibility that they might reform. El Mhamdi et al. (2018) point out that powerful adversaries may steer convergence to bad local minimisers. We see majority vote as a natural way to protect against less malign faults such as network errors, and thus satisfy ourselves with convergence guarantees to critical points without placing guarantees on their quality.
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Figure 2: Gradient distributions for resnet18 on Cifar-10 at mini-batch size 128. At the start of epochs 0, 1 and 5, we do a full pass over the data and collect the gradients for three randomly chosen weights (left, middle, right). In all cases the distribution is close to unimodal and symmetric.
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# 3 THEORY
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# 3.1 ASSUMPTIONS
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We aim to develop an optimisation theory that is relevant for real problems in deep learning. For this reason, we are careful about the assumptions we make. For example, we do not assume convexity because neural network loss functions are typically not convex. Though we allow our objective function to be non-convex, we insist on a lower bound to enable meaningful convergence results.
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+
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Assumption 1 (Lower bound). For all $x$ and some constant $f ^ { * }$ , we have objective value $f ( x ) \geq f ^ { * }$
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+
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+
Our next two assumptions of Lipschitz smoothness and bounded variance are standard in the stochastic optimisation literature (Allen-Zhu, 2017). That said, we give them in a component-wise form. This allows our convergence results to encode information not just about the total noise level and overall smoothness, but also about how these quantities are distributed across dimension.
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+
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Assumption 2 (Smooth). Let $g ( x )$ denote the gradient of the objective $f ( . )$ evaluated at point $x$ . Then $\forall x , y$ we require that for some non-negative constant $\vec { L } : = [ L _ { 1 } , . . . , L _ { d } ]$
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+
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| 60 |
+
$$
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+
\Big | f ( y ) - \big [ f ( x ) + g ( x ) ^ { T } ( y - x ) \big ] \Big | \leq \frac { 1 } { 2 } \sum _ { i } L _ { i } ( y _ { i } - x _ { i } ) ^ { 2 } .
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+
$$
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+
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+
Assumption 3 (Variance bound). Upon receiving query $x \in \mathbb { R } ^ { d }$ , the stochastic gradient oracle gives us an independent, unbiased estimate $\tilde { g }$ that has coordinate bounded variance:
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+
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+
$$
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\begin{array} { r } { \mathbb { E } [ \tilde { g } ( x ) ] = g ( x ) , \qquad \mathbb { E } \left[ ( \tilde { g } ( x ) _ { i } - g ( x ) _ { i } ) ^ { 2 } \right] \leq \sigma _ { i } ^ { 2 } } \end{array}
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+
$$
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| 69 |
+
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+
for a vector of non-negative constants $\vec { \sigma } : = [ \sigma _ { 1 } , . . , \sigma _ { d } ]$ .
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+
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Our final assumption is non-standard. We assume that the gradient noise is unimodal and symmetric. Clearly, Gaussian noise is a special case. Note that even for a moderate mini-batch size, we expect the central limit theorem to kick in rendering typical gradient noise distributions close to Gaussian. See Figure 2 for noise distributions measured whilst training resnet18 on Cifar-10.
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+
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Assumption 4 (Unimodal, symmetric gradient noise). At any given point $x$ , each component of the stochastic gradient vector $\tilde { g } ( x )$ has a unimodal distribution that is also symmetric about the mean.
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+
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Showing how to work with this assumption is a key theoretical contribution of this work. Combining Assumption 4 with an old tail bound of Gauss (1823) yields Lemma 1, which will be crucial for guaranteeing mini-batch convergence of SIGNSGD. As will be explained in Section 3.3, this result also constitutes a convergence proof for a parameter regime of ADAM. This suggests that Assumption 4 may more generally be a theoretical fix for Reddi et al. (2018)’s non-convergence proof of mini-batch ADAM, a fix which does not involve modifying the ADAM algorithm itself.
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+
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+
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Figure 3: Signal-to-noise ratio (SNR) whilst training resnet18 on Cifar-10 at batch size 128. At the start of each epoch we compute the SNR for every gradient component. We plot summary statistics like the mean over weights and the max. By roughly epoch 40, all gradient components have passed below the critical line (see Theorem 1) and remain there for the rest of training.
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+
# 3.2 MINI-BATCH CONVERGENCE OF SIGNSGD
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With our assumptions in place, we move on to presenting our theoretical results, which are all proved in Appendix C. Our first result establishes the mini-batch convergence behaviour of SIGNSGD. We will first state the result and make some remarks. We provide intuition for the proof in Section 3.3.
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Theorem 1 (Non-convex convergence rate of mini-batch SIGNSGD). Run the following algorithm for $K$ iterations under Assumptions $^ { l }$ to $^ { 4 }$ : $x _ { k + 1 } = x _ { k } - \eta \mathrm { s i g n } ( \tilde { g } _ { k } )$ . Set the learning rate, $\eta _ { ; }$ , and mini-batch size, $n$ , as
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+
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+
$$
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+
\eta = \sqrt { \frac { f _ { 0 } - f _ { * } } { \| \vec { L } \| _ { 1 } K } } , \qquad n = 1 .
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+
$$
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+
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+
Let $H _ { k }$ be the set of gradient components at step $k$ with large signal-to-noise ratio $\begin{array} { r } { S _ { i } : = \frac { | g _ { k , i } | } { \sigma _ { i } } } \end{array}$ , i.e. $H _ { k } : = { \Big \{ } i { \Big | } S _ { i } > { \frac { 2 } { \sqrt { 3 } } } { \Big \} }$ . We refer to $\textstyle { \frac { 2 } { \sqrt { 3 } } }$ as the ‘critical SNR’. Then we have
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+
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+
$$
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+
\frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left[ \sum _ { i \in { \cal H } _ { k } } \left. g _ { k , i } \right. + \sum _ { i \notin { \cal H } _ { k } } \frac { g _ { k , i } ^ { 2 } } { \sigma _ { i } } \right] \leq 3 \sqrt { \frac { \| \vec { L } \| _ { 1 } ( f _ { 0 } - f _ { * } ) } { N } } .
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+
$$
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+
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+
where $N = K$ is the total number of stochastic gradient calls up to step $K$ .
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+
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+
Theorem 1 provides a bound on the average gradient norm. The right hand side of the bound decays like O $\left( \textstyle { \frac { 1 } { \sqrt { N } } } \right)$ , establishing convergence to points of the objective where the gradient vanishes.
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+
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+
Remark 1: mini-batch SIGNSGD attains the same O $\scriptstyle \left( { \frac { 1 } { \sqrt { N } } } \right)$ non-convex convergence rate as SGD.
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+
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+
Remark 2: the gradient appears as a mixed norm: an $\ell _ { 1 }$ norm for high SNR components, and a weighted $\ell _ { 2 }$ norm for low SNR compoenents.
|
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+
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| 105 |
+
Remark 3: we wish to understand the dimension dependence of our bound. We may simplify matters by assuming that, during the entire course of optimisation, every gradient component lies in the low SNR regime. Figure 3 shows that this is almost true when training a resnet18 model. In this limit, the bound becomes:
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+
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+
$$
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+
\frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left[ \sum _ { i = 1 } ^ { d } \frac { g _ { k , i } ^ { 2 } } { \sigma _ { i } } \right] \leq 3 \sqrt { \frac { \| \vec { L } \| _ { 1 } ( f _ { 0 } - f _ { * } ) } { N } } .
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+
$$
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+
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+
Further assume that we are in a well-conditioned setting, meaning that the variance is distributed uniformly across dimension $\begin{array} { r } ( \sigma _ { i } ^ { 2 } = \frac { \sigma ^ { 2 } } { d } \ \end{array}$ ), and every weight has the same smoothness constant $\boldsymbol { L } _ { i } =$ $L$ ). $\sigma ^ { 2 }$ is the total variance bound, and $L$ is the conventional Lipschitz smoothness. These are the
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+
|
| 113 |
+
quantities which appear in the standard analysis of SGD. Then we get
|
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+
|
| 115 |
+
$$
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+
\frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \| g _ { k } \| _ { 2 } ^ { 2 } \leq 3 \sigma \sqrt { \frac { L ( f _ { 0 } - f _ { * } ) } { N } } .
|
| 117 |
+
$$
|
| 118 |
+
|
| 119 |
+
The factors of dimension $d$ have conveniently cancelled. This illustrates that there are problem geometries where mini-batch SIGNSGD does not pick up an unfavourable dimension dependence.
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+
|
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+
# 3.3 THE SUBTLETIES OF MINI-BATCH CONVERGENCE
|
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+
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+
Intuitively, the convergence analysis of SIGNSGD depends on the probability that a given bit of the sign stochastic gradient vector is incorrect, or $\mathbb { P } [ \mathrm { s i g n } ( \tilde { g } _ { i } ) \neq \mathrm { s i g n } ( g _ { i } ) ]$ . Lemma 1 provides a bound on this quantity under Assumption 4 (unimodal symmetric gradient noise).
|
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+
|
| 125 |
+
Lemma 1 (Bernstein et al. (2018)). Let $\tilde { g } _ { i }$ be an unbiased stochastic approximation to gradient component $g _ { i }$ , with variance bounded by $\sigma _ { i } ^ { 2 }$ . Further assume that the noise distribution is unimodal and symmetric. Define signal-to-noise ratio $\begin{array} { r } { S _ { i } : = \frac { | g _ { i } | } { \sigma _ { i } } } \end{array}$ |gi| . Then we have that
|
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+
|
| 127 |
+
$$
|
| 128 |
+
\begin{array} { r } { \mathbb { P } [ \mathrm { s i g n } ( \tilde { g } _ { i } ) \neq \mathrm { s i g n } ( g _ { i } ) ] \leq \left\{ \begin{array} { l l } { \frac { 2 } { 9 } \frac { 1 } { S _ { i } ^ { 2 } } \quad } & { i f S _ { i } > \frac { 2 } { \sqrt { 3 } } , } \\ { \frac { 1 } { 2 } - \frac { S _ { i } } { 2 \sqrt { 3 } } \quad } & { o t h e r w i s e } \end{array} \right. } \end{array}
|
| 129 |
+
$$
|
| 130 |
+
|
| 131 |
+
which is in all cases less than or equal to $\frac { 1 } { 2 }$
|
| 132 |
+
|
| 133 |
+
The bound characterises how the failure probability of a sign bit depends on the signal-to-noise ratio (SNR) of that gradient component. Intuitively as the SNR decreases, the quality of the sign estimate should degrade. The bound is important since it tells us that, under conditions of unimodal symmetric gradient noise, even at extremely low SNR we still have that $\mathbb { P } [ \mathrm { s i g n } ( \tilde { g } _ { i } ) \neq \mathrm { s i g n } ( g _ { i } ) ] \leq \frac { 1 } { 2 }$ . This means that even when the gradient is very small compared to the noise, the sign stochastic gradient still tells us, on average, useful information about the true gradient direction, allowing us to guarantee convergence as in Theorem 1.
|
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+
|
| 135 |
+
Without Assumption 4, the mini-batch algorithm may not converge. This is best appreciated with a simple example. Consider a stochastic gradient component $\tilde { g }$ with bimodal noise:
|
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+
|
| 137 |
+
$$
|
| 138 |
+
\tilde { g } = \left\{ \begin{array} { l l } { 5 0 } & { \mathrm { w i t h ~ p r o b a b i l i t y ~ 0 . 1 ; } } \\ { - 1 } & { \mathrm { w i t h ~ p r o b a b i l i t y ~ 0 . 9 . } } \end{array} \right.
|
| 139 |
+
$$
|
| 140 |
+
|
| 141 |
+
The true gradient $g = \mathbb { E } [ \tilde { g } ] = 4 . 1$ is positive. But the sign gradient $\mathrm { s i g n } ( \tilde { g } )$ is negative with probability 0.9. Therefore SIGNSGD will tend to move in the wrong direction for this noise distribution.
|
| 142 |
+
|
| 143 |
+
Note that SIGNSGD is a special case of the ADAM algorithm (Balles & Hennig, 2018). To see this, set $\beta _ { 1 } = \beta _ { 2 } = \epsilon = 0$ in ADAM, and the ADAM update becomes:
|
| 144 |
+
|
| 145 |
+
$$
|
| 146 |
+
- \frac { \tilde { g } } { \sqrt { \tilde { g } ^ { 2 } } } = - \frac { \tilde { g } } { | \tilde { g } | } = - \mathrm { s i g n } ( \tilde { g } )
|
| 147 |
+
$$
|
| 148 |
+
|
| 149 |
+
This correspondence suggests that Assumption 4 should be useful for obtaining mini-batch convergence guarantees for ADAM. Note that when Reddi et al. (2018) construct toy divergence examples for ADAM, they rely on bimodal noise distributions which violate Assumption 4.
|
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+
|
| 151 |
+
We conclude this section by noting that without Assumption 4, SIGNSGD can still be guaranteed to converge. The trick is to use a “large” batch size that grows with the number of iterations. This will ensure that the algorithm stays in the high SNR regime where the failure probability of the sign bit is low. This is the approach taken by both Carlson et al. (2016) and Bernstein et al. (2018).
|
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+
|
| 153 |
+
# 3.4 ROBUSTNESS OF CONVERGENCE
|
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+
|
| 155 |
+
We will now study SIGNSGD’s robustness when distributed by majority vote. We model adversaries as machines that may manipulate their stochastic gradient as follows.
|
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+
|
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+
Definition 1 (Blind multiplicative adversary). A blind multiplicative adversary may manipulate their stochastic gradient estimate $\tilde { g } _ { t }$ at iteration $t$ by element-wise multiplying $\tilde { g } _ { t }$ with any vector $v _ { t }$ of their choice. The vector $v _ { t }$ must be chosen before observing $\tilde { g } _ { t }$ , so the adversary is ‘blind’. Some interesting members of this class are:
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+
|
| 159 |
+
(i) adversaries that arbitrarily rescale their stochastic gradient estimate;
|
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+
|
| 161 |
+
(ii) adversaries that randomise the sign of each coordinate of the stochastic gradient;
|
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+
|
| 163 |
+
(iii) adversaries that invert their stochastic gradient estimate.
|
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+
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+
SGD is certainly not robust to rescaling since an adversary could set the gradient to infinity and corrupt the entire model. Our algorithm, on the other hand, is robust to all adversaries in this class. For ease of analysis, here we derive large batch results. We make sure to give results in terms of sample complexity $N$ (and not iteration number $K$ ) to enable fair comparison with other algorithms.
|
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+
|
| 167 |
+
Theorem 2 (Non-convex convergence rate of majority vote with adversarial workers). Run algorithm 1 for $K$ iterations under Assumptions $^ { l }$ to 4. Switch off momentum and weight decay $\beta = \lambda = 0$ ). Set the learning rate, $\eta$ , and mini-batch size, $n ,$ , for each worker as
|
| 168 |
+
|
| 169 |
+
$$
|
| 170 |
+
\eta = \sqrt { \frac { f _ { 0 } - f _ { * } } { \| L \| _ { 1 } K } } , \qquad n = K .
|
| 171 |
+
$$
|
| 172 |
+
|
| 173 |
+
Assume that a fraction $\alpha < \textstyle { \frac { 1 } { 2 } }$ of the $M$ workers behave adversarially according to Definition 1. Then majority vote converges at rate:
|
| 174 |
+
|
| 175 |
+
$$
|
| 176 |
+
\left[ \frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left\| g _ { k } \right\| _ { 1 } \right] ^ { 2 } \leq \frac { 4 } { \sqrt { N } } \left[ \frac { 1 } { 1 - 2 \alpha } \frac { \| \vec { \sigma } \| _ { 1 } } { \sqrt { M } } + \sqrt { \| L \| _ { 1 } ( f _ { 0 } - f ^ { * } ) } \right] ^ { 2 }
|
| 177 |
+
$$
|
| 178 |
+
|
| 179 |
+
where $N = K ^ { 2 }$ is the total number of stochastic gradient calls per worker up to step $K$
|
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+
|
| 181 |
+
The result is intuitive: provided there are more machines sending honest gradients than adversarial gradients, we expect that the majority vote should come out correct on average.
|
| 182 |
+
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| 183 |
+
Remark 1: if we switch off adversaries by setting the proportion of adversaries $\alpha = 0$ , this result reduces to Theorem 2 in (Bernstein et al., 2018). In this case, we note the $\textstyle { \frac { 1 } { \sqrt { M } } }$ variance reduction that majority vote obtains by distributing over $M$ machines, similar to distributed SGD.
|
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+
|
| 185 |
+
Remark 2: the convergence rate degrades as we ramp up $\alpha$ from 0 to $\frac { 1 } { 2 }$
|
| 186 |
+
|
| 187 |
+
Remark 3: from an optimisation theory perspective, the large batch size is an advantage. This is because when using a large batch size, fewer iterations and rounds of communication are theoreti-√ cally needed to reach a desired accuracy, since only $\sqrt { N }$ iterations are needed to reach $N$ samples. But from a practical perspective, workers may be unable to handle such a large batch size in a timely manner. It should be possible to extend the result to the mini-batch setting by combining the techniques of Theorems 1 and 2, but we leave this for future work.
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+
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+
# 4 EXPERIMENTS
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For our experiments, we distributed SIGNUM (Algorithm 1) by majority vote. SIGNUM is the momentum counterpart of SIGNSGD, where each worker maintains a momentum and transmits the sign momentum to the parameter server at each step. The addition of momentum to SIGNSGD is proposed and studied in (Balles & Hennig, 2018; Bernstein et al., 2018).
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We built SIGNUM with majority vote in the Pytorch deep learning framework (Paszke et al., 2017) using the Gloo (2018) communication library. Unfortunately Pytorch and Gloo do not natively support 1-bit tensors, therefore we wrote our own compression code to bit-pack a sign tensor down to an efficient 1-bit representation. We obtained a performance boost by fusing together smaller tensors, which saved on compression and communication costs.
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Figure 4: Timing breakdown for distributing on the cloud. Left: comparing communication (including compression) for training resnet50. Right: comparing communication (including compression) and computation. resnet50 results use 7 p3.2xlarge machines for training Imagenet, each at batch size 128. alexnet uses $7 { \mathrm { p } } 3$ .2xlarge machines for Imagenet, each at batch size 64. QRNN uses $3 \mathrm { p } 3 . 1 6 \mathrm { x } 1$ arge machines for training WikiText $- 1 0 3$ , each at batch size 240.
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Figure 5: Imagenet comparison of SIGNUM with majority vote and SGD distributed with NCCL. We train resnet50 on Imagenet distributed over 7 to 15 AWS p3.2xlarge machines. Top: increasing the number of workers participating in the majority vote shows a similar convergence speedup to distributed SGD. But in terms of wall-clock time, majority vote training is roughly $2 5 \%$ faster for the same number of epochs. Bottom: in terms of generalisation accuracy, majority vote shows a slight degradation compared to SGD. Perhaps a better regularisation scheme can fix this.
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Figure 6: Training QRNN across three $_ { \mathrm { p 3 . 1 6 x 1 } }$ arge machines on WikiText-103. Each machine uses a batch size of 240. For ADAM, the gradient is aggregated with NCCL. SIGNUM with majority vote shows some degradation compared to ADAM, although an epoch is completed roughly three times faster. This means that after 2 hours of training, SIGNUM attains a similar perplexity to ADAM. Increasing the per-worker batch size improved SIGNUM’s performance (see Appendix A), and increasing it beyond 240 may further improve SIGNUM’s performance. Note: the test perplexity beats training perplexity because dropout was applied during training but not testing.
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# Bits sent per iteration
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Majority vote $O ( M d )$ L2 QSGD $\operatorname { O } ( M ^ { 2 } { \sqrt { d } } \log d )$ max QSGD $O ( M d )$
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For $d$ weights, $M$ machines.
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Derivations in Appendix B.
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Figure 7: Left: comparing convergence of majority vote to QSGD (Alistarh et al., 2017). resnet18 is trained on Cifar-10 across $M = 3$ machines, each at bach size 128. 1-bit QSGD stochastically snaps gradient components to $\{ 0 , \pm 1 \}$ . 2-way refers to the compression function $Q ( . )$ being applied in both directions of communication: machine $i$ sends $Q ( \tilde { g _ { i } } )$ to the server and gets $\begin{array} { r } { Q ( \sum _ { i = 1 } ^ { M } Q ( \tilde { g } _ { i } ) ) } \end{array}$ sent back. Alistarh et al. (2017) develop a theory for $L 2$ QSGD, but experimentally benchmark max QSGD which has much larger communication costs. For this experiment, 1-bit max QSGD gives roughly $5 \times$ more compression than the $3 2 \times$ compression of majority vote, but this further gain turns out to be small relative to the cost of backpropagation. See Appendix A for QSGD experiments at higher bit-precision. Right: the table gives a theoretical comparison of the compression cost of each algorithm—see Appendix B for derivations.
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Figure 8: Imagenet robustness experiments. We used majority vote to train resnet50 distributed across 7 AWS $\mathtt { p 3 }$ . $2 \times 1$ arge machines. Adversaries invert their sign stochastic gradient. Left: all experiments are run at identical hyperparameter settings, with weight decay switched off for simplicity. The network still learns even at $43 \%$ adversarial. Right: at $43 \%$ adversarial, learning became slightly unstable. We decreased the learning rate for this setting, and learning stabilised.
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Figure 9: Comparing the robustness of majority vote to MULTI-KRUM (Blanchard et al., 2017). We train resnet18 on Cifar-10 across 7 workers, each at batch size 64. Momentum and weight decay are switched off for simplicity, and for majority vote we divide the learning rate by 10 at epoch 100. Negative adversaries multiply their stochastic gradient estimate by $- 1 0$ . Random adversaries multiply their stochastic gradient estimate by 10 and then randomise the sign of each coordinate. For MULTI-KRUM, we use the maximum allowed security level of $f = 2$ . Notice that MULTI-KRUM fails catastrophically once the number of adversaries exceeds the security level, whereas majority vote fails more gracefully.
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We test against SGD distributed using the state of the art NCCL (2018) communication library. NCCL provides an efficient implementation of allreduce. Our framework is often $4 \times$ faster in communication (including the cost of compression) than NCCL, as can be seen in Figure 4. Further code optimisation should bring the speedup closer to the ideal $3 2 \times$ .
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# 4.1 COMMUNICATION EFFICIENCY
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We first benchmark majority vote on the Imagenet dataset. We train a resnet50 model and disitribute learning over 7 to 15 AWS p3.2xlarge machines. These machines each contain one Nvidia Tesla V100 GPU, and AWS lists the connection speed between machines as “up to 10 Gbps”. Results are plotted in Figure 5. Per epoch, distributing by majority vote is able to attain a similar speedup to distributed SGD. But per hour majority vote is able to process more epochs than NCCL, meaning it can complete the 80 epoch training job roughly $2 5 \%$ faster. In terms of overall generalisation, majority vote reaches a slightly degraded test set accuracy. We hypothesise that this may be fixed by inventing a better regularisation scheme or tuning momentum, which we did not do.
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In Figure 6 we compare majority vote to ADAM (distributed by NCCL) for training QRNN (Bradbury et al., 2017) on WikiText-103. Majority vote completes an epoch roughly 3 times faster than ADAM, but it reaches a degraded accuracy so that the overall test perplexity after 2 hours ends up being similar. In Figure 7 we show that majority vote has superior convergence to the ‘theory’ version of QSGD that Alistarh et al. (2017) develop. Convergence is similar for the ‘max’ version that Alistarh et al. (2017) use in their experiments. Additional results are given in Appendix A.
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# 4.2 ROBUSTNESS
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In this section we test the robustness of SIGNUM with majority vote to Byzantine faults. Again we run tests on the Imagenet dataset, training resnet50 across 7 AWS $\mathtt { p 3 }$ . $2 \times 1$ arge machines. Our adversarial workers take the sign of their stochastic gradient calculation, but send the negation to the parameter server. Our results are plotted in Figure 8. In the left hand plot, all experiments were carried out using hyperparameters tuned for the $0 \%$ adversarial case. Weight decay was not used in these experiments to simplify matters. We see that learning is tolerant of up to $43 \%$ (3 out of 7) machines behaving adversarially. The $43 \%$ adversarial case was slightly unstable (Figure 8, left), but re-tuning the learning rate for this specific case stabilised learning (Figure 8, right).
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In Figure 9 we compare majority vote to MULTI-KRUM (Blanchard et al., 2017) with a security level of $f = 2$ . When the number of adversaries exceeds $f$ , MULTI-KRUM fails catastrophically in our experiments, whereas SIGNSGD fails more gracefully. Note that MULTI-KRUM requires $2 f + 2 < M$ , therefore $f = 2$ is the maximum possible security level for these experiments with $M = 7$ workers.
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# 5 DISCUSSION AND CONCLUSION
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We have analysed the theoretical and empirical properties of a very simple algorithm for distributed, stochastic optimisation. We have shown that SIGNSGD with majority vote aggregation is robust and communication efficient, whilst its per-iteration convergence rate is competitive with SGD for training large-scale convolutional neural nets on image datasets. We believe that it is important to understand this simple algorithm before going on to devise more complex learning algorithms.
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An important takeaway from our theory is that mini-batch SIGNSGD should converge if the gradient noise is Gaussian. This means that the performance of SIGNSGD may be improved by increasing the per-worker mini-batch size, since this should make the noise ‘more Gaussian’ according to the Central Limit Theorem.
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We will now give some possible directions for future work. Our implementation of majority vote may be further optimised by breaking up the parameter server and distributing it across machines. This would prevent a single machine from becoming a communication bottleneck as in our experiments. Though our framework speeds up Imagenet training, we still have a test set gap. Future work could attempt to devise new regularisation schemes for signed updates to close this gap. Promising future work could also explore the link between SIGNSGD and model compression. Signed updates force the weights to live on a lattice, facilitating compression of the resulting model.
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# ACKNOWLEDGMENTS
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We would like to thank Yu-Xiang Wang, Alexander Sergeev, Soumith Chintala, Pieter Noordhuis, Hongyi Wang, Scott Sievert and El Mahdi El Mhamdi for useful discussions.
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KA is supported in part by NSF Career Award CCF-1254106. AA is supported in part by a Microsoft Faculty Fellowship, Google Faculty Award, Adobe Grant, NSF Career Award CCF-1254106, and AFOSR YIP FA9550-15-1-0221.
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# REFERENCES
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Dan Alistarh, Demjan Grubic, Jerry Li, Ryota Tomioka, and Milan Vojnovic. QSGD: Communication-Efficient SGD via Gradient Quantization and Encoding. In Advances in Neural Information Processing Systems (NIPS-17), 2017.
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Dan Alistarh, Zeyuan Allen-Zhu, and Jerry Li. Byzantine Stochastic Gradient Descent. arXiv:1803.08917, 2018.
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Zeyuan Allen-Zhu. Natasha 2: Faster Non-Convex Optimization Than SGD. arXiv:1708.08694, 2017.
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Lukas Balles and Philipp Hennig. Dissecting Adam: The Sign, Magnitude and Variance of Stochastic Gradients. In International Conference on Machine Learning (ICML-18), 2018.
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Jeremy Bernstein, Yu-Xiang Wang, Kamyar Azizzadenesheli, and Animashree Anandkumar. signSGD: Compressed Optimisation for Non-Convex Problems. In International Conference on Machine Learning (ICML-18), 2018.
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Peva Blanchard, El Mahdi El Mhamdi, Rachid Guerraoui, and Julien Stainer. Machine Learning with Adversaries: Byzantine Tolerant Gradient Descent. In Advances in Neural Information Processing Systems (NIPS-17), 2017.
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James Bradbury, Stephen Merity, Caiming Xiong, and Richard Socher. Quasi-Recurrent Neural Networks. In International Conference on Learning Representations (ICLR-17), 2017.
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Francesco Paolo Cantelli. Sui confini della probabilit. Atti del Congresso Internazionale dei Matematici, 1928.
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David Carlson, Ya-Ping Hsieh, Edo Collins, Lawrence Carin, and Volkan Cevher. Stochastic spectral descent for discrete graphical models. IEEE Journal of Selected Topics in Signal Processing, 10 (2):296–311, 2016.
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El Mahdi El Mhamdi, Rachid Guerraoui, and Sebastien Rouault. The Hidden Vulnerability of ´ Distributed Learning in Byzantium. In International Conference on Machine Learning (ICML18), 2018.
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Carl Friedrich Gauss. Theoria combinationis observationum erroribus minimis obnoxiae, pars prior. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores, 1823.
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Gloo. Gloo Collective Communications Library, 2018. URL https://github.com/ facebookincubator/gloo. Accessed on 9/27/18.
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J J Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences, 1982.
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Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization. In International Conference on Learning Representations (ICLR-15), 2015.
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Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 521(7553):436, 2015.
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Yujun Lin, Song Han, Huizi Mao, Yu Wang, and Bill Dally. Deep gradient compression: Reducing the communication bandwidth for distributed training. In International Conference on Learning Representations (ICLR-18), 2018.
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NCCL. Nvidia Collective Communications Library, 2018. URL https://developer. nvidia.com/nccl. Accessed on 9/27/18.
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Yurii Nesterov and B.T. Polyak. Cubic Regularization of Newton Method and its Global Performance. Mathematical Programming, 2006.
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Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic Differentiation in PyTorch. In Advances in Neural Information Processing Systems, Autodiff Workshop (NIPS-17), 2017.
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Friedrich Pukelsheim. The Three Sigma Rule. The American Statistician, 1994.
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Sashank J. Reddi, Satyen Kale, and Sanjiv Kumar. On the Convergence of Adam and Beyond. In International Conference on Learning Representations (ICLR-18), 2018.
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M. Riedmiller and H. Braun. A Direct Adaptive Method for Faster Backpropagation Learning: the RPROP Algorithm. In International Conference on Neural Networks (ICNN-93), pp. 586–591. IEEE, 1993.
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Herbert Robbins and Sutton Monro. A Stochastic Approximation Method. The Annals of Mathematical Statistics, 1951.
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Frank Seide, Hao Fu, Jasha Droppo, Gang Li, and Dong Yu. 1-Bit Stochastic Gradient Descent and Application to Data-Parallel Distributed Training of Speech DNNs. In Conference of the International Speech Communication Association (INTERSPEECH-14), 2014.
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Tijmen Tieleman and Geoffrey Hinton. RMSprop. Coursera: Neural Networks for Machine Learning, Lecture 6.5, 2012.
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TOP500. IBM Summit Supercomputer, 2018. URL https://www.top500.org/system/ 179397. Accessed on 9/19/18.
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Hongyi Wang, Scott Sievert, Shengchao Liu, Zachary B. Charles, Dimitris S. Papailiopoulos, and Stephen Wright. ATOMO: Communication-efficient Learning via Atomic Sparsification. In Advances in Neural Information Processing Systems (NIPS-18), 2018.
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Wei Wen, Cong Xu, Feng Yan, Chunpeng Wu, Yandan Wang, Yiran Chen, and Hai Li. TernGrad: Ternary Gradients to Reduce Communication in Distributed Deep Learning. In Advances in Neural Information Processing Systems (NIPS-17), 2017.
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Dong Yin, Yudong Chen, Ramchandran Kannan, and Peter Bartlett. Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates. In International Conference on Machine Learning (ICML-18), 2018.
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Figure 10: QSGD at varying levels of precision. Top row: L2 QSGD. Bottom row: max QSGD. resnet18 is trained on Cifar-10 across $M = 3$ machines, each at bach size 128. The one-way version of QSGD is used, meaning that the compression function is not re-applied after aggregation. The comparison is given in terms of number of epochs. A comparison in terms of wall-clock time will depend on details of the systems implementation.
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Figure 11: SIGNUM at varying batch sizes. We use a single worker and train a QRNN model on WikiText $- 1 0 3$ . ADAM is shown for comparison, at batch size 60. The performance of SIGNUM is seen to improve with increasing batch size.
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# B BITS SENT PER ITERATION: SIGNSGD VS. QSGD
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In this section, we perform theoretical calculations of the number of bits sent per iteration in distributed training. We compare SIGNSGD using majority vote aggregation to two forms of QSGD (Alistarh et al., 2017). These calculations give the numbers in the table in Figure 7.
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The communication cost of SIGNSGD with majority vote is trivially $2 M d$ bits per iterations, since at each iteration $M$ machines send $d$ -dimensional sign vectors up to the server, and the server sends back one $d$ -dimensional sign vector to all $M$ machines.
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There are two variants of QSGD given in (Alistarh et al., 2017). The first we refer to as $L 2$ QSGD which is the version developed in the theory section of (Alistarh et al., 2017). The second we refer to as max QSGD which is the version actually used in their experiments. For each version we compute the number of bits sent for the highest compression version of the algorithm, which is a ternary quantisation (snapping gradient components into $\{ 0 , \pm 1 \} ,$ ). We refer to this as $^ { l }$ -bit QSGD. The higher precision versions of QSGD will send more bits per iteration.
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1-bit L2 QSGD takes a gradient vector $g$ and snaps $i ^ { t h }$ coordinate $g _ { i }$ to $\mathrm { s i g n } ( g _ { i } )$ with probability $\frac { \left| g _ { i } \right| } { \left\| g \right\| _ { 2 } }$ and sets it to zero otherwise. Therefore the expected number of bits set to $\pm 1$ is bounded by
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+
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$$
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\mathbb { E } [ \# \mathrm { b i t s } ] = \sum _ { i = 1 } { \frac { | g _ { i } | } { \| g \| _ { 2 } } } = { \frac { \| g \| _ { 1 } } { \| g \| _ { 2 } } } \leq { \sqrt { d } } .
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$$
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+
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To send a vector compressed in this way, for each non-zero component 1 bit is needed to send the sign and $\log d$ bits are needed to send the index. Therefore sending a vector compressed by 1-bit L2√ QSGD requires at most ${ \sqrt { d } } ( 1 + \log d )$ bits.
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In the experiments in Figure 7 we see that the ‘2-way’ version of 1-bit L2 QSGD (which recompresses the aggregated compressed gradients) converges very poorly. Therefore it makes sense to use the 1-way version where the aggregated compressed gradient is not recompressed. A sensible way to enact this is to have each of the $M$ workers broadcast their compressed gradient vector to all√ other workers. This has a cost of $( M - 1 ) { \sqrt { d } } ( 1 + \log d )$ bits for each of the $M$ workers, and from this we get the total cost of $\operatorname { O } ( M ^ { 2 } { \sqrt { d } } \log d )$ for 1-bit L2 QSGD.
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The final algorithm to characterise is 1-bit max QSGD. 1-bit max QSGD takes a gradient vector $g$ and snaps $i ^ { t h }$ coordinate $g _ { i }$ to $\mathrm { s i g n } ( g _ { i } )$ with probability $\frac { | g _ { i } | } { \| g \| _ { \infty } }$ and sets it to zero otherwise. As noted in (Alistarh et al., 2017), there are no sparsity guarantees for this algorithm, so compression will generally be much lower than for 1-bit L2 QSGD.
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It is easy to see that 1-bit max QSGD requires no more than ${ \mathrm { O } } ( d )$ bits to compress a $d$ -dimensional vector, since $2 d$ bits can always store $d$ numbers in $\{ 0 , \pm 1 \}$ . To see that we can’t generally do better than ${ \mathrm { O } } ( d )$ bits, notice that 1-bit max QSGD leaves sign vectors invariant, and thus the compressed form of a sign vector requires exactly $d$ bits. The natural way to enact 1-bit max QSGD is with a two-way compression where the $M$ workers each send an ${ \mathrm { O } } ( d )$ -bit compressed gradient up to the server, and the server sends back an ${ \mathrm { O } } ( d )$ -bit compressed aggregated result back to the $M$ workers. This gives a number of bits sent per iteration of ${ \bf O } ( M d )$ .
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For very sparse vectors 1-bit max QSGD will compress much better than indicated above. For a vector $g$ with a single non-zero entry, 1-bit max QSGD will set this entry to 1 and keep the rest zero, thus requiring only $\log d$ bits to send the index of the non-zero entry. But it is not clear whether these extremely sparse vectors appear in deep learning problems. In the experiments in Figure 7, 1-bit max QSGD led to compressed vectors that were $5 \times$ more compressed than SIGNSGD—in our experimental setting this additional improvement turned out to be small relative to the time cost of backpropagation.
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# C PROOFS
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C.1 ACCURACY OF THE SIGN STOCHASTIC GRADIENT
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Lemma 1 (Bernstein et al. (2018)). Let $\tilde { g } _ { i }$ be an unbiased stochastic approximation to gradient component $g _ { i }$ , with variance bounded by $\sigma _ { i } ^ { 2 }$ . Further assume that the noise distribution is unimodal and symmetric. Define signal-to-noise ratio $\begin{array} { r } { S _ { i } : = \frac { | g _ { i } | } { \sigma _ { i } } } \end{array}$ |gi| . Then we have that
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+
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$$
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\begin{array} { r } { \mathbb { P } [ \mathrm { s i g n } ( \tilde { g } _ { i } ) \neq \mathrm { s i g n } ( g _ { i } ) ] \leq \left\{ \begin{array} { l l } { \frac { 2 } { 9 } \frac { 1 } { S _ { i } ^ { 2 } } \quad } & { i f S _ { i } > \frac { 2 } { \sqrt { 3 } } , } \\ { \frac { 1 } { 2 } - \frac { S _ { i } } { 2 \sqrt { 3 } } \quad } & { o t h e r w i s e } \end{array} \right. } \end{array}
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$$
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+
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which is in all cases less than or equal to $\frac { 1 } { 2 }$
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Proof. Recall Gauss’ inequality for unimodal random variable $\mathrm { X }$ with mode $\nu$ and expected squared deviation from the mode $\dot { \tau } ^ { 2 }$ (Gauss, 1823; Pukelsheim, 1994):
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+
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$$
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\mathbb { P } [ | X - \nu | > k ] \leq \left\{ \begin{array} { l l } { \frac { 4 } { 9 } \frac { \tau ^ { 2 } } { k ^ { 2 } } \quad } & { \mathrm { i f ~ } \frac { k } { \tau } > \frac { 2 } { \sqrt { 3 } } , } \\ { 1 - \frac { k } { \sqrt { 3 } \tau } \quad } & { \mathrm { o t h e r w i s e } } \end{array} \right.
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+
$$
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+
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+
By the symmetry assumption, the mode is equal to the mean, so we replace mean $\mu = \nu$ and variance $\sigma ^ { \bar { 2 } } = \tau ^ { \bar { 2 } }$ .
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+
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+
$$
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+
\mathbb { P } [ | X - \mu | > k ] \leq \left\{ { \begin{array} { l l } { \frac { 4 } { 9 } \frac { \sigma ^ { 2 } } { k ^ { 2 } } \quad } & { \mathrm { ~ i f ~ } \frac { k } { \sigma } > \frac { 2 } { \sqrt { 3 } } , } \\ { 1 - \frac { k } { \sqrt { 3 } \sigma } \quad } & { \mathrm { ~ o t h e r w i s e } } \end{array} } \right.
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+
$$
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+
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+
Without loss of generality assume that $g _ { i }$ is negative. Then applying symmetry followed by Gauss, the failure probability for the sign bit satisfies:
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+
|
| 349 |
+
$$
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+
\begin{array} { r l } { \mathbb { P } [ \mathrm { s i g n } ( \tilde { g } _ { i } ) \neq \mathrm { s i g n } ( g _ { i } ) ] = \mathbb { P } [ \tilde { g } _ { i } - g _ { i } \geq | g _ { i } | ] } & { } \\ { = \frac { 1 } { 2 } \mathbb { P } [ | \tilde { g } _ { i } - g _ { i } | \geq | g _ { i } | ] } & { } \\ { \leq \left\{ \begin{array} { l l } { \frac { 2 } { g } \frac { \sigma _ { i } ^ { 2 } } { g _ { i } ^ { 2 } } } & { \mathrm { ~ i f ~ } \frac { | g _ { i } | } { \sigma } > \frac { 2 } { \sqrt { 3 } } , } \\ { \frac { 1 } { 2 } - \frac { | g _ { i } | } { 2 \sqrt { 3 } \sigma _ { i } } } & { \mathrm { ~ o t h e r w i s e } } \end{array} \right. } & { } \\ { = \left\{ \begin{array} { l l } { \frac { 2 } { 9 } \frac { 1 } { S _ { i } ^ { 2 } } } & { \mathrm { ~ i f ~ } S _ { i } > \frac { 2 } { \sqrt { 3 } } , } \\ { \frac { 1 } { 2 } - \frac { S _ { i } } { 2 \sqrt { 3 } } } & { \mathrm { ~ o t h e r w i s e } } \end{array} \right. } \end{array}
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$$
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+
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# C.2 MINI-BATCH CONVERGENCE GUARANTEES
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| 355 |
+
Theorem 1 (Non-convex convergence rate of mini-batch SIGNSGD). Run the following algorithm for $K$ iterations under Assumptions $^ { l }$ to $^ { 4 }$ : $x _ { k + 1 } = x _ { k } - \eta \mathrm { s i g n } ( \tilde { g } _ { k } )$ . Set the learning rate, $\eta _ { ; }$ , and mini-batch size, $n$ , as
|
| 356 |
+
|
| 357 |
+
$$
|
| 358 |
+
\eta = \sqrt { \frac { f _ { 0 } - f _ { * } } { \| \vec { L } \| _ { 1 } K } } , \qquad n = 1 .
|
| 359 |
+
$$
|
| 360 |
+
|
| 361 |
+
Let i.e. $H _ { k }$ ent compone. We refer to at stepas the $k$ with large signal-to-noise ratio critical SNR’. Then we have $\begin{array} { r } { S _ { i } : = \frac { | g _ { k , i } | } { \sigma _ { i } } } \end{array}$ , $H _ { k } : = { \Big \{ } i { \Big | } S _ { i } > { \frac { 2 } { \sqrt { 3 } } } { \Big \} }$ $\frac { 2 } { \sqrt { 3 } }$
|
| 362 |
+
|
| 363 |
+
$$
|
| 364 |
+
\frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left[ \sum _ { i \in { \cal H } _ { k } } \left. g _ { k , i } \right. + \sum _ { i \notin { \cal H } _ { k } } \frac { g _ { k , i } ^ { 2 } } { \sigma _ { i } } \right] \leq 3 \sqrt { \frac { \| \vec { L } \| _ { 1 } ( f _ { 0 } - f _ { * } ) } { N } } .
|
| 365 |
+
$$
|
| 366 |
+
|
| 367 |
+
where $N = K$ is the total number of stochastic gradient calls up to step $K$ .
|
| 368 |
+
|
| 369 |
+
Proof. First let’s bound the improvement of the objective during a single step of the algorithm for one instantiation of the noise. I[.] is the indicator function, $g _ { k , i }$ denotes the $\cdot ^ { \mathit { t h } }$ component of the true gradient $g ( x _ { k } )$ and $\tilde { g } _ { k }$ is a stochastic sample obeying Assumption 3.
|
| 370 |
+
|
| 371 |
+
First take Assumption 2, plug in the algorithmic step, and decompose the improvement to expose the stochasticity-induced error:
|
| 372 |
+
|
| 373 |
+
$$
|
| 374 |
+
\begin{array} { r l } { f _ { k + 1 } - f _ { k } \le g _ { k } ^ { T } ( x _ { k + 1 } - x _ { k } ) + \displaystyle \sum _ { i = 1 } ^ { d } \frac { L _ { i } } { 2 } ( x _ { k + 1 } - x _ { k } ) _ { i } ^ { 2 } } & { } \\ { \displaystyle = - \eta g _ { k } ^ { T } \mathbb { s i g n } ( \tilde { g } _ { k } ) + \eta ^ { 2 } \sum _ { i = 1 } ^ { d } \frac { L _ { i } } { 2 } } & { } \\ { \displaystyle = - \eta \| g _ { k } \| _ { 1 } + \frac { \eta ^ { 2 } } { 2 } \| \vec { L } \| _ { 1 } + 2 \eta \displaystyle \sum _ { i = 1 } ^ { d } | g _ { k , i } | \mathbb { I } [ \mathrm { s i g n } ( \tilde { g } _ { k , i } ) \neq \mathrm { s i g n } ( g _ { k , i } ) ] } \end{array}
|
| 375 |
+
$$
|
| 376 |
+
|
| 377 |
+
Next we find the expected improvement at time $k + 1$ conditioned on the previous iterate.
|
| 378 |
+
|
| 379 |
+
$$
|
| 380 |
+
\mathbb { E } [ f _ { k + 1 } - f _ { k } | x _ { k } ] \leq - \eta \| g _ { k } \| _ { 1 } + \frac { \eta ^ { 2 } } { 2 } \| \vec { L } \| _ { 1 } + 2 \eta \sum _ { i = 1 } ^ { d } | g _ { k , i } | \operatorname { \mathbb { P } } [ \mathrm { s i g n } ( \tilde { g } _ { k , i } ) \neq \mathrm { s i g n } ( g _ { k , i } ) ]
|
| 381 |
+
$$
|
| 382 |
+
|
| 383 |
+
By Assumption 4 and Lemma 1 we have the following bound on the failure probability of the sign:
|
| 384 |
+
|
| 385 |
+
$$
|
| 386 |
+
\begin{array} { r l } { \mathbb { P } [ \mathrm { s i g n } ( \tilde { g } _ { i } ) \neq \mathrm { s i g n } ( g _ { i } ) ] \leq \left\{ \frac { 2 } { 9 } \frac { 1 } { S _ { i } ^ { 2 } } \right. } & { \mathrm { ~ i f ~ } S _ { i } > \frac { 2 } { \sqrt { 3 } } , } \\ { \frac { 1 } { 2 } - \frac { S _ { i } } { 2 \sqrt { 3 } } ~ } & { \mathrm { ~ o t h e r w i s e ~ } } \\ { \leq \left\{ \begin{array} { l l } { \frac { 1 } { 6 } ~ } & { \mathrm { ~ i f ~ } S _ { i } > \frac { 2 } { \sqrt { 3 } } , } \\ { \frac { 1 } { 2 } - \frac { S _ { i } } { 2 \sqrt { 3 } } ~ } & { \mathrm { ~ o t h e r w i s e ~ } } \end{array} \right. } \end{array}
|
| 387 |
+
$$
|
| 388 |
+
|
| 389 |
+
Substituting this in, we get that
|
| 390 |
+
|
| 391 |
+
$$
|
| 392 |
+
\begin{array} { r l } & { \mathbb { E } [ f _ { k + 1 } - f _ { k } | x _ { k } ] \le - \eta \| g _ { k } \| _ { 1 } + \frac { \eta ^ { 2 } } { 2 } \| \vec { L } \| _ { 1 } + 2 \eta \displaystyle \sum _ { i \in { \cal H } _ { k } } \frac { | g _ { k , i } | } { 6 } + 2 \eta \displaystyle \sum _ { i \notin { \cal H } _ { k } } | g _ { k , i } | \left[ \frac { 1 } { 2 } - \frac { | g _ { k , i } | } { 2 \sqrt { 3 } \sigma _ { i } } \right] } \\ & { \qquad = - \eta \displaystyle \sum _ { i = 1 } ^ { d } | g _ { k , i } | + \frac { \eta ^ { 2 } } { 2 } \| \vec { L } \| _ { 1 } + \eta \displaystyle \sum _ { i \in { \cal H } _ { k } } \frac { | g _ { k , i } | } { 3 } + \eta \displaystyle \sum _ { i \notin { \cal H } _ { k } } | g _ { k , i } | - \eta \displaystyle \sum _ { i \notin { \cal H } _ { k } } \frac { g _ { k , i } ^ { 2 } } { \sqrt { 3 } \sigma _ { i } } } \\ & { \qquad = - \frac { 2 \eta } { 3 } \displaystyle \sum _ { i \in { \cal H } _ { k } } | g _ { k , i } | - \eta \displaystyle \sum _ { i \notin { \cal H } _ { k } } \frac { g _ { k , i } ^ { 2 } } { \sqrt { 3 } \sigma _ { i } } + \frac { \eta ^ { 2 } } { 2 } \| \vec { L } \| _ { 1 } } \end{array}
|
| 393 |
+
$$
|
| 394 |
+
|
| 395 |
+
Interestingly a mixture between an $\ell _ { 1 }$ and a variance weighted $\ell _ { 2 }$ norm has appeared. Now substitute in the learning rate schedule, and we get:
|
| 396 |
+
|
| 397 |
+
$$
|
| 398 |
+
\begin{array} { l } { \displaystyle \mathbb { E } [ f _ { k + 1 } - f _ { k } | x _ { k } ] \leq - \sqrt { \frac { f _ { 0 } - f _ { * } } { \| \vec { L } \| _ { 1 } K } } \Bigg [ \frac { 2 } { 3 } \sum _ { i \in H _ { k } } | g _ { k , i } | + \frac { 1 } { \sqrt { 3 } } \sum _ { i \notin H _ { k } } \frac { g _ { k , i } ^ { 2 } } { \sigma _ { i } } \Bigg ] + \frac { f _ { 0 } - f _ { * } } { 2 K } } \\ { \leq - \sqrt { \frac { f _ { 0 } - f _ { * } } { 3 \| \vec { L } \| _ { 1 } K } } \left[ \displaystyle \sum _ { i \in H _ { k } } | g _ { k , i } | + \displaystyle \sum _ { i \notin H _ { k } } \frac { g _ { k , i } ^ { 2 } } { \sigma _ { i } } \right] + \frac { f _ { 0 } - f _ { * } } { 2 K } } \end{array}
|
| 399 |
+
$$
|
| 400 |
+
|
| 401 |
+
Now extend the expectation over the randomness in the trajectory and telescope over the iterations:
|
| 402 |
+
|
| 403 |
+
$$
|
| 404 |
+
\begin{array} { r l } & { f _ { 0 } - f ^ { * } \geq f _ { 0 } - \mathbb { E } [ f _ { K } ] } \\ & { \qquad = \mathbb { E } \left[ \displaystyle \sum _ { k = 0 } ^ { K - 1 } f _ { k } - f _ { k + 1 } \right] } \\ & { \qquad \geq \sqrt { \displaystyle \frac { f _ { 0 } - f _ { * } } { 3 \| \vec { L } \| _ { 1 } K } } \displaystyle \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left[ \displaystyle \sum _ { i \in H _ { k } } | g _ { k , i } | + \displaystyle \sum _ { i \notin H _ { k } } \frac { g _ { k , i } ^ { 2 } } { \sigma _ { i } } \right] - \frac { f _ { 0 } - f _ { * } } { 2 } } \end{array}
|
| 405 |
+
$$
|
| 406 |
+
|
| 407 |
+
Finally, rearrange and substitute in $N = K$ to yield the bound
|
| 408 |
+
|
| 409 |
+
$$
|
| 410 |
+
\frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left[ \sum _ { i \in { \cal H } _ { k } } | g _ { k , i } | + \sum _ { i \notin { \cal H } _ { k } } \frac { g _ { k , i } ^ { 2 } } { \sigma _ { i } } \right] \leq \frac { 3 \sqrt { 3 } } { 2 } \sqrt { \frac { \| \vec { L } \| _ { 1 } ( f _ { 0 } - f _ { * } ) } { N } } \leq 3 \sqrt { \frac { \| \vec { L } \| _ { 1 } ( f _ { 0 } - f _ { * } ) } { N } } .
|
| 411 |
+
$$
|
| 412 |
+
|
| 413 |
+
# C.3 ROBUSTNESS OF MAJORITY VOTE
|
| 414 |
+
|
| 415 |
+
Theorem 2 (Non-convex convergence rate of majority vote with adversarial workers). Run algorithm 1 for $K$ iterations under Assumptions $^ { l }$ to 4. Switch off momentum and weight decay $\beta = \lambda = 0$ ). Set the learning rate, $\eta$ , and mini-batch size, $n _ { : }$ , for each worker as
|
| 416 |
+
|
| 417 |
+
$$
|
| 418 |
+
\eta = \sqrt { \frac { f _ { 0 } - f _ { * } } { \| L \| _ { 1 } K } } , \qquad n = K .
|
| 419 |
+
$$
|
| 420 |
+
|
| 421 |
+
Assume that a fraction $\alpha < \textstyle { \frac { 1 } { 2 } }$ of the $M$ workers behave adversarially according to Definition 1. Then majority vote converges at rate:
|
| 422 |
+
|
| 423 |
+
$$
|
| 424 |
+
\left[ \frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left\| g _ { k } \right\| _ { 1 } \right] ^ { 2 } \leq \frac { 4 } { \sqrt { N } } \left[ \frac { 1 } { 1 - 2 \alpha } \frac { \| \vec { \sigma } \| _ { 1 } } { \sqrt { M } } + \sqrt { \| L \| _ { 1 } ( f _ { 0 } - f ^ { * } ) } \right] ^ { 2 }
|
| 425 |
+
$$
|
| 426 |
+
|
| 427 |
+
where $N = K ^ { 2 }$ is the total number of stochastic gradient calls per worker up to step $K$ .
|
| 428 |
+
|
| 429 |
+
Proof. We need to bound the failure probability of the vote. We can then use this bound to derive a convergence rate. We will begin by showing this bound is worst when the adversary inverts the signs of the sign stochastic gradient.
|
| 430 |
+
|
| 431 |
+
Given an adversary from the class of blind multiplicative adversaries (Definition 1), the adversary may manipulate their stochastic gradient estimate $\tilde { g } _ { t }$ into the form $v _ { t } \otimes \tilde { g } _ { t }$ . Here $v _ { t }$ is a vector of the adversary’s choosing, and $\otimes$ denotes element-wise multiplication. The sign of this quantity obeys:
|
| 432 |
+
|
| 433 |
+
$$
|
| 434 |
+
\operatorname { s i g n } ( v _ { t } \otimes { \tilde { g } } _ { t } ) = \operatorname { s i g n } ( v _ { t } ) \otimes \operatorname { s i g n } ( { \tilde { g } } _ { t } ) .
|
| 435 |
+
$$
|
| 436 |
+
|
| 437 |
+
Therefore, the only thing that matters is the sign of $v _ { t }$ , and rescaling attacks are immediately nullified. For each component of the stochastic gradient, the adversary must decide (without observing $g _ { t }$ , since the adversary is blind) whether or not they would like to invert the sign of that component. We will now show that the failure probability of the vote is always larger when the adversary decides to invert (by setting every component of $\mathrm { s i g n } ( v _ { t } )$ to $^ { - 1 }$ ). Our analysis will then proceed under this worst case.
|
| 438 |
+
|
| 439 |
+
For a gradient component with true value $g$ , let random variable $Z \in [ 0 , M ]$ denote the number of correct sign bits received by the parameter server. For a given adversary, we may decompose $Z$ into the contribution from that adversary and a residual term $X$ from the remaining workers (both regular and adversarial):
|
| 440 |
+
|
| 441 |
+
$$
|
| 442 |
+
Z ( \operatorname { s i g n } ( v ) ) = X + \mathbb { I } [ \operatorname { s i g n } ( v ) \operatorname { s i g n } ( { \tilde { g } } ) = \operatorname { s i g n } ( \operatorname { g } ) ] ,
|
| 443 |
+
$$
|
| 444 |
+
|
| 445 |
+
where $\tilde { g }$ is the adversary’s stochastic gradient estimate for that component, $v$ is the adversary’s chosen scalar for that component, and $\mathbb { I }$ is the 0-1 indicator function. We are considering $\textsf { Z }$ to be a function of $\mathrm { s i g n } ( v )$ .
|
| 446 |
+
|
| 447 |
+
But by Assumption 4 and Lemma 1, we see that $\mathbb { I } [ + 1 \times \mathrm { s i g n } ( \tilde { g } ) = \mathrm { s i g n } ( \mathrm { g } ) ]$ is a Bernoulli random variable with success probability $\begin{array} { r } { p \geq \frac { 1 } { 2 } } \end{array}$ . On the other hand, $\mathbb { I } [ - 1 \times \mathrm { s i g n } ( \tilde { g } ) = \mathrm { s i g n } ( \mathrm { g } ) ]$ is a Bernoulli random variable with success probability $\begin{array} { r } { q = 1 - p \leq \frac { 1 } { 2 } } \end{array}$ .
|
| 448 |
+
|
| 449 |
+
The essential quantity in our analysis is the probability that more than half the workers provide the correct sign bit. But from the preceding discussion, this clearly obeys:
|
| 450 |
+
|
| 451 |
+
$$
|
| 452 |
+
\begin{array} { r l } { { \mathbb { P } \bigg [ Z ( + 1 ) \leq \frac { M } { 2 } \bigg ] = \mathbb { P } \bigg [ X + \mathbb { I } [ + 1 \times \mathrm { s i g n } ( \tilde { g } ) = \mathrm { s i g n } ( \mathrm { g } ) ] \leq \frac { M } { 2 } \bigg ] } } \\ & { \leq \mathbb { P } \bigg [ X + \mathbb { I } [ - 1 \times \mathrm { s i g n } ( \tilde { g } ) = \mathrm { s i g n } ( \mathrm { g } ) ] \leq \frac { M } { 2 } \bigg ] } \\ & { = \mathbb { P } \bigg [ Z ( - 1 ) \leq \frac { M } { 2 } \bigg ] . } \end{array}
|
| 453 |
+
$$
|
| 454 |
+
|
| 455 |
+
As we will see below, the implication of this is that our bounds always worse under the setting $v = - 1$ , and so we will adopt $v = - 1$ hereon. It is worth remarking that blindness in the definition of blind multiplicative adversaries is important to ensure that $\mathbb { I } [ \mathrm { s i g n } ( v ) \mathrm { s i g n } ( \tilde { g } ) = \mathrm { s i g n } ( \mathrm { g } ) ]$ is indeed a random variable as described above. Were the adversary not blind, then the adversary could in effect deterministically set $\mathrm { s i g n } ( v ) \mathrm { s i g n } ( \tilde { g } )$ . Cooperative adversaries, for example, could use this power to control the vote.
|
| 456 |
+
|
| 457 |
+
Now we restrict to our worst case blind multiplicative adversaries, that always choose to invert their sign stochastic gradient estimate. So, we have $( 1 - \alpha ) M$ good machines and $\alpha M$ adversaries. The good workers each compute a stochastic gradient estimate, take its sign and transmit this to the server. The bad workers follow an identical procedure except they negate their sign bits prior to transmission to the server. It is intuitive that because the proportion of adversaries $\begin{array} { r } { \bar { \alpha ^ { } } < \frac { 1 } { 2 } } \end{array}$ , the good workers will win the vote on average. To make this rigorous, we will need Lemma 1 and Cantelli’s inequality. Cantelli (1928) tells us that for a random variable $X$ with mean $\mu$ and variance $\sigma ^ { 2 }$ :
|
| 458 |
+
|
| 459 |
+
$$
|
| 460 |
+
\mathbb { P } [ \mu - X \geq | \lambda | ] \leq \frac { 1 } { 1 + \frac { \lambda ^ { 2 } } { \sigma ^ { 2 } } }
|
| 461 |
+
$$
|
| 462 |
+
|
| 463 |
+
For a given gradient component, again let random variable $Z \in [ 0 , M ]$ denote the number of correct sign bits received by the parameter server. Let random variables $G$ and $B$ denote the number of good and bad workers (respectively) who (possibly inadvertently) sent the correct sign bit. Then, letting $p$ be the probability that a good worker computed the correct sign bit, $q : = 1 - p$ and $\begin{array} { r } { \epsilon : = p - \frac { 1 } { 2 } } \end{array}$ we can decompose $Z$ as follows:
|
| 464 |
+
|
| 465 |
+
$$
|
| 466 |
+
\begin{array} { l } { { \displaystyle Z = G + B } } \\ { { \displaystyle G \sim \mathrm { b i n o m i a l } [ ( 1 - \alpha ) M , p ] } } \\ { { \displaystyle B \sim \mathrm { b i n o m i a l } [ \alpha M , q ] } } \\ { { \displaystyle \mathbb { E } [ Z ] = ( 1 - \alpha ) M p + \alpha M q = \frac { M } { 2 } + ( 1 - 2 \alpha ) M \epsilon } } \\ { { \displaystyle \mathrm { V a r } [ Z ] = ( 1 - \alpha ) M p q + \alpha M p q = M \biggl ( \frac { 1 } { 4 } - \epsilon ^ { 2 } \biggr ) . } } \end{array}
|
| 467 |
+
$$
|
| 468 |
+
|
| 469 |
+
The vote only fails if $Z < \frac { M } { 2 }$ which happens with probability
|
| 470 |
+
|
| 471 |
+
since $1 + x ^ { 2 } \geq 2 x$
|
| 472 |
+
|
| 473 |
+
$$
|
| 474 |
+
\begin{array} { r l } { \mathbb { P } \bigg [ Z \leq \frac { M } { 2 } \bigg ] = \mathbb { P } \bigg [ \mathbb { E } [ Z ] - Z \geq \mathbb { E } [ Z ] - \frac { M } { 2 } \bigg ] } & { } \\ { \leq \frac { 1 } { 1 + \frac { ( \frac { 2 } { \sqrt { | Z | } - \frac { M } { \delta } \big ] ^ { 2 } } } { \sqrt { \kappa ( \frac { \sqrt { | Z | } } { \delta } ) ^ { 2 } } } } } & { } \\ { \leq \frac { 1 } { 2 } \sqrt { \frac { \mathbb { V } \times [ Z ] } { ( \mathbb { E } [ Z ] - \frac { M } { 2 } ) ^ { 2 } } } } & { } \\ { = \frac { 1 } { 2 } \sqrt { \frac { M \big ( \frac { 1 } { 4 } - \epsilon ^ { 2 } \big ) } { ( 1 - 2 \alpha ) ^ { 2 } M ^ { 2 } \epsilon ^ { 2 } } } } & { } \\ { = \frac { 1 } { 2 } \frac { \sqrt { \frac { 1 } { 4 \epsilon ^ { 2 } } - 1 } } { ( 1 - 2 \alpha ) \sqrt { M } } } & { } \end{array}
|
| 475 |
+
$$
|
| 476 |
+
|
| 477 |
+
by Cantelli’s inequality
|
| 478 |
+
|
| 479 |
+
We now need to substitute in a bound on $\epsilon$ . Assumption 4 and Lemma 1 tell us that
|
| 480 |
+
|
| 481 |
+
$$
|
| 482 |
+
\epsilon = \frac { 1 } { 2 } - q \geq \left\{ \begin{array} { l l } { \frac { 1 } { 2 } - \frac { 2 } { 9 } \frac { 1 } { S ^ { 2 } } \quad } & { \mathrm { i f } \ : S > \frac { 2 } { \sqrt { 3 } } , } \\ { \frac { S } { 2 \sqrt { 3 } } \quad } & { \mathrm { o t h e r w i s e . } } \end{array} \right.
|
| 483 |
+
$$
|
| 484 |
+
|
| 485 |
+
Froand $\begin{array} { r } { \frac { 1 } { 4 \epsilon ^ { 2 } } - 1 \le \frac { 3 } { S ^ { 2 } } - 1 < \frac { 4 } { S ^ { 2 } } } \end{array}$ t 142 − 1 < 4S2 as follows. First take the case S ≤ the case $S > \frac { 2 } { \sqrt { 3 } }$ . Then $\epsilon \geq \frac { 1 } { 2 } - \frac { 2 } { 9 } \frac { 1 } { S ^ { 2 } }$ Then and $\begin{array} { r } { \epsilon ^ { 2 } \geq \frac { S ^ { 2 } } { 1 2 } } \end{array}$ $\begin{array} { r } { \frac { 1 } { 4 \epsilon ^ { 2 } } - 1 \leq \frac { 1 } { S ^ { 2 } } \frac { \frac { 8 } { 9 } - \frac { 1 6 } { 8 1 } \frac { 1 } { S ^ { 2 } } } { 1 - \frac { 8 } { 9 } \frac { 1 } { S ^ { 2 } } + \frac { 1 6 } { 8 1 } \frac { 1 } { S ^ { 4 } } } < \frac { 1 } { S ^ { 2 } } \frac { \frac { 8 } { 9 } } { 1 - \frac { 8 } { 9 } \frac { 1 } { S ^ { 2 } } } < \frac { 4 } { S ^ { 2 } } } \end{array}$ by the condition on $S$ .
|
| 486 |
+
|
| 487 |
+
We have now completed the first part of the proof by showing the key statement that for the $i ^ { t h }$ gradient component with signal to noise ratio $\begin{array} { r } { S _ { i } : = \frac { | g _ { i } | } { \sigma _ { i } } } \end{array}$ |gi| , the failure probability of the majority vote is bounded by
|
| 488 |
+
|
| 489 |
+
$$
|
| 490 |
+
\mathbb { P } [ \mathrm { v o t e ~ f a i l s ~ f o r ~ } i ^ { t h } \mathrm { \ c o o r d i n a t e } ] = \mathbb { P } \bigg [ Z _ { i } \leq \frac { M } { 2 } \bigg ] \leq \frac { 1 } { ( 1 - 2 \alpha ) \sqrt { M } S _ { i } }
|
| 491 |
+
$$
|
| 492 |
+
|
| 493 |
+
The second stage of the proof will proceed by straightforwardly substituting this bound into the convergence analysis of SIGNSGD from Bernstein et al. (2018).
|
| 494 |
+
|
| 495 |
+
First let’s bound the improvement of the objective during a single step of the algorithm for one instantiation of the noise. I[.] is the indicator function, ${ g } _ { k , i }$ denotes the $i ^ { t h }$ component of the true gradient $g ( x _ { k } )$ and $\mathrm { s i g n } ( V _ { k } )$ is the outcome of the vote at the $k ^ { t h }$ iteration.
|
| 496 |
+
|
| 497 |
+
First take Assumption 2, plug in the step from Algorithm 1, and decompose the improvement to expose the error induced by stochasticity and adversarial workers:
|
| 498 |
+
|
| 499 |
+
$$
|
| 500 |
+
\begin{array} { r l } { f _ { k + 1 } - f _ { k } \le g _ { k } ^ { T } ( x _ { k + 1 } - x _ { k } ) + \displaystyle \sum _ { i = 1 } ^ { d } \frac { L _ { i } } { 2 } ( x _ { k + 1 } - x _ { k } ) _ { i } ^ { 2 } } & { } \\ { \displaystyle = - \eta g _ { k } ^ { T } \mathrm { s i g n } ( V _ { k } ) + \eta ^ { 2 } \sum _ { i = 1 } ^ { d } \frac { L _ { i } } { 2 } } & { } \\ { \displaystyle = - \eta \| g _ { k } \| _ { 1 } + \frac { \eta ^ { 2 } } { 2 } \| \vec { L } \| _ { 1 } + 2 \eta \displaystyle \sum _ { i = 1 } ^ { d } | g _ { k , i } | \mathbb { I } [ \mathrm { s i g n } ( V _ { k , i } ) \neq \mathrm { s i g n } ( g _ { k , i } ) ] } & { } \end{array}
|
| 501 |
+
$$
|
| 502 |
+
|
| 503 |
+
Next we find the expected improvement at time $k + 1$ conditioned on the previous iterate.
|
| 504 |
+
|
| 505 |
+
$$
|
| 506 |
+
\mathbb { E } [ f _ { k + 1 } - f _ { k } | x _ { k } ] \leq - \eta \| g _ { k } \| _ { 1 } + \frac { \eta ^ { 2 } } { 2 } \| \vec { L } \| _ { 1 } + 2 \eta \sum _ { i = 1 } ^ { d } | g _ { k , i } | \operatorname { \mathbb { P } } [ \mathrm { s i g n } ( V _ { k , i } ) \neq \mathrm { s i g n } ( g _ { k , i } ) ]
|
| 507 |
+
$$
|
| 508 |
+
|
| 509 |
+
From $( \star )$ , we have that the probability of the vote failing for the $i ^ { t h }$ coordinate is bounded by
|
| 510 |
+
|
| 511 |
+
$$
|
| 512 |
+
\mathbb { P } [ \mathrm { s i g n } ( V _ { k , i } ) \neq \mathrm { s i g n } ( g _ { k , i } ) ] \leq \frac { \sigma _ { k , i } } { ( 1 - 2 \alpha ) \sqrt { M } | g _ { k , i } | }
|
| 513 |
+
$$
|
| 514 |
+
|
| 515 |
+
where $\sigma _ { k , i }$ refers to the variance of the $k ^ { t h }$ stochastic gradient estimate, computed over a mini-batch√ of size $n$ . Therefore, by Assumption 3, we have that $\bar { \sigma } _ { k , i } \leq \sigma _ { i } / \sqrt { n }$ .
|
| 516 |
+
|
| 517 |
+
We now substitute these results and our learning rate and mini-batch settings into the expected improvement:
|
| 518 |
+
|
| 519 |
+
$$
|
| 520 |
+
\begin{array} { l } { \displaystyle \mathbb { E } [ f _ { k + 1 } - f _ { k } | x _ { k } ] \leq - \eta \| g _ { k } \| _ { 1 } + \frac { 2 \eta } { \sqrt { n } } \frac { \| \vec { \sigma } \| _ { 1 } } { ( 1 - 2 \alpha ) \sqrt { M } } + \frac { \eta ^ { 2 } } { 2 } \| \vec { L } \| _ { 1 } } \\ { \displaystyle \qquad = - \sqrt { \frac { f _ { 0 } - f _ { * } } { \| L \| _ { 1 } K } } \| g _ { k } \| _ { 1 } + 2 \sqrt { \frac { f _ { 0 } - f _ { * } } { \| L \| _ { 1 } K ^ { 2 } } } \frac { \| \vec { \sigma } \| _ { 1 } } { ( 1 - 2 \alpha ) \sqrt { M } } + \frac { f _ { 0 } - f _ { * } } { 2 K } } \end{array}
|
| 521 |
+
$$
|
| 522 |
+
|
| 523 |
+
Now extend the expectation over randomness in the trajectory, and perform a telescoping sum over the iterations:
|
| 524 |
+
|
| 525 |
+
$$
|
| 526 |
+
\begin{array} { r l } { f _ { 0 } - f ^ { * } \geq f _ { 0 } - \mathbb { E } [ f _ { K } ] } & { } \\ & { \qquad = \displaystyle \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } [ f _ { k } - f _ { k + 1 } ] } \\ & { \qquad \geq \displaystyle \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left[ \sqrt { \frac { f _ { 0 } - f _ { * } } { \| L \| _ { 1 } K } } \| g _ { k } \| _ { 1 } - 2 \sqrt { \frac { f _ { 0 } - f _ { * } } { \| L \| _ { 1 } K ^ { 2 } } } \frac { \| \vec { \sigma } \| _ { 1 } } { ( 1 - 2 \alpha ) \sqrt { M } } - \frac { f _ { 0 } - f _ { * } } { 2 K } \right] } \\ & { \qquad = \displaystyle \sqrt { \frac { K ( f _ { 0 } - f _ { * } ) } { \| L \| _ { 1 } } } \mathbb { E } \left[ \frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \| g _ { k } \| _ { 1 } \right] - 2 \sqrt { \frac { f _ { 0 } - f _ { * } } { \| L \| _ { 1 } } } \frac { \| \vec { \sigma } \| _ { 1 } } { ( 1 - 2 \alpha ) \sqrt { M } } - \frac { f _ { 0 } - f _ { * } } { 2 } } \end{array}
|
| 527 |
+
$$
|
| 528 |
+
|
| 529 |
+
We can rearrange this inequality to yield the rate:
|
| 530 |
+
|
| 531 |
+
$$
|
| 532 |
+
\frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \| g _ { k } \| _ { 1 } \leq \frac { 1 } { \sqrt { K } } \bigg [ 2 \frac { \| \vec { \sigma } \| _ { 1 } } { ( 1 - 2 \alpha ) \sqrt { M } } + \frac { 3 } { 2 } \sqrt { \| L \| _ { 1 } ( f _ { 0 } - f ^ { * } ) } \bigg ]
|
| 533 |
+
$$
|
| 534 |
+
|
| 535 |
+
$$
|
| 536 |
+
\mathbb { E } \bigg [ \frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \| g _ { k } \| _ { 1 } \bigg ] \leq \frac { 1 } { \sqrt { K } } \bigg [ \frac { 3 } { 2 } \sqrt { \| L \| _ { 1 } } ( f _ { 0 } - f _ { * } ) + 2 \| \vec { \sigma } \| _ { 1 } \bigg ]
|
| 537 |
+
$$
|
| 538 |
+
|
| 539 |
+
Since we are growing our mini-batch size, it will take $N = O ( K ^ { 2 } )$ gradient calls to reach step $K$ Substitute this in on the right hand side, square the result, use that ${ \frac { 3 } { 2 } } < 2$ , and we are done:
|
| 540 |
+
|
| 541 |
+
$$
|
| 542 |
+
\left[ \frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left\| g _ { k } \right\| _ { 1 } \right] ^ { 2 } \leq \frac { 4 } { \sqrt { N } } \left[ \frac { \| \vec { \sigma } \| _ { 1 } } { ( 1 - 2 \alpha ) \sqrt { M } } + \sqrt { \| L \| _ { 1 } ( f _ { 0 } - f ^ { * } ) } \right] ^ { 2 }
|
| 543 |
+
$$
|
parse/train/BJxhijAcY7/BJxhijAcY7_content_list.json
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parse/train/BJxhijAcY7/BJxhijAcY7_middle.json
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|
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parse/train/BJxhijAcY7/BJxhijAcY7_model.json
ADDED
|
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|
|
|
parse/train/ByIAPUcee/ByIAPUcee.md
ADDED
|
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|
| 1 |
+
# FRUSTRATINGLY SHORT ATTENTION SPANS IN NEURAL LANGUAGE MODELING
|
| 2 |
+
|
| 3 |
+
Michał Daniluk, Tim Rocktaschel, Johannes Welbl & Sebastian Riedel ¨
|
| 4 |
+
|
| 5 |
+
Department of Computer Science
|
| 6 |
+
University College London
|
| 7 |
+
michal.daniluk.15@ucl.ac.uk,
|
| 8 |
+
{t.rocktaschel,j.welbl,s.riedel}@cs.ucl.ac.uk
|
| 9 |
+
|
| 10 |
+
# ABSTRACT
|
| 11 |
+
|
| 12 |
+
Neural language models predict the next token using a latent representation of the immediate token history. Recently, various methods for augmenting neural language models with an attention mechanism over a differentiable memory have been proposed. For predicting the next token, these models query information from a memory of the recent history which can facilitate learning mid- and long-range dependencies. However, conventional attention mechanisms used in memoryaugmented neural language models produce a single output vector per time step. This vector is used both for predicting the next token as well as for the key and value of a differentiable memory of a token history. In this paper, we propose a neural language model with a key-value attention mechanism that outputs separate representations for the key and value of a differentiable memory, as well as for encoding the next-word distribution. This model outperforms existing memoryaugmented neural language models on two corpora. Yet, we found that our method mainly utilizes a memory of the five most recent output representations. This led to the unexpected main finding that a much simpler model based only on the concatenation of recent output representations from previous time steps is on par with more sophisticated memory-augmented neural language models.
|
| 13 |
+
|
| 14 |
+
# 1 INTRODUCTION
|
| 15 |
+
|
| 16 |
+
At the core of language models (LMs) is their ability to infer the next word given a context. This requires representing context-specific dependencies in a sequence across different time scales. On the one hand, classical $N$ -gram language models capture relevant dependencies between words in short time distances explicitly, but suffer from data sparsity. Neural language models, on the other hand, maintain and update a dense vector representation over a sequence where time dependencies are captured implicitly (Mikolov et al., 2010). A recent extension of neural sequence models are attention mechanisms (Bahdanau et al., 2015), which can capture long-range connections more directly. However, we argue that applying such an attention mechanism directly to neural language models requires output vectors to fulfill several purposes at the same time: they need to (i) encode a distribution for predicting the next token, (ii) serve as a key to compute the attention vector, as well as (iii) encode relevant content to inform future predictions.
|
| 17 |
+
|
| 18 |
+
We hypothesize that such overloaded use of output representations makes training the model difficult and propose a modification to the attention mechanism which separates these functions explicitly, inspired by Miller et al. (2016); Ba et al. (2016); Reed & de Freitas (2015); Gulcehre et al. (2016). Specifically, at every time step our neural language model outputs three vectors. The first is used to encode the next-word distribution, the second serves as key, and the third as value for an attention mechanism. We term the model key-value-predict attention and show that it outperforms existing memory-augmented neural language models on the Children’s Book Test (CBT, Hill et al., 2016) and a new corpus of 7500 Wikipedia articles. However, we observed that this model pays attention mainly to the previous five memories. We thus also experimented with a much simpler model that only uses a concatenation of output vectors from the previous time steps for predicting the next token. This simple model is on par with more sophisticated memory-augmented neural language models. Thus, our main finding is that modeling short attention spans properly works well and provides notable improvements over a neural language model with attention. Conversely, it seems to be notoriously hard to train neural language models to leverage long-range dependencies.
|
| 19 |
+
|
| 20 |
+
|
| 21 |
+
Figure 1: Memory-augmented neural language modelling architectures.
|
| 22 |
+
|
| 23 |
+
In this paper, we investigate various memory-augmented neural language models and compare them against previous architectures. Our contributions are threefold: (i) we propose a key-value attention mechanism that uses specific output representations for querying a sliding-window memory of previous token representations, (ii) we demonstrate that while this new architecture outperforms previous memory-augmented neural language models, it mainly utilizes a memory of the previous five representations, and finally (iii) based on this observation we experiment with a much simpler but effective model that uses the concatenation of three previous output representations to predict the next word.
|
| 24 |
+
|
| 25 |
+
# 2 METHODS
|
| 26 |
+
|
| 27 |
+
In the following, we discuss methods for extending neural language models with differentiable memory. We first present a standard attention mechanism for language modeling (§2.1). Subsequently, we introduce two methods for separating the usage of output vectors in the attention mechanism: (i) using a dedicated key and value (§2.2), and (ii) further separating the value into a memory value and a representation that encodes the next-word distribution (§2.3). Finally, we describe a very simple method that concatenates previous output representations for predicting the next token (§2.4).
|
| 28 |
+
|
| 29 |
+
# 2.1 ATTENTION FOR NEURAL LANGUAGE MODELING
|
| 30 |
+
|
| 31 |
+
Augmenting a neural language model with attention (Bahdanau et al., 2015) is straight-forward. We simply take the previous $L$ output vectors as memory $Y _ { t } = [ \pmb { h } _ { t - L } \ \cdot \ \cdot \ \pmb { h } _ { t - 1 } ] \in \mathbb { R } ^ { k \times \tilde { L } }$ where $k$ is the output dimension of a Long Short-Term Memory (LSTM) unit (Hochreiter & Schmidhuber, 1997). This memory could in principle contain all previous output representations, but for practical reasons we only keep a sliding window of the previous $L$ outputs. Let $\boldsymbol { h } _ { t } \in \mathbb { R } ^ { k }$ be the output representation at time step $t$ and $\mathbf { 1 } \in \mathbb { R } ^ { L }$ be a vector of ones.
|
| 32 |
+
|
| 33 |
+
The attention weights $\pmb { \alpha } \in \mathbb { R } ^ { L }$ are computed from a comparison of the current and previous LSTM outputs. Subsequently, the context vector $\boldsymbol { r } _ { t } \in \mathbb { R } ^ { k }$ is calculated from a sum over previous output vectors weighted by their respective attention value. This can be formulated as
|
| 34 |
+
|
| 35 |
+
$$
|
| 36 |
+
\begin{array} { r l } & { M _ { t } = \operatorname { t a n h } ( W ^ { Y } Y _ { t } + ( W ^ { h } h _ { t } ) \mathbf { 1 } ^ { T } ) } \\ & { ~ \alpha _ { t } = \operatorname { s o f t m a x } ( \pmb { w } ^ { T } M _ { t } ) } \\ & { ~ { \pmb { r } _ { t } } = { \pmb { Y } _ { t } } \pmb { \alpha } ^ { T } } \end{array}
|
| 37 |
+
$$
|
| 38 |
+
|
| 39 |
+
$$
|
| 40 |
+
\begin{array} { l } { \in \mathbb { R } ^ { k \times L } } \\ { \in \mathbb { R } ^ { 1 \times L } } \\ { \in \mathbb { R } ^ { k } } \end{array}
|
| 41 |
+
$$
|
| 42 |
+
|
| 43 |
+
where $W ^ { Y }$ , $W ^ { h } \in \mathbb { R } ^ { k \times k }$ are trainable projection matrices and $\mathbf { \boldsymbol { w } } \in \mathbb { R } ^ { k }$ is a trainable vector. The final representation that encodes the next-word distribution is computed from a non-linear combination of the attention-weighted representation $\mathbf { \nabla } _ { \mathbf { \boldsymbol { r } } _ { t } }$ of previous outputs and the final output vector $\boldsymbol { h } _ { t }$ via
|
| 44 |
+
|
| 45 |
+
$$
|
| 46 |
+
\begin{array} { r } { \pmb { h } _ { t } ^ { * } = \operatorname { t a n h } ( \pmb { W } ^ { r } \pmb { r } _ { t } + \pmb { W } ^ { x } \pmb { h } _ { t } ) } \end{array}
|
| 47 |
+
$$
|
| 48 |
+
|
| 49 |
+
$$
|
| 50 |
+
\in \mathbb { R } ^ { k }
|
| 51 |
+
$$
|
| 52 |
+
|
| 53 |
+
where $W ^ { r }$ , $W ^ { x } \in \mathbb { R } ^ { k \times k }$ are trainable projection matrices. An overview of this architecture is depicted in Figure 1a. Lastly, the probablity distribution ${ \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \Xi } _ { \mathbf { } } \mathbf { \Lambda } _ { \mathbf { } } \mathbf { \Lambda } _ { \mathbf { } } \textbf { } _ { \mathbf { } } \textbf { } \textbf { } _ { \mathrm { } }$ for the next word is represented by
|
| 54 |
+
|
| 55 |
+
$$
|
| 56 |
+
\begin{array} { r } { y _ { t } = \mathrm { s o f t m a x } ( W ^ { * } h _ { t } ^ { * } + b ) \qquad \in \mathbb { R } ^ { | V | } } \end{array}
|
| 57 |
+
$$
|
| 58 |
+
|
| 59 |
+
where $W ^ { \ast } \in \mathbb { R } ^ { | V | \times k }$ and $\pmb { b } \in \mathbb { R } ^ { | V | }$ are a trainable projection matrix and bias, respectively.
|
| 60 |
+
|
| 61 |
+
# 2.2 KEY-VALUE ATTENTION
|
| 62 |
+
|
| 63 |
+
Inspired by Miller et al. (2016); Ba et al. (2016); Reed & de Freitas (2015); Gulcehre et al. (2016), we introduce a key-value attention model that separates output vectors into keys used for calculating the attention distribution $\pmb { \alpha } _ { t }$ , and a value part used for encoding the next-word distribution and context representation. This model is depicted in Figure 1b. Formally, we rewrite Equations 1-4 as follows:
|
| 64 |
+
|
| 65 |
+
$$
|
| 66 |
+
\begin{array} { r l r l } & { \Big [ \pmb { k } _ { t } \Big ] = \pmb { h } _ { t } } & { \qquad } & & { \in \mathbb { R } ^ { 2 k } } \\ & { M _ { t } = \operatorname { t a n h } ( \pmb { W } ^ { Y } [ \pmb { k } _ { t - L } \cdot \cdot \cdot \pmb { k } _ { t - 1 } ] + ( \pmb { W } ^ { h } \pmb { k } _ { t } ) \mathbf { 1 } ^ { T } ) } & & { \qquad \in \mathbb { R } ^ { k \times L } } \\ & { \pmb { \alpha } _ { t } = \operatorname { s o f t m a x } ( \pmb { w } ^ { T } M _ { t } ) } & & { \in \mathbb { R } ^ { 1 \times L } } \\ & { r _ { t } = [ \pmb { v } _ { t - L } \cdot \cdot \cdot \cdot \pmb { v } _ { t - 1 } ] \pmb { \alpha } ^ { T } } & & { \in \mathbb { R } ^ { k } } \\ & { \pmb { h } _ { t } ^ { * } = \operatorname { t a n h } ( \pmb { W } ^ { r } r _ { t } + \pmb { W } ^ { x } \pmb { v } _ { t } ) } & & { \in \mathbb { R } ^ { k } } \end{array}
|
| 67 |
+
$$
|
| 68 |
+
|
| 69 |
+
In essence, Equation 7 compares the key at time step $t$ with the previous $L$ keys to calculate the attention distribution $\pmb { \alpha } _ { t }$ which is then used in Equation 9 to obtain a weighted context representation from values associated with these keys.
|
| 70 |
+
|
| 71 |
+
# 2.3 KEY-VALUE-PREDICT ATTENTION
|
| 72 |
+
|
| 73 |
+
Even with a key-value separation, a potential problem is that the same representation ${ \mathbf { } } v _ { t }$ is still used both for encoding the probability distribution of the next word and for retrieval from the memory via the attention later. Thus, we experimented with another extension of this model where we further separate $\boldsymbol { h } _ { t }$ into a $k e y$ , a value and a predict representation where the latter is only used for encoding the next-word distribution (see Figure 1c). To this end, equations 6 and 10 are replaced by
|
| 74 |
+
|
| 75 |
+
$$
|
| 76 |
+
\begin{array} { r l } & { \left\lceil \begin{array} { l } { k _ { t } } \\ { v _ { t } } \\ { p _ { t } } \end{array} \right\rceil = h _ { t } } \\ & { ~ \left\lceil \begin{array} { l } { k _ { t } } \end{array} \right\rceil } \\ & { ~ h _ { t } ^ { * } = \operatorname { t a n h } ( W ^ { r } r _ { t } + W ^ { x } p _ { t } ) } \end{array}
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+
$$
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+
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$$
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\begin{array} { l } { \in \mathbb { R } ^ { 3 k } } \\ { \ } \\ { \in \mathbb { R } ^ { k } } \end{array}
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$$
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+
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More precisely, the output vector $\boldsymbol { h } _ { t }$ is divided into three equal parts: key, value and predict. In our implementation we simply split the output vector $h _ { t }$ into $k _ { t }$ , ${ \mathbf { } } v _ { t }$ and ${ \pmb p } _ { t }$ . To this end the hidden dimension of the key-value-predict attention model needs to be a multiplicative of three. Consequently, the dimensions of $k _ { t }$ , ${ \mathbf { } } v _ { t }$ and ${ \mathbf { } } p _ { t }$ are 100 for a hidden dimension of 300.
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# 2.4 $N$ -GRAM RECURRENT NEURAL NETWORK
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Neural language models often work best in combination with traditional $N$ -gram models (Mikolov et al., 2011; Chelba et al., 2013; Williams et al., 2015; Ji et al., 2016; Shazeer et al., 2015), since the former excel at generalization while the latter ensure memorization. In addition, from initial experiments with memory-augmented neural language models, we found that usually only the previous five output representations are utilized. This is in line with observations by Tran et al. (2016). Hence, we experiment with a much simpler architecture depicted in Figure 1d. Instead of an attention mechanism, the output representations from the previous $N - 1$ time steps are directly used to calculate next-word probabilities. Specifically, at every time step we split the LSTM output into $N - 1$ vectors $[ { \pmb h } _ { t } ^ { 1 } , \dots , { \pmb h } _ { t } ^ { N - 1 } ]$ and replace Equation 4 with
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$$
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\begin{array} { r } { h _ { t } ^ { * } = \operatorname { t a n h } \left( W ^ { N } \left[ \begin{array} { c } { \ h _ { t } ^ { 1 } } \\ { \vdots } \\ { h _ { t - N + 1 } ^ { N - 1 } } \end{array} \right] \right) } \end{array}
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$$
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where $W ^ { N } \in \mathbb { R } ^ { k \times ( N - 1 ) k }$ is a trainable projection matrix. This model is related to higher-order RNNs (Soltani $\&$ Jiang, 2016) with the difference that we do not incorporate output vectors from the previous steps into the hidden state but only use them for predicting the next word. Furthermore, note that at time step $t$ the first part of the output vector $ { \boldsymbol { h } } _ { t } ^ { 1 }$ will contribute to predicting the next word, the second part $h _ { t } ^ { 2 }$ will contribute to predicting the second word thereafter, and so on. As the output vectors from the $N - 1$ previous time-steps are used to score the next word, we call the resulting model an $N$ -gram RNN.
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# 3 RELATED WORK
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Early attempts of using memory in neural networks have been undertaken by Taylor (1959) and Steinbuch $\&$ Piske (1963) by performing nearest-neighbor operations on input vectors and fitting parametric models to the retrieved sets. The dedicated use of external memory in neural architectures has more recently witnessed increased interest. Weston et al. (2015) introduced Memory Networks to explicitly segregate memory storage from the computation of the neural network, and Sukhbaatar et al. (2015) trained this model end-to-end with an attention-based memory addressing mechanism. The Neural Turing Machines by Graves et al. (2014) add an external differentiable memory with read-write functions to a controller recurrent neural network, and has shown promising results in simple sequence tasks such as copying and sorting. These models make use of external memory, whereas our model directly uses a short sequence from the history of tokens to dynamically populate an addressable memory.
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In sequence modeling, RNNs such as LSTMs (Hochreiter & Schmidhuber, 1997) maintain an internal memory state as they process an input sequence. Attending over previous state outputs on top of an RNN encoder has improved performances in a wide range of tasks, including machine translation (Bahdanau et al., 2015), recognizing textual entailment (Rocktaschel et al. ¨ , 2016), sentence summarization (Rush et al., 2015), image captioning (Xu et al., 2015) and speech recognition (Chorowski et al., 2015).
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Recently, Cheng et al. (2016) proposed an architecture that modifies the standard LSTM by replacing the memory cell with a memory network (Weston et al., 2015). Another proposal for conditioning on previous output representations are Higher-order Recurrent Neural Networks (HORNNs, Soltani & Jiang, 2016). Soltani & Jiang found it useful to include information from multiple preceding RNN states when computing the next state. This previous work centers around preceding state vectors, whereas we investigate attention mechanisms on top of RNN outputs, i.e. the vectors used for predicting the next word. Furthermore, instead of pooling we use attention vectors to calculate a context representation of previous memories.
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Yang et al. (2016) introduced a reference-aware neural language model where at every position a latent variable determines from which source a target token is generated, e.g., by copying entries from a table or referencing entities that were mentioned earlier.
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Another class of models that include memory into sequence modeling are Recurrent Memory Networks (RMNs) (Tran et al., 2016). Here, a memory block accesses the most recent input words to selectively attend over relevant word representations from a global vocabulary. RMNs use a global memory with two input word vector look-up tables for the attention mechanism, and consequently have a large number of trainable parameters. Instead, we proposed models that need much fewer parameters by producing the vectors that will be attended over in the future, which can be seen as a memory that is dynamically populated by the language model.
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Finally, the functional separation of look-up keys and memory content has been found useful for Memory Networks (Miller et al., 2016), Neural Programmer-Interpreters (Reed & de Freitas, 2015), Dynamic Neural Turing Machines (Gulcehre et al., 2016), and Fast Associative Memory (Ba et al., 2016). We apply and extend this principle to neural language models.
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# 4 EXPERIMENTS
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We evaluate models on two different corpora for language modeling. The first is a subset of the Wikipedia corpus.1 It consists of 7500 English Wikipedia articles (dump from 6 Feb 2015) belonging to one of the following categories: People, Cities, Countries, Universities, and Novels. We chose these categories as we expect articles in these categories to often contain references to previously mentioned entities. Subsequently, we split this corpus into a train, development, and test part, resulting in corpora of $2 2 . 5 \mathbf { M }$ words, 1.2M and 1.2M words, respectively. We map all numbers to a dedicated numerical symbol $N$ and restrict the vocabulary to the 77K most frequent words, encompassing $9 7 \%$ of the training vocabulary. All other words are replaced by the UNK symbol. The average length of sentences is 25 tokens. In addition to this Wikipedia corpus, we also run experiments on the Children’s Book Test (CBT Hill et al., 2016). While this corpus is designed for cloze-style question-answering, in this paper we use it to test how well language models can exploit wider linguistic context.
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# 4.1 TRAINING PROCEDURE
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We use ADAM (Kingma & Ba, 2015) with an initial learning rate of 0.001 and a mini-batch size of 64 for optimization. Furthermore, we apply gradient clipping at a gradient norm of 5 (Pascanu et al., 2013). The bias of the LSTM’s forget gate is initialized to 1 (Jozefowicz et al., 2016), while other parameters are initialized uniformly from the range $( - 0 . 1 , 0 . 1 )$ . Backpropagation Through Time (Rumelhart et al., 1985; Werbos, 1990) was used to train the network with 20 steps of unrolling. We reset the hidden states between articles for the Wikipedia corpus and between stories for CBT, respectively. We take the best configuration based on performance on the validation set and evaluate it on the test set.
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# 5 RESULTS
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In the first set of experiments we explore how well the proposed models and Tran et al.’s Recurrentmemory Model can make use of histories of varying lengths. Perplexity results for different attention window sizes on the Wikipedia corpus are summarized in Figure 2a. The average attention these models pay to specific positions in the history is illustrated in Figure 3. We observed that although our models attend over tokens further in the past more often than the Recurrent-memory Model, attending over a longer history does not significantly improve the perplexity of any attentive model.
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The much simpler $N$ -gram RNN model achieves comparable results (Figure 2b) and seems to work best with a history of the previous three output vectors (4-gram RNN). As a result, we choose the 4-gram model for the following $N$ -gram RNN experiments.
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Figure 2: Perplexities of memory-augmented neural language models on the Wikipedia corpus (a-c) and accuracies on the CBT test set (d).
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<table><tr><td>Model</td><td colspan="4">Attention Window Size</td></tr><tr><td></td><td>1</td><td>5</td><td>10</td><td>15</td></tr><tr><td>RM(+tM-g) (Tran et al.,2016)</td><td>83.5</td><td>80.5</td><td>80.3</td><td>80.1</td></tr><tr><td>Attention</td><td>82.2</td><td>82.2</td><td>82.0</td><td>82.8</td></tr><tr><td>Key-Value</td><td>78.7</td><td>79.0</td><td>78.2</td><td>78.9</td></tr><tr><td>Key-Value-Predict</td><td>76.1</td><td>75.8</td><td>76.0</td><td>75.8</td></tr></table>
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(a) Test perplexity of different attention architectures with varying attention window sizes. Best perplexity per model is italic.
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(b) Comparison of $N$ -gram neural language models. $w$ denotes the input size, $k$ the hidden size and $\theta _ { M }$ the total number of model parameters.
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<table><tr><td>Model</td><td>w</td><td>k</td><td>0M</td><td>Dev</td><td>Test</td></tr><tr><td>2-gram RNN</td><td>300</td><td>564</td><td>23.9M</td><td>76.0</td><td>77.1</td></tr><tr><td>3-gram RNN</td><td>300</td><td>786</td><td>23.9M</td><td>74.9</td><td>75.9</td></tr><tr><td>4-gram RNN</td><td>300</td><td>968</td><td>23.9M</td><td>74.8</td><td>75.9</td></tr><tr><td>5-gram RNN</td><td>300</td><td>1120</td><td>23.9M</td><td>76.0</td><td>77.3</td></tr></table>
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(c) Summary of models with best attention window size $a$ . The total number of model parameters, including word representations, is denoted by $\theta _ { W + M }$ (without word representations $\theta _ { M }$ ).
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<table><tr><td>Model</td><td>w</td><td>k</td><td>a</td><td>0W+M</td><td>0M</td><td>Dev</td><td>Test</td></tr><tr><td>RNN</td><td>300</td><td>307</td><td>1</td><td>47.0M</td><td>23.9M</td><td>121.7</td><td>125.7</td></tr><tr><td>LSTM</td><td>300</td><td>300</td><td>-</td><td>47.0M</td><td>23.9M</td><td>83.2</td><td>85.2</td></tr><tr><td>FOFE HORNN (3-rd order) (Soltani & Jiang,2016)</td><td>300</td><td>303</td><td>-</td><td>47.0M</td><td>23.9M</td><td>116.7</td><td>120.5</td></tr><tr><td>Gated HORNN (3-rd order) (Soltani & Jiang, 2016)</td><td>300</td><td>297</td><td>-</td><td>47.0M</td><td>23.9M</td><td>93.9</td><td>97.1</td></tr><tr><td>RM(+tM-g) (Tran et al.,2016)</td><td>300</td><td>300</td><td>15</td><td>93.7M</td><td>70.6M</td><td>78.2</td><td>80.1</td></tr><tr><td>Attention</td><td>300</td><td>296</td><td>10</td><td>47.0M</td><td>23.9M</td><td>80.6</td><td>82.0</td></tr><tr><td>Key-Value</td><td>300</td><td>560</td><td>10</td><td>47.0M</td><td>23.9M</td><td>77.1</td><td>78.2</td></tr><tr><td>Key-Value-Predict</td><td>300</td><td>834</td><td>5</td><td>47.0M</td><td>23.9M</td><td>74.2</td><td>75.8</td></tr><tr><td>4-gram RNN</td><td>300</td><td>968</td><td>-</td><td>47.0M</td><td>23.9M</td><td>74.8</td><td>75.9</td></tr></table>
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(d) Results on CBT; those marked with ‡ are taken from Hill et al. (2016).
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<table><tr><td>Model</td><td>Named Entities</td><td>Common Nouns</td><td>Verbs</td><td>Prepositions</td></tr><tr><td>Humans (context+query) ‡</td><td>0.816</td><td>0.816</td><td>0.828</td><td>0.708</td></tr><tr><td>Kneser-Ney LM ‡</td><td>0.390</td><td>0.544</td><td>0.778</td><td>0.768</td></tr><tr><td>Kneser-Ney LM + cache ‡</td><td>0.439</td><td>0.577</td><td>0.772</td><td>0.679</td></tr><tr><td>LSTM (context+query) ‡</td><td>0.418</td><td>0.560</td><td>0.818</td><td>0.791</td></tr><tr><td>Memory Network ‡</td><td>0.666</td><td>0.630</td><td>0.690</td><td>0.703</td></tr><tr><td>AS Reader, avg ensemble (Kadlec et al., 2016)</td><td>0.706</td><td>0.689</td><td>一</td><td>1</td></tr><tr><td>AS Reader, greedy ensemble (Kadlec et al., 2016)</td><td>0.710</td><td>0.675</td><td></td><td></td></tr><tr><td>QANN, 4 hops, GloVe (Weissenborn, 2016)</td><td>0.729</td><td>1</td><td></td><td></td></tr><tr><td>AoA Reader, single model (Cui et al.,2016a)</td><td>0.720</td><td>0.694</td><td></td><td></td></tr><tr><td>CAS Reader, mode avg (Cui et al.,2016b)</td><td>0.692</td><td>0.657</td><td></td><td></td></tr><tr><td>GA Reader, ensemble (Dhingra et al., 2016)</td><td>0.719</td><td>0.694</td><td></td><td></td></tr><tr><td>EpiReader, ensemble (Trischler et al., 2016)</td><td>0.718</td><td>0.706</td><td></td><td></td></tr><tr><td>FOFE HORNN (3-rd order) (Soltani & Jiang,2016)</td><td>0.465</td><td>0.497</td><td>0.774</td><td>0.741</td></tr><tr><td>Gated HORNN (3-rd order) (Soltani & Jiang,2016)</td><td>0.508</td><td>0.547</td><td>0.790</td><td>0.774</td></tr><tr><td>RM(+tM-g) (Tran et al.,2016)</td><td>0.525</td><td>0.597</td><td>0.817</td><td>0.797</td></tr><tr><td>LSTM</td><td>0.523</td><td>0.604</td><td>0.819</td><td>0.786</td></tr><tr><td>Attention</td><td>0.538</td><td>0.595</td><td>0.826</td><td>0.803</td></tr><tr><td>Key-Value</td><td>0.528</td><td>0.601</td><td>0.822</td><td>0.813</td></tr><tr><td>Key-Value-Predict</td><td>0.528</td><td>0.599</td><td>0.829</td><td>0.803</td></tr><tr><td>4-gram RNN</td><td>0.532</td><td>0.598</td><td>0.815</td><td>0.800</td></tr></table>
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# 5.1 COMPARISON WITH STATE-OF-THE-ART MODELS
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In the next set of experiments, we compared our proposed models against a variety of state-of-the-art models on the Wikipedia and CBT corpora. Results are shown in Figure 2c and 2d, respectively. Note that the models presented here do not achieve state-of-the-art on CBT as they are language models and not tailored towards cloze-sytle question answering. Thus, we merely use this corpus for comparing different neural language model architectures. We reimplemented the Recurrent-Memory model by Tran et al. (2016) with the temporal matrix and gating composition function $\mathbf { \left( R M + t M - g \right) }$ ).
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Figure 3: Attention weights of the Key-Value-Predict model on a randomly sampled Wikipedia article (a) and average attention weight distribution on the whole Wikipedia test set for $\mathbf { R M } ( + \mathbf { t M } - \mathbf { g } )$ , Attention, Key-Value and Key-Value-Predict models (b). The rightmost positions represent the most recent history.
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+
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+
Furthermore, we reimplemented Higher Order Recurrent Neural Networks (HORNNs) by Soltani & Jiang (2016).
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+
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+
To ensure a comparable number of parameters to a vanilla LSTM model, we adjusted the hidden size of all models to have roughly the same total number of model parameters. The attention window size $N$ for the $N$ -gram RNN model was set to 4 according to the best validation set perplexity on the Wikipedia corpus. Below we discuss the results in detail.
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Attention By using a neural language model with an attention mechanism over a dynamically populated memory, we observed a 3.2 points lower perplexity over a vanilla LSTM on Wikipedia, but only notable differences for predicting verbs and prepositions in CBT. This indicates that incorporating mechanisms for querying previous output vectors is useful for neural language modeling.
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Key-Value Decomposing the output vector into a key-value paired memory improves the perplexity by 7.0 points compared to a baseline LSTM, and by 1.9 points compared to the $\mathbf { R M } ( + \mathbf { t M } - \mathbf { g } )$ model. Again, for CBT we see only small improvements.
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+
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Key-Value-Predict By further separating the output vector into a key, value and next-word prediction part, we get the lowest perplexity and gain 9.4 points over a baseline LSTM, a 4.3 points compared to $\mathbf { R M } ( + \mathbf { t M } - \mathbf { g } )$ , and 2.4 points compared to only splitting the output into a key and value. For CBT, we see an accuracy increase of 1.0 percentage points for verbs, and 1.7 for prepositions. As stated earlier, the performance of the Key-Value-Predict model does not improve significantly when increasing the attention window size. This leads to the conclusion that none of the attentive models investigated in this paper can utilize a large memory of previous token representations. Moreover, none of the presented methods differ significantly for predicting common nouns and named entities in CBT.
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$N$ -gram RNN Our main finding is that the simple modification of using output vectors from the previous time steps for the next-word prediction leads to perplexities that are on par with or better than more complicated neural language models with attention. Specifically, the 4-gram RNN achieves only slightly worse perplexities than the Key-Value-Predict architecture.
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# 6 CONCLUSION
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In this paper, we observed that using an attention mechanism for neural language modeling where we separate output vectors into a key, value and predict part outperform simpler attention mechanisms on a Wikipedia corpus and the Children Book Test (CBT, Hill et al., 2016). However, we found that all attentive neural language models mainly utilize a memory of only the most recent history and fail to exploit long-range dependencies. In fact, a much simpler $N$ -gram RNN model, which only uses a concatenation of output representations from the previous three time steps, is on par with more sophisticated memory-augmented neural language models. Training neural language models that take long-range dependencies into account seems notoriously hard and needs further investigation. Thus, for future work we want to investigate ways to encourage attending over a longer history, for instance by forcing the model to ignore the local context and only allow attention over output representations further behind the local history.
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# ACKNOWLEDGMENTS
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This work was supported by Microsoft Research and the Engineering and Physical Sciences Research Council through PhD Scholarship Programmes, an Allen Distinguished Investigator Award, and a Marie Curie Career Integration Award.
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|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "FRUSTRATINGLY SHORT ATTENTION SPANS IN NEURAL LANGUAGE MODELING ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
176,
|
| 8 |
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98,
|
| 9 |
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|
| 10 |
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146
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Michał Daniluk, Tim Rocktaschel, Johannes Welbl & Sebastian Riedel ¨ ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
184,
|
| 19 |
+
170,
|
| 20 |
+
673,
|
| 21 |
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185
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "Department of Computer Science \nUniversity College London \nmichal.daniluk.15@ucl.ac.uk, \n{t.rocktaschel,j.welbl,s.riedel}@cs.ucl.ac.uk ",
|
| 28 |
+
"bbox": [
|
| 29 |
+
184,
|
| 30 |
+
186,
|
| 31 |
+
624,
|
| 32 |
+
239
|
| 33 |
+
],
|
| 34 |
+
"page_idx": 0
|
| 35 |
+
},
|
| 36 |
+
{
|
| 37 |
+
"type": "text",
|
| 38 |
+
"text": "ABSTRACT ",
|
| 39 |
+
"text_level": 1,
|
| 40 |
+
"bbox": [
|
| 41 |
+
454,
|
| 42 |
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276,
|
| 43 |
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544,
|
| 44 |
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291
|
| 45 |
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],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "Neural language models predict the next token using a latent representation of the immediate token history. Recently, various methods for augmenting neural language models with an attention mechanism over a differentiable memory have been proposed. For predicting the next token, these models query information from a memory of the recent history which can facilitate learning mid- and long-range dependencies. However, conventional attention mechanisms used in memoryaugmented neural language models produce a single output vector per time step. This vector is used both for predicting the next token as well as for the key and value of a differentiable memory of a token history. In this paper, we propose a neural language model with a key-value attention mechanism that outputs separate representations for the key and value of a differentiable memory, as well as for encoding the next-word distribution. This model outperforms existing memoryaugmented neural language models on two corpora. Yet, we found that our method mainly utilizes a memory of the five most recent output representations. This led to the unexpected main finding that a much simpler model based only on the concatenation of recent output representations from previous time steps is on par with more sophisticated memory-augmented neural language models. ",
|
| 51 |
+
"bbox": [
|
| 52 |
+
233,
|
| 53 |
+
306,
|
| 54 |
+
766,
|
| 55 |
+
542
|
| 56 |
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],
|
| 57 |
+
"page_idx": 0
|
| 58 |
+
},
|
| 59 |
+
{
|
| 60 |
+
"type": "text",
|
| 61 |
+
"text": "1 INTRODUCTION ",
|
| 62 |
+
"text_level": 1,
|
| 63 |
+
"bbox": [
|
| 64 |
+
176,
|
| 65 |
+
566,
|
| 66 |
+
336,
|
| 67 |
+
582
|
| 68 |
+
],
|
| 69 |
+
"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "text",
|
| 73 |
+
"text": "At the core of language models (LMs) is their ability to infer the next word given a context. This requires representing context-specific dependencies in a sequence across different time scales. On the one hand, classical $N$ -gram language models capture relevant dependencies between words in short time distances explicitly, but suffer from data sparsity. Neural language models, on the other hand, maintain and update a dense vector representation over a sequence where time dependencies are captured implicitly (Mikolov et al., 2010). A recent extension of neural sequence models are attention mechanisms (Bahdanau et al., 2015), which can capture long-range connections more directly. However, we argue that applying such an attention mechanism directly to neural language models requires output vectors to fulfill several purposes at the same time: they need to (i) encode a distribution for predicting the next token, (ii) serve as a key to compute the attention vector, as well as (iii) encode relevant content to inform future predictions. ",
|
| 74 |
+
"bbox": [
|
| 75 |
+
174,
|
| 76 |
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|
| 77 |
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|
| 78 |
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|
| 79 |
+
],
|
| 80 |
+
"page_idx": 0
|
| 81 |
+
},
|
| 82 |
+
{
|
| 83 |
+
"type": "text",
|
| 84 |
+
"text": "We hypothesize that such overloaded use of output representations makes training the model difficult and propose a modification to the attention mechanism which separates these functions explicitly, inspired by Miller et al. (2016); Ba et al. (2016); Reed & de Freitas (2015); Gulcehre et al. (2016). Specifically, at every time step our neural language model outputs three vectors. The first is used to encode the next-word distribution, the second serves as key, and the third as value for an attention mechanism. We term the model key-value-predict attention and show that it outperforms existing memory-augmented neural language models on the Children’s Book Test (CBT, Hill et al., 2016) and a new corpus of 7500 Wikipedia articles. However, we observed that this model pays attention mainly to the previous five memories. We thus also experimented with a much simpler model that only uses a concatenation of output vectors from the previous time steps for predicting the next token. This simple model is on par with more sophisticated memory-augmented neural language models. Thus, our main finding is that modeling short attention spans properly works well and provides notable improvements over a neural language model with attention. Conversely, it seems to be notoriously hard to train neural language models to leverage long-range dependencies. ",
|
| 85 |
+
"bbox": [
|
| 86 |
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| 87 |
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| 88 |
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| 89 |
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| 90 |
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],
|
| 91 |
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"page_idx": 0
|
| 92 |
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},
|
| 93 |
+
{
|
| 94 |
+
"type": "image",
|
| 95 |
+
"img_path": "images/302874ddb64e68d1aa6e176ed09a638b131fc709b70c9e1392500c16fa42bdec.jpg",
|
| 96 |
+
"image_caption": [
|
| 97 |
+
"Figure 1: Memory-augmented neural language modelling architectures. "
|
| 98 |
+
],
|
| 99 |
+
"image_footnote": [],
|
| 100 |
+
"bbox": [
|
| 101 |
+
168,
|
| 102 |
+
101,
|
| 103 |
+
826,
|
| 104 |
+
556
|
| 105 |
+
],
|
| 106 |
+
"page_idx": 1
|
| 107 |
+
},
|
| 108 |
+
{
|
| 109 |
+
"type": "text",
|
| 110 |
+
"text": "",
|
| 111 |
+
"bbox": [
|
| 112 |
+
174,
|
| 113 |
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621,
|
| 114 |
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823,
|
| 115 |
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648
|
| 116 |
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],
|
| 117 |
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"page_idx": 1
|
| 118 |
+
},
|
| 119 |
+
{
|
| 120 |
+
"type": "text",
|
| 121 |
+
"text": "In this paper, we investigate various memory-augmented neural language models and compare them against previous architectures. Our contributions are threefold: (i) we propose a key-value attention mechanism that uses specific output representations for querying a sliding-window memory of previous token representations, (ii) we demonstrate that while this new architecture outperforms previous memory-augmented neural language models, it mainly utilizes a memory of the previous five representations, and finally (iii) based on this observation we experiment with a much simpler but effective model that uses the concatenation of three previous output representations to predict the next word. ",
|
| 122 |
+
"bbox": [
|
| 123 |
+
173,
|
| 124 |
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|
| 125 |
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825,
|
| 126 |
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|
| 127 |
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],
|
| 128 |
+
"page_idx": 1
|
| 129 |
+
},
|
| 130 |
+
{
|
| 131 |
+
"type": "text",
|
| 132 |
+
"text": "2 METHODS ",
|
| 133 |
+
"text_level": 1,
|
| 134 |
+
"bbox": [
|
| 135 |
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174,
|
| 136 |
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| 137 |
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| 138 |
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816
|
| 139 |
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],
|
| 140 |
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"page_idx": 1
|
| 141 |
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},
|
| 142 |
+
{
|
| 143 |
+
"type": "text",
|
| 144 |
+
"text": "In the following, we discuss methods for extending neural language models with differentiable memory. We first present a standard attention mechanism for language modeling (§2.1). Subsequently, we introduce two methods for separating the usage of output vectors in the attention mechanism: (i) using a dedicated key and value (§2.2), and (ii) further separating the value into a memory value and a representation that encodes the next-word distribution (§2.3). Finally, we describe a very simple method that concatenates previous output representations for predicting the next token (§2.4). ",
|
| 145 |
+
"bbox": [
|
| 146 |
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174,
|
| 147 |
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|
| 148 |
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825,
|
| 149 |
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|
| 150 |
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],
|
| 151 |
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"page_idx": 1
|
| 152 |
+
},
|
| 153 |
+
{
|
| 154 |
+
"type": "text",
|
| 155 |
+
"text": "2.1 ATTENTION FOR NEURAL LANGUAGE MODELING ",
|
| 156 |
+
"text_level": 1,
|
| 157 |
+
"bbox": [
|
| 158 |
+
173,
|
| 159 |
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103,
|
| 160 |
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560,
|
| 161 |
+
118
|
| 162 |
+
],
|
| 163 |
+
"page_idx": 2
|
| 164 |
+
},
|
| 165 |
+
{
|
| 166 |
+
"type": "text",
|
| 167 |
+
"text": "Augmenting a neural language model with attention (Bahdanau et al., 2015) is straight-forward. We simply take the previous $L$ output vectors as memory $Y _ { t } = [ \\pmb { h } _ { t - L } \\ \\cdot \\ \\cdot \\ \\pmb { h } _ { t - 1 } ] \\in \\mathbb { R } ^ { k \\times \\tilde { L } }$ where $k$ is the output dimension of a Long Short-Term Memory (LSTM) unit (Hochreiter & Schmidhuber, 1997). This memory could in principle contain all previous output representations, but for practical reasons we only keep a sliding window of the previous $L$ outputs. Let $\\boldsymbol { h } _ { t } \\in \\mathbb { R } ^ { k }$ be the output representation at time step $t$ and $\\mathbf { 1 } \\in \\mathbb { R } ^ { L }$ be a vector of ones. ",
|
| 168 |
+
"bbox": [
|
| 169 |
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173,
|
| 170 |
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128,
|
| 171 |
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825,
|
| 172 |
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213
|
| 173 |
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],
|
| 174 |
+
"page_idx": 2
|
| 175 |
+
},
|
| 176 |
+
{
|
| 177 |
+
"type": "text",
|
| 178 |
+
"text": "The attention weights $\\pmb { \\alpha } \\in \\mathbb { R } ^ { L }$ are computed from a comparison of the current and previous LSTM outputs. Subsequently, the context vector $\\boldsymbol { r } _ { t } \\in \\mathbb { R } ^ { k }$ is calculated from a sum over previous output vectors weighted by their respective attention value. This can be formulated as ",
|
| 179 |
+
"bbox": [
|
| 180 |
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174,
|
| 181 |
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219,
|
| 182 |
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825,
|
| 183 |
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262
|
| 184 |
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],
|
| 185 |
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"page_idx": 2
|
| 186 |
+
},
|
| 187 |
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{
|
| 188 |
+
"type": "equation",
|
| 189 |
+
"img_path": "images/ad40128e070864c0b9f8bc745a9ec7cd031746e7ae4cad21c0ebbc84a2c2ad02.jpg",
|
| 190 |
+
"text": "$$\n\\begin{array} { r l } & { M _ { t } = \\operatorname { t a n h } ( W ^ { Y } Y _ { t } + ( W ^ { h } h _ { t } ) \\mathbf { 1 } ^ { T } ) } \\\\ & { ~ \\alpha _ { t } = \\operatorname { s o f t m a x } ( \\pmb { w } ^ { T } M _ { t } ) } \\\\ & { ~ { \\pmb { r } _ { t } } = { \\pmb { Y } _ { t } } \\pmb { \\alpha } ^ { T } } \\end{array}\n$$",
|
| 191 |
+
"text_format": "latex",
|
| 192 |
+
"bbox": [
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| 193 |
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| 196 |
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| 197 |
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],
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| 198 |
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"page_idx": 2
|
| 199 |
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},
|
| 200 |
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{
|
| 201 |
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"type": "equation",
|
| 202 |
+
"img_path": "images/391665ff7bb435179cffe57a900fbb1547df2eedbb68dced9e8167d6b9237132.jpg",
|
| 203 |
+
"text": "$$\n\\begin{array} { l } { \\in \\mathbb { R } ^ { k \\times L } } \\\\ { \\in \\mathbb { R } ^ { 1 \\times L } } \\\\ { \\in \\mathbb { R } ^ { k } } \\end{array}\n$$",
|
| 204 |
+
"text_format": "latex",
|
| 205 |
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"bbox": [
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"page_idx": 2
|
| 212 |
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},
|
| 213 |
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{
|
| 214 |
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"type": "text",
|
| 215 |
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"text": "where $W ^ { Y }$ , $W ^ { h } \\in \\mathbb { R } ^ { k \\times k }$ are trainable projection matrices and $\\mathbf { \\boldsymbol { w } } \\in \\mathbb { R } ^ { k }$ is a trainable vector. The final representation that encodes the next-word distribution is computed from a non-linear combination of the attention-weighted representation $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { r } } _ { t } }$ of previous outputs and the final output vector $\\boldsymbol { h } _ { t }$ via ",
|
| 216 |
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"bbox": [
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"img_path": "images/256e0959299e5b7ee251b778713642fb126040d7ad2d69abb9b085165bf51fb6.jpg",
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| 227 |
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"text": "$$\n\\begin{array} { r } { \\pmb { h } _ { t } ^ { * } = \\operatorname { t a n h } ( \\pmb { W } ^ { r } \\pmb { r } _ { t } + \\pmb { W } ^ { x } \\pmb { h } _ { t } ) } \\end{array}\n$$",
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| 228 |
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"text_format": "latex",
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"type": "equation",
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| 239 |
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"img_path": "images/d492a9e757cf4befd39935df2ead07efd406c15fceb8c6e557b4ca25f4acbe38.jpg",
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| 240 |
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"text": "$$\n\\in \\mathbb { R } ^ { k }\n$$",
|
| 241 |
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"text_format": "latex",
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| 242 |
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"bbox": [
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| 243 |
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"type": "text",
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| 252 |
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"text": "where $W ^ { r }$ , $W ^ { x } \\in \\mathbb { R } ^ { k \\times k }$ are trainable projection matrices. An overview of this architecture is depicted in Figure 1a. Lastly, the probablity distribution ${ \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\Xi } _ { \\mathbf { } } \\mathbf { \\Lambda } _ { \\mathbf { } } \\mathbf { \\Lambda } _ { \\mathbf { } } \\textbf { } _ { \\mathbf { } } \\textbf { } \\textbf { } _ { \\mathrm { } }$ for the next word is represented by ",
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"type": "equation",
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"img_path": "images/7c3a9f0c5ccb96cf2c038d5a5ae26ab7ccab6834f80da85a04131f48df751e48.jpg",
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"text": "$$\n\\begin{array} { r } { y _ { t } = \\mathrm { s o f t m a x } ( W ^ { * } h _ { t } ^ { * } + b ) \\qquad \\in \\mathbb { R } ^ { | V | } } \\end{array}\n$$",
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"text_format": "latex",
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"type": "text",
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"text": "where $W ^ { \\ast } \\in \\mathbb { R } ^ { | V | \\times k }$ and $\\pmb { b } \\in \\mathbb { R } ^ { | V | }$ are a trainable projection matrix and bias, respectively. ",
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"type": "text",
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| 287 |
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"text": "2.2 KEY-VALUE ATTENTION ",
|
| 288 |
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"text_level": 1,
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"type": "text",
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"text": "Inspired by Miller et al. (2016); Ba et al. (2016); Reed & de Freitas (2015); Gulcehre et al. (2016), we introduce a key-value attention model that separates output vectors into keys used for calculating the attention distribution $\\pmb { \\alpha } _ { t }$ , and a value part used for encoding the next-word distribution and context representation. This model is depicted in Figure 1b. Formally, we rewrite Equations 1-4 as follows: ",
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| 300 |
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"type": "equation",
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"img_path": "images/10d5db40bd26ac4614d51624dac12543688c10c882493be1c75f17867dff1152.jpg",
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"text": "$$\n\\begin{array} { r l r l } & { \\Big [ \\pmb { k } _ { t } \\Big ] = \\pmb { h } _ { t } } & { \\qquad } & & { \\in \\mathbb { R } ^ { 2 k } } \\\\ & { M _ { t } = \\operatorname { t a n h } ( \\pmb { W } ^ { Y } [ \\pmb { k } _ { t - L } \\cdot \\cdot \\cdot \\pmb { k } _ { t - 1 } ] + ( \\pmb { W } ^ { h } \\pmb { k } _ { t } ) \\mathbf { 1 } ^ { T } ) } & & { \\qquad \\in \\mathbb { R } ^ { k \\times L } } \\\\ & { \\pmb { \\alpha } _ { t } = \\operatorname { s o f t m a x } ( \\pmb { w } ^ { T } M _ { t } ) } & & { \\in \\mathbb { R } ^ { 1 \\times L } } \\\\ & { r _ { t } = [ \\pmb { v } _ { t - L } \\cdot \\cdot \\cdot \\cdot \\pmb { v } _ { t - 1 } ] \\pmb { \\alpha } ^ { T } } & & { \\in \\mathbb { R } ^ { k } } \\\\ & { \\pmb { h } _ { t } ^ { * } = \\operatorname { t a n h } ( \\pmb { W } ^ { r } r _ { t } + \\pmb { W } ^ { x } \\pmb { v } _ { t } ) } & & { \\in \\mathbb { R } ^ { k } } \\end{array}\n$$",
|
| 312 |
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"text_format": "latex",
|
| 313 |
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"bbox": [
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| 321 |
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| 322 |
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"type": "text",
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| 323 |
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"text": "In essence, Equation 7 compares the key at time step $t$ with the previous $L$ keys to calculate the attention distribution $\\pmb { \\alpha } _ { t }$ which is then used in Equation 9 to obtain a weighted context representation from values associated with these keys. ",
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"type": "text",
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"text": "2.3 KEY-VALUE-PREDICT ATTENTION ",
|
| 335 |
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"text_level": 1,
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"type": "text",
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"text": "Even with a key-value separation, a potential problem is that the same representation ${ \\mathbf { } } v _ { t }$ is still used both for encoding the probability distribution of the next word and for retrieval from the memory via the attention later. Thus, we experimented with another extension of this model where we further separate $\\boldsymbol { h } _ { t }$ into a $k e y$ , a value and a predict representation where the latter is only used for encoding the next-word distribution (see Figure 1c). To this end, equations 6 and 10 are replaced by ",
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| 354 |
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| 355 |
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| 356 |
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"type": "equation",
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"img_path": "images/fdaef764ab84d1568b3fa5a8f9e4f8cd5a4de73486cec6d71f5ba3c460164b96.jpg",
|
| 358 |
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"text": "$$\n\\begin{array} { r l } & { \\left\\lceil \\begin{array} { l } { k _ { t } } \\\\ { v _ { t } } \\\\ { p _ { t } } \\end{array} \\right\\rceil = h _ { t } } \\\\ & { ~ \\left\\lceil \\begin{array} { l } { k _ { t } } \\end{array} \\right\\rceil } \\\\ & { ~ h _ { t } ^ { * } = \\operatorname { t a n h } ( W ^ { r } r _ { t } + W ^ { x } p _ { t } ) } \\end{array}\n$$",
|
| 359 |
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"text_format": "latex",
|
| 360 |
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"page_idx": 2
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| 367 |
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},
|
| 368 |
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{
|
| 369 |
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"type": "equation",
|
| 370 |
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"img_path": "images/0675974f60f411ba0419d0aefd6375cc0ae37c94067f5c78f497e53dfd108819.jpg",
|
| 371 |
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"text": "$$\n\\begin{array} { l } { \\in \\mathbb { R } ^ { 3 k } } \\\\ { \\ } \\\\ { \\in \\mathbb { R } ^ { k } } \\end{array}\n$$",
|
| 372 |
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"text_format": "latex",
|
| 373 |
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"bbox": [
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| 379 |
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| 380 |
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| 381 |
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| 382 |
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"type": "text",
|
| 383 |
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"text": "More precisely, the output vector $\\boldsymbol { h } _ { t }$ is divided into three equal parts: key, value and predict. In our implementation we simply split the output vector $h _ { t }$ into $k _ { t }$ , ${ \\mathbf { } } v _ { t }$ and ${ \\pmb p } _ { t }$ . To this end the hidden dimension of the key-value-predict attention model needs to be a multiplicative of three. Consequently, the dimensions of $k _ { t }$ , ${ \\mathbf { } } v _ { t }$ and ${ \\mathbf { } } p _ { t }$ are 100 for a hidden dimension of 300. ",
|
| 384 |
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| 391 |
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},
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| 392 |
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{
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| 393 |
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"type": "text",
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| 394 |
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"text": "2.4 $N$ -GRAM RECURRENT NEURAL NETWORK",
|
| 395 |
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"text_level": 1,
|
| 396 |
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"type": "text",
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"text": "Neural language models often work best in combination with traditional $N$ -gram models (Mikolov et al., 2011; Chelba et al., 2013; Williams et al., 2015; Ji et al., 2016; Shazeer et al., 2015), since the former excel at generalization while the latter ensure memorization. In addition, from initial experiments with memory-augmented neural language models, we found that usually only the previous five output representations are utilized. This is in line with observations by Tran et al. (2016). Hence, we experiment with a much simpler architecture depicted in Figure 1d. Instead of an attention mechanism, the output representations from the previous $N - 1$ time steps are directly used to calculate next-word probabilities. Specifically, at every time step we split the LSTM output into $N - 1$ vectors $[ { \\pmb h } _ { t } ^ { 1 } , \\dots , { \\pmb h } _ { t } ^ { N - 1 } ]$ and replace Equation 4 with ",
|
| 407 |
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"bbox": [
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| 410 |
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| 413 |
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| 414 |
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},
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| 415 |
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{
|
| 416 |
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"type": "equation",
|
| 417 |
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"img_path": "images/230d20800bae3fec41fa56e4d79790fb3785a124e0e08f73ae7566045267c957.jpg",
|
| 418 |
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"text": "$$\n\\begin{array} { r } { h _ { t } ^ { * } = \\operatorname { t a n h } \\left( W ^ { N } \\left[ \\begin{array} { c } { \\ h _ { t } ^ { 1 } } \\\\ { \\vdots } \\\\ { h _ { t - N + 1 } ^ { N - 1 } } \\end{array} \\right] \\right) } \\end{array}\n$$",
|
| 419 |
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"text_format": "latex",
|
| 420 |
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"bbox": [
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| 421 |
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| 423 |
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| 425 |
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| 426 |
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{
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| 429 |
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"type": "text",
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| 430 |
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"text": "where $W ^ { N } \\in \\mathbb { R } ^ { k \\times ( N - 1 ) k }$ is a trainable projection matrix. This model is related to higher-order RNNs (Soltani $\\&$ Jiang, 2016) with the difference that we do not incorporate output vectors from the previous steps into the hidden state but only use them for predicting the next word. Furthermore, note that at time step $t$ the first part of the output vector $ { \\boldsymbol { h } } _ { t } ^ { 1 }$ will contribute to predicting the next word, the second part $h _ { t } ^ { 2 }$ will contribute to predicting the second word thereafter, and so on. As the output vectors from the $N - 1$ previous time-steps are used to score the next word, we call the resulting model an $N$ -gram RNN. ",
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| 431 |
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},
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| 440 |
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"type": "text",
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| 441 |
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"text": "3 RELATED WORK ",
|
| 442 |
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"text_level": 1,
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| 443 |
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"type": "text",
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"text": "Early attempts of using memory in neural networks have been undertaken by Taylor (1959) and Steinbuch $\\&$ Piske (1963) by performing nearest-neighbor operations on input vectors and fitting parametric models to the retrieved sets. The dedicated use of external memory in neural architectures has more recently witnessed increased interest. Weston et al. (2015) introduced Memory Networks to explicitly segregate memory storage from the computation of the neural network, and Sukhbaatar et al. (2015) trained this model end-to-end with an attention-based memory addressing mechanism. The Neural Turing Machines by Graves et al. (2014) add an external differentiable memory with read-write functions to a controller recurrent neural network, and has shown promising results in simple sequence tasks such as copying and sorting. These models make use of external memory, whereas our model directly uses a short sequence from the history of tokens to dynamically populate an addressable memory. ",
|
| 454 |
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"type": "text",
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| 464 |
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"text": "In sequence modeling, RNNs such as LSTMs (Hochreiter & Schmidhuber, 1997) maintain an internal memory state as they process an input sequence. Attending over previous state outputs on top of an RNN encoder has improved performances in a wide range of tasks, including machine translation (Bahdanau et al., 2015), recognizing textual entailment (Rocktaschel et al. ¨ , 2016), sentence summarization (Rush et al., 2015), image captioning (Xu et al., 2015) and speech recognition (Chorowski et al., 2015). ",
|
| 465 |
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"bbox": [
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| 468 |
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| 469 |
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| 471 |
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| 472 |
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|
| 473 |
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| 474 |
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"type": "text",
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| 475 |
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"text": "Recently, Cheng et al. (2016) proposed an architecture that modifies the standard LSTM by replacing the memory cell with a memory network (Weston et al., 2015). Another proposal for conditioning on previous output representations are Higher-order Recurrent Neural Networks (HORNNs, Soltani & Jiang, 2016). Soltani & Jiang found it useful to include information from multiple preceding RNN states when computing the next state. This previous work centers around preceding state vectors, whereas we investigate attention mechanisms on top of RNN outputs, i.e. the vectors used for predicting the next word. Furthermore, instead of pooling we use attention vectors to calculate a context representation of previous memories. ",
|
| 476 |
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| 482 |
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| 483 |
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|
| 484 |
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{
|
| 485 |
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"type": "text",
|
| 486 |
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"text": "Yang et al. (2016) introduced a reference-aware neural language model where at every position a latent variable determines from which source a target token is generated, e.g., by copying entries from a table or referencing entities that were mentioned earlier. ",
|
| 487 |
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| 493 |
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|
| 494 |
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},
|
| 495 |
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|
| 496 |
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"type": "text",
|
| 497 |
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"text": "Another class of models that include memory into sequence modeling are Recurrent Memory Networks (RMNs) (Tran et al., 2016). Here, a memory block accesses the most recent input words to selectively attend over relevant word representations from a global vocabulary. RMNs use a global memory with two input word vector look-up tables for the attention mechanism, and consequently have a large number of trainable parameters. Instead, we proposed models that need much fewer parameters by producing the vectors that will be attended over in the future, which can be seen as a memory that is dynamically populated by the language model. ",
|
| 498 |
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"bbox": [
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| 504 |
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| 505 |
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| 506 |
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| 507 |
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"type": "text",
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| 508 |
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"text": "Finally, the functional separation of look-up keys and memory content has been found useful for Memory Networks (Miller et al., 2016), Neural Programmer-Interpreters (Reed & de Freitas, 2015), Dynamic Neural Turing Machines (Gulcehre et al., 2016), and Fast Associative Memory (Ba et al., 2016). We apply and extend this principle to neural language models. ",
|
| 509 |
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| 510 |
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"type": "text",
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| 519 |
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"text": "4 EXPERIMENTS ",
|
| 520 |
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| 521 |
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"bbox": [
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| 522 |
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"type": "text",
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"text": "We evaluate models on two different corpora for language modeling. The first is a subset of the Wikipedia corpus.1 It consists of 7500 English Wikipedia articles (dump from 6 Feb 2015) belonging to one of the following categories: People, Cities, Countries, Universities, and Novels. We chose these categories as we expect articles in these categories to often contain references to previously mentioned entities. Subsequently, we split this corpus into a train, development, and test part, resulting in corpora of $2 2 . 5 \\mathbf { M }$ words, 1.2M and 1.2M words, respectively. We map all numbers to a dedicated numerical symbol $N$ and restrict the vocabulary to the 77K most frequent words, encompassing $9 7 \\%$ of the training vocabulary. All other words are replaced by the UNK symbol. The average length of sentences is 25 tokens. In addition to this Wikipedia corpus, we also run experiments on the Children’s Book Test (CBT Hill et al., 2016). While this corpus is designed for cloze-style question-answering, in this paper we use it to test how well language models can exploit wider linguistic context. ",
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"page_idx": 4
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{
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"type": "text",
|
| 542 |
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"text": "4.1 TRAINING PROCEDURE ",
|
| 543 |
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"text_level": 1,
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| 544 |
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"bbox": [
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"type": "text",
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"text": "We use ADAM (Kingma & Ba, 2015) with an initial learning rate of 0.001 and a mini-batch size of 64 for optimization. Furthermore, we apply gradient clipping at a gradient norm of 5 (Pascanu et al., 2013). The bias of the LSTM’s forget gate is initialized to 1 (Jozefowicz et al., 2016), while other parameters are initialized uniformly from the range $( - 0 . 1 , 0 . 1 )$ . Backpropagation Through Time (Rumelhart et al., 1985; Werbos, 1990) was used to train the network with 20 steps of unrolling. We reset the hidden states between articles for the Wikipedia corpus and between stories for CBT, respectively. We take the best configuration based on performance on the validation set and evaluate it on the test set. ",
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"type": "text",
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"text": "5 RESULTS ",
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"text_level": 1,
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"type": "text",
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"text": "In the first set of experiments we explore how well the proposed models and Tran et al.’s Recurrentmemory Model can make use of histories of varying lengths. Perplexity results for different attention window sizes on the Wikipedia corpus are summarized in Figure 2a. The average attention these models pay to specific positions in the history is illustrated in Figure 3. We observed that although our models attend over tokens further in the past more often than the Recurrent-memory Model, attending over a longer history does not significantly improve the perplexity of any attentive model. ",
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"type": "text",
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| 588 |
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"text": "The much simpler $N$ -gram RNN model achieves comparable results (Figure 2b) and seems to work best with a history of the previous three output vectors (4-gram RNN). As a result, we choose the 4-gram model for the following $N$ -gram RNN experiments. ",
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"type": "table",
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"img_path": "images/a2b07665c1c901aed126e3f5284078eaeefd88a9bedf4b76fad451ca63518a6f.jpg",
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"table_caption": [
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| 601 |
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"Figure 2: Perplexities of memory-augmented neural language models on the Wikipedia corpus (a-c) and accuracies on the CBT test set (d). "
|
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],
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"table_footnote": [],
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"table_body": "<table><tr><td>Model</td><td colspan=\"4\">Attention Window Size</td></tr><tr><td></td><td>1</td><td>5</td><td>10</td><td>15</td></tr><tr><td>RM(+tM-g) (Tran et al.,2016)</td><td>83.5</td><td>80.5</td><td>80.3</td><td>80.1</td></tr><tr><td>Attention</td><td>82.2</td><td>82.2</td><td>82.0</td><td>82.8</td></tr><tr><td>Key-Value</td><td>78.7</td><td>79.0</td><td>78.2</td><td>78.9</td></tr><tr><td>Key-Value-Predict</td><td>76.1</td><td>75.8</td><td>76.0</td><td>75.8</td></tr></table>",
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| 605 |
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"bbox": [
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"type": "table",
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"img_path": "images/ceedcb64e0c3e1d18aec1ac3e80dd86338c815affc386bef6777c32dacd288f6.jpg",
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"table_caption": [
|
| 617 |
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"(a) Test perplexity of different attention architectures with varying attention window sizes. Best perplexity per model is italic. ",
|
| 618 |
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"(b) Comparison of $N$ -gram neural language models. $w$ denotes the input size, $k$ the hidden size and $\\theta _ { M }$ the total number of model parameters. "
|
| 619 |
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],
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| 620 |
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"table_footnote": [],
|
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"table_body": "<table><tr><td>Model</td><td>w</td><td>k</td><td>0M</td><td>Dev</td><td>Test</td></tr><tr><td>2-gram RNN</td><td>300</td><td>564</td><td>23.9M</td><td>76.0</td><td>77.1</td></tr><tr><td>3-gram RNN</td><td>300</td><td>786</td><td>23.9M</td><td>74.9</td><td>75.9</td></tr><tr><td>4-gram RNN</td><td>300</td><td>968</td><td>23.9M</td><td>74.8</td><td>75.9</td></tr><tr><td>5-gram RNN</td><td>300</td><td>1120</td><td>23.9M</td><td>76.0</td><td>77.3</td></tr></table>",
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{
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"type": "text",
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"text": "(c) Summary of models with best attention window size $a$ . The total number of model parameters, including word representations, is denoted by $\\theta _ { W + M }$ (without word representations $\\theta _ { M }$ ). ",
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"bbox": [
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"type": "table",
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"img_path": "images/3f480394fefa0d7831679c9a460745c751e0377da6e0e21f7c9fb38adf5a97e7.jpg",
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"table_caption": [],
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"table_footnote": [],
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"table_body": "<table><tr><td>Model</td><td>w</td><td>k</td><td>a</td><td>0W+M</td><td>0M</td><td>Dev</td><td>Test</td></tr><tr><td>RNN</td><td>300</td><td>307</td><td>1</td><td>47.0M</td><td>23.9M</td><td>121.7</td><td>125.7</td></tr><tr><td>LSTM</td><td>300</td><td>300</td><td>-</td><td>47.0M</td><td>23.9M</td><td>83.2</td><td>85.2</td></tr><tr><td>FOFE HORNN (3-rd order) (Soltani & Jiang,2016)</td><td>300</td><td>303</td><td>-</td><td>47.0M</td><td>23.9M</td><td>116.7</td><td>120.5</td></tr><tr><td>Gated HORNN (3-rd order) (Soltani & Jiang, 2016)</td><td>300</td><td>297</td><td>-</td><td>47.0M</td><td>23.9M</td><td>93.9</td><td>97.1</td></tr><tr><td>RM(+tM-g) (Tran et al.,2016)</td><td>300</td><td>300</td><td>15</td><td>93.7M</td><td>70.6M</td><td>78.2</td><td>80.1</td></tr><tr><td>Attention</td><td>300</td><td>296</td><td>10</td><td>47.0M</td><td>23.9M</td><td>80.6</td><td>82.0</td></tr><tr><td>Key-Value</td><td>300</td><td>560</td><td>10</td><td>47.0M</td><td>23.9M</td><td>77.1</td><td>78.2</td></tr><tr><td>Key-Value-Predict</td><td>300</td><td>834</td><td>5</td><td>47.0M</td><td>23.9M</td><td>74.2</td><td>75.8</td></tr><tr><td>4-gram RNN</td><td>300</td><td>968</td><td>-</td><td>47.0M</td><td>23.9M</td><td>74.8</td><td>75.9</td></tr></table>",
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},
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{
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"type": "table",
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"img_path": "images/a46c0774d10b32d9443dd9e064be58e5b897adb8f721d8bd869ab291cabdf7ed.jpg",
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| 658 |
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"table_caption": [
|
| 659 |
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"(d) Results on CBT; those marked with ‡ are taken from Hill et al. (2016). "
|
| 660 |
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],
|
| 661 |
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"table_footnote": [],
|
| 662 |
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"table_body": "<table><tr><td>Model</td><td>Named Entities</td><td>Common Nouns</td><td>Verbs</td><td>Prepositions</td></tr><tr><td>Humans (context+query) ‡</td><td>0.816</td><td>0.816</td><td>0.828</td><td>0.708</td></tr><tr><td>Kneser-Ney LM ‡</td><td>0.390</td><td>0.544</td><td>0.778</td><td>0.768</td></tr><tr><td>Kneser-Ney LM + cache ‡</td><td>0.439</td><td>0.577</td><td>0.772</td><td>0.679</td></tr><tr><td>LSTM (context+query) ‡</td><td>0.418</td><td>0.560</td><td>0.818</td><td>0.791</td></tr><tr><td>Memory Network ‡</td><td>0.666</td><td>0.630</td><td>0.690</td><td>0.703</td></tr><tr><td>AS Reader, avg ensemble (Kadlec et al., 2016)</td><td>0.706</td><td>0.689</td><td>一</td><td>1</td></tr><tr><td>AS Reader, greedy ensemble (Kadlec et al., 2016)</td><td>0.710</td><td>0.675</td><td></td><td></td></tr><tr><td>QANN, 4 hops, GloVe (Weissenborn, 2016)</td><td>0.729</td><td>1</td><td></td><td></td></tr><tr><td>AoA Reader, single model (Cui et al.,2016a)</td><td>0.720</td><td>0.694</td><td></td><td></td></tr><tr><td>CAS Reader, mode avg (Cui et al.,2016b)</td><td>0.692</td><td>0.657</td><td></td><td></td></tr><tr><td>GA Reader, ensemble (Dhingra et al., 2016)</td><td>0.719</td><td>0.694</td><td></td><td></td></tr><tr><td>EpiReader, ensemble (Trischler et al., 2016)</td><td>0.718</td><td>0.706</td><td></td><td></td></tr><tr><td>FOFE HORNN (3-rd order) (Soltani & Jiang,2016)</td><td>0.465</td><td>0.497</td><td>0.774</td><td>0.741</td></tr><tr><td>Gated HORNN (3-rd order) (Soltani & Jiang,2016)</td><td>0.508</td><td>0.547</td><td>0.790</td><td>0.774</td></tr><tr><td>RM(+tM-g) (Tran et al.,2016)</td><td>0.525</td><td>0.597</td><td>0.817</td><td>0.797</td></tr><tr><td>LSTM</td><td>0.523</td><td>0.604</td><td>0.819</td><td>0.786</td></tr><tr><td>Attention</td><td>0.538</td><td>0.595</td><td>0.826</td><td>0.803</td></tr><tr><td>Key-Value</td><td>0.528</td><td>0.601</td><td>0.822</td><td>0.813</td></tr><tr><td>Key-Value-Predict</td><td>0.528</td><td>0.599</td><td>0.829</td><td>0.803</td></tr><tr><td>4-gram RNN</td><td>0.532</td><td>0.598</td><td>0.815</td><td>0.800</td></tr></table>",
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},
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"type": "text",
|
| 673 |
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"text": "5.1 COMPARISON WITH STATE-OF-THE-ART MODELS ",
|
| 674 |
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"text_level": 1,
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| 675 |
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"type": "text",
|
| 685 |
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"text": "In the next set of experiments, we compared our proposed models against a variety of state-of-the-art models on the Wikipedia and CBT corpora. Results are shown in Figure 2c and 2d, respectively. Note that the models presented here do not achieve state-of-the-art on CBT as they are language models and not tailored towards cloze-sytle question answering. Thus, we merely use this corpus for comparing different neural language model architectures. We reimplemented the Recurrent-Memory model by Tran et al. (2016) with the temporal matrix and gating composition function $\\mathbf { \\left( R M + t M - g \\right) }$ ). ",
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{
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"type": "image",
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"img_path": "images/20cbcc35b6f2dc20b08c3b338545fa6758915ab6c15ff82b3db714f15c2813c6.jpg",
|
| 697 |
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"image_caption": [
|
| 698 |
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"Figure 3: Attention weights of the Key-Value-Predict model on a randomly sampled Wikipedia article (a) and average attention weight distribution on the whole Wikipedia test set for $\\mathbf { R M } ( + \\mathbf { t M } - \\mathbf { g } )$ , Attention, Key-Value and Key-Value-Predict models (b). The rightmost positions represent the most recent history. "
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{
|
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"type": "text",
|
| 711 |
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"text": "Furthermore, we reimplemented Higher Order Recurrent Neural Networks (HORNNs) by Soltani & Jiang (2016). ",
|
| 712 |
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"bbox": [
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{
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"type": "text",
|
| 722 |
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"text": "To ensure a comparable number of parameters to a vanilla LSTM model, we adjusted the hidden size of all models to have roughly the same total number of model parameters. The attention window size $N$ for the $N$ -gram RNN model was set to 4 according to the best validation set perplexity on the Wikipedia corpus. Below we discuss the results in detail. ",
|
| 723 |
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"type": "text",
|
| 733 |
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"text": "Attention By using a neural language model with an attention mechanism over a dynamically populated memory, we observed a 3.2 points lower perplexity over a vanilla LSTM on Wikipedia, but only notable differences for predicting verbs and prepositions in CBT. This indicates that incorporating mechanisms for querying previous output vectors is useful for neural language modeling. ",
|
| 734 |
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|
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{
|
| 743 |
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"type": "text",
|
| 744 |
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"text": "Key-Value Decomposing the output vector into a key-value paired memory improves the perplexity by 7.0 points compared to a baseline LSTM, and by 1.9 points compared to the $\\mathbf { R M } ( + \\mathbf { t M } - \\mathbf { g } )$ model. Again, for CBT we see only small improvements. ",
|
| 745 |
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"page_idx": 7
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},
|
| 753 |
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{
|
| 754 |
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"type": "text",
|
| 755 |
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"text": "Key-Value-Predict By further separating the output vector into a key, value and next-word prediction part, we get the lowest perplexity and gain 9.4 points over a baseline LSTM, a 4.3 points compared to $\\mathbf { R M } ( + \\mathbf { t M } - \\mathbf { g } )$ , and 2.4 points compared to only splitting the output into a key and value. For CBT, we see an accuracy increase of 1.0 percentage points for verbs, and 1.7 for prepositions. As stated earlier, the performance of the Key-Value-Predict model does not improve significantly when increasing the attention window size. This leads to the conclusion that none of the attentive models investigated in this paper can utilize a large memory of previous token representations. Moreover, none of the presented methods differ significantly for predicting common nouns and named entities in CBT. ",
|
| 756 |
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| 763 |
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},
|
| 764 |
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{
|
| 765 |
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"type": "text",
|
| 766 |
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"text": "$N$ -gram RNN Our main finding is that the simple modification of using output vectors from the previous time steps for the next-word prediction leads to perplexities that are on par with or better than more complicated neural language models with attention. Specifically, the 4-gram RNN achieves only slightly worse perplexities than the Key-Value-Predict architecture. ",
|
| 767 |
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"bbox": [
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"page_idx": 7
|
| 774 |
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|
| 775 |
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{
|
| 776 |
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"type": "text",
|
| 777 |
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"text": "6 CONCLUSION ",
|
| 778 |
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"text_level": 1,
|
| 779 |
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"bbox": [
|
| 780 |
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| 787 |
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|
| 788 |
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"type": "text",
|
| 789 |
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"text": "In this paper, we observed that using an attention mechanism for neural language modeling where we separate output vectors into a key, value and predict part outperform simpler attention mechanisms on a Wikipedia corpus and the Children Book Test (CBT, Hill et al., 2016). However, we found that all attentive neural language models mainly utilize a memory of only the most recent history and fail to exploit long-range dependencies. In fact, a much simpler $N$ -gram RNN model, which only uses a concatenation of output representations from the previous three time steps, is on par with more sophisticated memory-augmented neural language models. Training neural language models that take long-range dependencies into account seems notoriously hard and needs further investigation. Thus, for future work we want to investigate ways to encourage attending over a longer history, for instance by forcing the model to ignore the local context and only allow attention over output representations further behind the local history. ",
|
| 790 |
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|
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| 797 |
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},
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| 798 |
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{
|
| 799 |
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"type": "text",
|
| 800 |
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"text": "ACKNOWLEDGMENTS ",
|
| 801 |
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| 802 |
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"bbox": [
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"type": "text",
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"text": "This work was supported by Microsoft Research and the Engineering and Physical Sciences Research Council through PhD Scholarship Programmes, an Allen Distinguished Investigator Award, and a Marie Curie Career Integration Award. ",
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parse/train/ByIAPUcee/ByIAPUcee_model.json
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parse/train/HJGXzmspb/HJGXzmspb.md
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| 1 |
+
# TRAINING AND INFERENCE WITH INTEGERS IN DEEP NEURAL NETWORKS
|
| 2 |
+
|
| 3 |
+
Shuang $\mathbf { W } \mathbf { u } ^ { 1 }$ , Guoqi $\mathbf { L i } ^ { 1 }$ , Feng Chen2, Luping Shi1
|
| 4 |
+
|
| 5 |
+
1Department of Precision Instrument
|
| 6 |
+
2Department of Automation
|
| 7 |
+
Center for Brain Inspired Computing Research
|
| 8 |
+
Beijing Innovation Center for Future Chip
|
| 9 |
+
Tsinghua University
|
| 10 |
+
{lpshi,chenfeng}@mail.tsinghua.edu.cn
|
| 11 |
+
|
| 12 |
+
# ABSTRACT
|
| 13 |
+
|
| 14 |
+
Researches on deep neural networks with discrete parameters and their deployment in embedded systems have been active and promising topics. Although previous works have successfully reduced precision in inference, transferring both training and inference processes to low-bitwidth integers has not been demonstrated simultaneously. In this work, we develop a new method termed as “WAGE” to discretize both training and inference, where weights (W), activations (A), gradients (G) and errors (E) among layers are shifted and linearly constrained to low-bitwidth integers. To perform pure discrete dataflow for fixed-point devices, we further replace batch normalization by a constant scaling layer and simplify other components that are arduous for integer implementation. Improved accuracies can be obtained on multiple datasets, which indicates that WAGE somehow acts as a type of regularization. Empirically, we demonstrate the potential to deploy training in hardware systems such as integer-based deep learning accelerators and neuromorphic chips with comparable accuracy and higher energy efficiency, which is crucial to future AI applications in variable scenarios with transfer and continual learning demands.
|
| 15 |
+
|
| 16 |
+
# 1 INTRODUCTION
|
| 17 |
+
|
| 18 |
+
Recently deep neural networks (DNNs) are being widely used for numerous AI applications (Krizhevsky et al., 2012; Hinton et al., 2012; Silver et al., 2016). Depending on the massive tunable parameters, DNNs are considered to have powerful multi-level feature extraction and representation abilities. However, training DNNs needs energy-intensive devices such as GPU and CPU with high precision (float32) processing units and abundant memory, which has greatly challenged their extensive applications for portable devices. In addition, a state-of-art network often has far more weights and effective capacity to shatter all training samples (Zhang et al., 2016), leading to overfitting easily.
|
| 19 |
+
|
| 20 |
+
As a result, there is much interest in reducing the size of network during inference (Hubara et al., 2016; Rastegari et al., 2016; Li et al., 2016), as well as dedicated hardware for commercial solutions (Jouppi et al., 2017; Chen et al., 2017; Shi et al., 2015). Due to the accumulation in stochastic gradient descent (SGD) optimization, the precision demand for training is usually higher than inference (Hubara et al., 2016; Li et al., 2017). Therefore, most of the existing techniques only focus on the deployment of a well-trained compressed network, while still keeping high precision and computational complexity during training. In this work, we address this problem as how to process both training and inference with low-bitwidth integers, which is essential for implementing DNNs in dedicated hardware. To this end, two fundamental issues are addressed for discretely training DNNs: i) how to quantize all the operands and operations, and ii) how many bits or states are needed for SGD computation and accumulation.
|
| 21 |
+
|
| 22 |
+
With respect to the issues, we propose a framework termed as “WAGE” that constrains weights (W), activations (A), gradients (G) and errors (E) among all layers to low-bitwidth integers in both training and inference. Firstly, for operands, linear mapping and orientation-preserved shifting are applied to achieve ternary weights, 8-bit integers for activations and gradients accumulation. Secondly, for operations, batch normalization (Ioffe & Szegedy, 2015) is replaced by a constant scaling factor. Other techniques for fine-tuning such as SGD optimizer with momentum and L2 regularization are simplified or abandoned with little performance degradation. Considering the overall bidirectional propagation, we completely streamline inference into accumulate-compare cycles and training into low-bitwidth multiply-accumulate (MAC) cycles with alignment operations, respectively.
|
| 23 |
+
|
| 24 |
+
We heuristically explore the bitwidth requirements of integers for error computation and gradient accumulation, which have rarely been discussed in previous works. Experiments indicate that it is the relative values (orientations) rather than absolute values (orders of magnitude) in error that guides previous layers to converge. Moreover, small values have negligible effects on previous orientations though propagated layer by layer, which can be partially discarded in quantization. We leverage these phenomena and use an orientation-preserved shifting operation to constrain errors. As for the gradient accumulation, though weights are quantized to ternary values in inference, a relatively higher bitwidth is indispensable to store and accumulate gradient updates.
|
| 25 |
+
|
| 26 |
+
The proposed framework is evaluated on MNIST, CIFAR10, SVHN, ImageNet datasets. Comparing to those who only discretize weights and activations at inference time, it has comparable accuracy and can further alleviate overfitting, indicating some type of regularization. WAGE produces pure bidirectional low-precision integer dataflow for DNNs, which can be applied for training and inference in dedicated hardware neatly. We publish the code on GitHub1.
|
| 27 |
+
|
| 28 |
+
# 2 RELATED WORK
|
| 29 |
+
|
| 30 |
+
We mainly focus on reducing precision of operands and operations in both training and inference. Orthogonal and complementary techniques for reducing complexity like network compression, pruning (Han et al., 2015; Zhou et al., 2017) and compact architectures (Howard et al., 2017) are impressively efficient but outside the scope this paper.
|
| 31 |
+
|
| 32 |
+
Weight and activation Courbariaux et al. (2015); Hubara et al. (2016) propose methods to train DNNs with binary weights (BC) and activations (BNN) successively. They add noises to weights and activations as a form of regularization but real-valued gradients are accumulated in real-valued variables, suggesting that high precision accumulation is likely required for SGD optimization. XNOR-Net (Rastegari et al., 2016) has a filter-wise scaling factor for weights to improve the performance. Convolutions in XNOR-Net can be implemented efficiently using XNOR logical units and bit-count operations. However, these floating-point factors are calculated simultaneously during training, which generally aggravates the training effort. In TWN (Li et al., 2016) and TTQ (Zhu et al., 2016) two symmetric thresholds are introduced to constrain the weights to be ternary-valued: $\{ + 1 , 0 , - 1 \}$ . They claimed a tradeoff between model complexity and expressive ability.
|
| 33 |
+
|
| 34 |
+
Gradient computation and accumulation DoReFa-Net (Zhou et al., 2016) quantizes gradients to low-bitwidth floating-point numbers with discrete states in the backward pass. TernGrad (Wen et al., 2017) quantizes gradient updates to ternary values to reduce the overhead of gradient synchronization in distributed training. Nevertheless, weights in DoReFa-Net and TernGrad are stored and updated with float32 during training like previous works. Besides, the quantization of batch normalization and its derivative is ignored. Thus, the overall computation graph for the training process is still presented with float32 and more complex with external quantization. Generally, it is difficult to apply DoReFa-Net training in an integer-based hardware directly, but it shows potential for exploring high-dimensional discrete spaces with discrete gradient descent directions.
|
| 35 |
+
|
| 36 |
+
# 3 WAGE QUANTIZATION
|
| 37 |
+
|
| 38 |
+
The main idea of WAGE quantization is to constrain four operands to low-bitwidth integers: weight $W$ and activation $^ { a }$ in inference, error $e$ and gradient $\textbf { { g } }$ in backpropagation training, see Figure 1. We extend the original definition of errors to multi-layer: error $e$ is the gradient of activation $\textbf { \em a }$ for the perspective of each convolution or fully-connected layer, while gradient $\textbf { { g } }$ particularly refers to the gradient accumulation of weight $W$ . Considering the $i$ -th layer of a feed-forward network, we
|
| 39 |
+
|
| 40 |
+
have:
|
| 41 |
+
|
| 42 |
+
$$
|
| 43 |
+
e ^ { i } = { \frac { \partial { \mathcal { L } } } { \partial a ^ { i } } } , \pmb { g } ^ { i } = { \frac { \partial { \mathcal { L } } } { \partial W ^ { i } } }
|
| 44 |
+
$$
|
| 45 |
+
|
| 46 |
+
where $\mathcal { L }$ is the loss function. We separate these two terms that are mixed up in most existing schemes. The gradient of weight $\textbf { { g } }$ and the gradient of activation $e$ flow to different paths in each layer, which is a fork both in inference and in backward training and generally acts as node of MAC operations.
|
| 47 |
+
|
| 48 |
+
For the forward propagation in the $i$ -th layer, assuming that weights are stored and accumulated with $k _ { G }$ -bit integers, then numerous works strive for a better quantization function $Q _ { W } ( \cdot )$ that maps higher precision weights to their $k _ { W }$ -bit reflections, for example, $[ - 0 . 9 , 0 . 1 , 0 . 7 ]$ to $[ - 1 , 0 , 1 ]$ . Although weights are accumulated with high precision like float32, the deployment of the reflections in dedicated hardware are much more memory efficient after training. Activations are quantized with function $Q _ { A } ( \cdot )$ to $k _ { A }$ bits to align the increased bitwidth caused by MACs. Weights and activations are discretized to even binary values in previous works, then MACs degrade into logical and bit-count operations that are extremely efficient (Rastegari et al., 2016).
|
| 49 |
+
|
| 50 |
+
For the backward propagation in the $i$ -th layer, the gradients of activations and weights are calculated by the derivatives of MACs that are generally considered to be in 16-bit floating-point precision at least. As illustrated in Figure 1, the MACs between $k _ { A }$ -bit inputs and $k _ { W }$ -bit weights will increase the bitwidth of outputs to $[ k _ { A } + k _ { W } - 1 ]$ in signed integer representation, and the similar broadening happens to errors $e$ as well. In consideration of training with only low-bitwidth integers, we propose additional functions $Q _ { E } ( \cdot )$ and $Q _ { G } ( \cdot )$ to constrain bitwidth of $e$ and $\textbf { { g } }$ to $k _ { E }$ bits and $k _ { G }$ bits, respectively. In general, where there is a MAC operation, there are quantization operators named $Q _ { W } ( \cdot ) , \bar { Q _ { A } ( \cdot ) } , \bar { Q _ { G } ( \cdot ) }$ and $Q _ { E } ( \cdot )$ in inference and backpropagation.
|
| 51 |
+
|
| 52 |
+
|
| 53 |
+
Figure 1: Four operators $Q _ { W } ( \cdot )$ , $Q _ { A } ( \cdot )$ , $Q _ { G } ( \cdot )$ , $Q _ { E } ( \cdot )$ added in WAGE computation dataflow to reduce precision, bitwidth of signed integers are below or on the right of arrows, activations are included in MAC for concision.
|
| 54 |
+
|
| 55 |
+
# 3.1 SHIFT-BASED LINEAR MAPPING AND STOCHASTIC ROUNDING
|
| 56 |
+
|
| 57 |
+
In WAGE quantization, we adopt a linear mapping with $k$ -bit integers for simplicity, where continuous and unbounded values are discretized with uniform distance $\sigma$ :
|
| 58 |
+
|
| 59 |
+
$$
|
| 60 |
+
\sigma ( k ) = 2 ^ { 1 - k } , k \in \mathbb { N } _ { + }
|
| 61 |
+
$$
|
| 62 |
+
|
| 63 |
+
Then the basic quantization function that converts a floating-point number $x$ to its $k$ -bitwidth signed integer representation can be formulated as:
|
| 64 |
+
|
| 65 |
+
$$
|
| 66 |
+
Q ( x , k ) = C l i p \left\{ \sigma ( k ) \cdot r o u n d \left[ \frac { x } { \sigma ( k ) } \right] , - 1 + \sigma ( k ) , 1 - \sigma ( k ) \right\}
|
| 67 |
+
$$
|
| 68 |
+
|
| 69 |
+
where round approximates continuous values to their nearest discrete states. Clip is the saturation function that clips unbounded values to $[ - 1 + \sigma , 1 - \sigma ]$ , where the negative maximum value $- 1$ is removed to maintain symmetry. For example, $Q ( x , 2 )$ quantizes $\{ - 1 , 0 . 2 , 0 . 6 \}$ to $\{ - 0 . 5 , 0 , 0 . 5 \}$ . Equation 3 is merely used for simulation in floating-point hardware like GPU, whereas in a fixedpoint device, quantization and saturation is satisfied automatically.
|
| 70 |
+
|
| 71 |
+
Before applying linear mapping in some operands (e.g., error), we introduce an additional monolithic scaling factor for shifting values distribution to an appropriate order of magnitude, otherwise values will be all saturated or cleared by Equation 3. The scaling factor is calculated by $S h i f t$ function and then divided in later steps:
|
| 72 |
+
|
| 73 |
+
$$
|
| 74 |
+
S h i f t ( x ) = 2 ^ { r o u n d ( \log _ { 2 } x ) }
|
| 75 |
+
$$
|
| 76 |
+
|
| 77 |
+
Finally, we propose stochastic rounding to substitute small and real-valued updates for gradient accumulation in training. Section 3.3.4 will detail the implementation of operator $Q _ { G } ( \cdot )$ , where high bitwidth gradients are constrained to $k _ { G }$ -bit integers stochastically by a 16-bit random number generator. Figure 2 summarizes quantization methods used in WAGE.
|
| 78 |
+
|
| 79 |
+
|
| 80 |
+
Figure 2: Quantization methods used in WAGE. The notation $P , \pmb { x } , \lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ denotes probability, vector, floor and ceil, respectively. $S h i f t ( \cdot )$ refers to Equation 4 with a certain argument.
|
| 81 |
+
|
| 82 |
+
# 3.2 WEIGHT INITIALIZATION
|
| 83 |
+
|
| 84 |
+
In previous works, weights are binarized directly by sgn function or ternarized by threshold parameters calculated during training. However, BNN fails to converge without batch normalization because weight values $\pm 1$ are rather big for a typical DNN. Batch normalization not only efficiently avoids the problem of exploding and vanishing gradients, but also alleviates the demand for proper initialization. However, normalizing outputs for each layer and computing their gradients are quite complex without floating point unit (FPU). Besides, the moving averages of batch outputs occupy external memory. BNN shows a shift-based variation of batch normalization but it is hard to transform all of the elements to the fixed-point representations. As a result, weights should be cautiously initialized in this work where batch normalization is simplified to a constant scaling layer. A modified initialization method based on MSRA (He et al., 2015) can be formulated as:
|
| 85 |
+
|
| 86 |
+
$$
|
| 87 |
+
W \sim U ( - L , + L ) , L = m a x \{ \sqrt { 6 / n _ { i n } } , L _ { m i n } \} , L _ { m i n } = \beta \sigma
|
| 88 |
+
$$
|
| 89 |
+
|
| 90 |
+
where $n _ { i n }$ is the layer fan-in number, and the original limit $\sqrt { 6 / n _ { i n } }$ in MSRA is calculated to keep same variance between inputs and outputs of the same layer theoretically. The additional limit $L _ { m i n }$ is a minimum value that the uniform distribution $U$ should reach, and $\beta$ is a constant greater than 1 to create overlaps between minimum step size $\sigma$ and maximum value $L$ . In case of $k _ { W }$ -bit linear mapping, if weights $W$ are quantized directly with original limits, we will get all-zero tensors when bitwidth $k _ { W }$ is small enough, e.g., 4, or fan-in $n _ { i n }$ is wide enough, where initialized weights may never reach the minimum step $\sigma$ presented by fixed-point integers. So $L _ { m i n }$ ensures that weights can go beyond $\sigma$ and quantized to non-zero values after $Q _ { W } ( \cdot )$ when initialized randomly.
|
| 91 |
+
|
| 92 |
+
# 3.3 QUANTIZATION DETAILS
|
| 93 |
+
|
| 94 |
+
# 3.3.1 WEIGHT $Q _ { W } ( \cdot )$
|
| 95 |
+
|
| 96 |
+
The modified initialization in Equation 5 will amplify weights holistically and guarantee their proper distribution, then $W$ is quantized directly with Equation 3:
|
| 97 |
+
|
| 98 |
+
$$
|
| 99 |
+
\boldsymbol { W _ { q } } = \boldsymbol { Q _ { W } } ( \boldsymbol { W } ) = \boldsymbol { Q } ( \boldsymbol { W } , \boldsymbol { k _ { W } } )
|
| 100 |
+
$$
|
| 101 |
+
|
| 102 |
+
It should be noted that the variance of weights is scaled compared to the original limit, which will cause exploding of network’s outputs. To alleviate the amplification effect, XNOR-Net proposed a filter-wise scaling factor calculated continuously with full precision. In consideration of integer implementation, we introduce a layer-wise shift-based scaling factor $\alpha$ to attenuate the amplification effect:
|
| 103 |
+
|
| 104 |
+
$$
|
| 105 |
+
\alpha = m a x \{ S h i f t ( L _ { m i n } / L ) , 1 \}
|
| 106 |
+
$$
|
| 107 |
+
|
| 108 |
+
where $\alpha$ is a pre-defined constant for each layer determined by the network structure. The modified initialization and attenuation factor $\alpha$ together approximates floating-point weights to their integer representations, except that $\alpha$ takes effect after activations to maintain precision of weights presented by $k _ { W }$ -bit integers.
|
| 109 |
+
|
| 110 |
+
# 3.3.2 ACTIVATION $Q _ { A } ( \cdot )$
|
| 111 |
+
|
| 112 |
+
As stated above, the bitwidth of operands increases after MACs. Then a typical CNN is usually followed with pooling, normalization and activation. Average pooling is avoided because mean operations will increase precision demand. Besides, we hypothesize that batch outputs of each hidden layer approximately have zero-mean, then batch normalization degenerates into to a scaling layer where trainable and batch-calculated scaling parameters are replaced by $\alpha$ mentioned in Equation 7. If activations are presented in $k _ { A }$ bits, the overall quantization of activations can be formulated as:
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$$
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\pmb { a } _ { q } = Q _ { A } ( \pmb { a } ) = Q ( \pmb { a } / \alpha , k _ { A } )
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$$
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# 3.3.3 ERROR $Q _ { E } ( \cdot )$
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Errors $e$ are calculated layer by layer using the chain rule during training. Although the computation graph of backpropagation is similar to the inference, the inputs are the gradients of $\mathcal { L }$ , which are relatively small compared to actual inputs for networks. More importantly, the errors are unbounded and might have significantly larger ranges than that of activations, e.g., $[ 1 0 ^ { - 9 } , 1 0 ^ { - 4 } ]$ . DoReFa-Net first applies an affine transform on $e$ to map them into $[ - 1 , 1 ]$ , and then inverts the transform after quantization. Thus, the quantized $e$ are still presented as float32 numbers with discrete states and mostly small values.
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However, experiments uncover that it is the orientations rather than orders of magnitude in errors that guides previous layers to converge, then the inverse transformation after quantization in DoReFa-Net is no longer needed. The orientation-only preservation prompts us to propagate errors with integer√ √ thoroughly, where error distribution is firstly scaled into $[ - { \sqrt { 2 } } , + { \sqrt { 2 } } ]$ by dividing a shift factor as shown in Figure 2 and then quantized by $Q ( e , k _ { E } )$ :
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$$
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\pmb { e _ { q } } = Q _ { E } ( \pmb { e } ) = Q ( \pmb { e } / S h i f t ( m a x \{ | \pmb { e } | \} ) , k _ { E } )
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$$
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where $m a x \{ | e | \}$ extracts the layer-wise maximum absolute value among all elements in error $e$ , multi-channel for convolution and multi-sample for batch training. The quantization of error discards large proportion of values smaller than $\sigma$ , we will discuss the influence on accuracy later.
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# 3.3.4 GRADIENT $Q _ { G } ( \cdot )$
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Since we only preserve relative values of error after shifting, the gradient updates $\textbf { { g } }$ derived from MACs between backward errors $e$ and forward activations $^ { a }$ are shifted consequently. We first rescale gradients $\textbf { { g } }$ with another scaling factor and then bring in shift-based learning rate $\eta$ :
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$$
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g _ { s } = \eta \cdot g / S h i f t ( m a x \{ | g | \} )
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$$
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where $\eta$ is an integer power of 2. The shifted gradients $\mathbf { \nabla } _ { \mathbf { { \boldsymbol { g } } } _ { s } }$ represent for minimum step numbers and directions for updating weights. If weights are stored with $k _ { G }$ -bit numbers, the minimum step of modification will be $\pm 1$ for integers and $\pm \sigma ( k _ { G } )$ for floating-point values, respectively. The implement of learning rate $\eta$ here is quite different from that in a vanilla DNN based on float32. In WAGE, there only remain directions for weights to change and the step sizes are integer multiples of minimum step $\sigma$ . Shifted gradients $\mathbf { \nabla } _ { \mathbf { { \boldsymbol { g } } } _ { s } }$ may get greater than 1 if $\eta$ is 2 or bigger to accelerate training at the beginning, or smaller than 0.5 during latter half of training when learning rate decay is usually applied. As illustrated in Figure 2, to substitute accumulation of small gradients in latter case, we separate $\mathbf { \nabla } _ { \mathbf { { \boldsymbol { g } } } _ { s } }$ into integer parts and decimal parts, then use a 16-bit random number generator to constrain high bitwidth $\mathbf { \nabla } _ { \mathbf { { \boldsymbol { g } } } _ { s } }$ to $k _ { G }$ -bit integers stochastically:
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$$
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\Delta W = Q _ { G } ( g ) = \sigma ( k _ { G } ) \cdot s g n ( g _ { s } ) \cdot \Big \{ \lfloor | g _ { s } | \rfloor + B e r n o u l l i ( | g _ { s } | - \lfloor | g _ { s } | \rfloor ) \Big \}
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$$
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where Bernoulli (Zhou et al., 2016) stochastically samples decimal parts to either 0 or 1. With proper setting of $k _ { G }$ , quantization of gradients will restrict the minimum step size, which may avoid local minimum and overfitting. Furthermore, the gradients will be ternary values when $\eta$ is not greater than 1, which reduces communication costs for distributed training (Wen et al., 2017). At last, weights $W$ might exceed the range $[ - 1 + \sigma , 1 - \sigma ]$ presented by $k _ { G }$ -bit integers after updating with discrete increments $\Delta \mathbf { W }$ . So $C l i p$ function is indispensable to saturate and make sure there are only $2 ^ { k _ { G } - 1 } - 1$ states for weights accumulation. In case of the $t$ -th iteration, we have:
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$$
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W _ { t + 1 } = C l i p \left\{ { \cal W } _ { t } - \Delta { \cal W } _ { t } , - 1 + \sigma ( k _ { G } ) , 1 - \sigma ( k _ { G } ) \right\}
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$$
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# 3.4 MISCELLANEOUS
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From the above, we have illustrated our quantization methods for weights, activations, gradients and errors. See Algorithm 1 for the detailed computation graph. There remain some issues to specify in an overall training process with only integers.
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Gradient descent optimizer like Momentum, RMSProp and Adam contains at least one copy of gradient updates $\Delta \mathbf { W }$ or their moving average, doubling memory consumption for weights during training, which is partially equivalent to use bigger $k _ { G }$ . Since the weight updates $\Delta \mathbf { W }$ are quantized to integer multiple of $\sigma$ and scaled by $\eta$ , we adopt pure mini-batch SGD without any form of momentum or adaptive learning rate to show the potential of reducing storage demands.
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Although L2 regularization works quite well for many large-scale DNNs where overfitting occurs commonly, WAGE removes small values in Equation 3 and introduces randomness in Equation 11, acting as certain types of regularization and can get comparable accuracy in later experiments. Thus, we remain L2 weight decay and dropout as supplementary regularization methods.
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The Softmax layer and cross-entropy criterion are widely adopted in classification tasks but the calculation of $e ^ { x }$ can hardly be applied in low-bitwidth linear mapping occasions. For tasks with small number of categories, we avoid Softmax layer and apply mean-square-error criterion but omit mean operation to form a sum-square-error (SSE) criterion since shifted errors will get the same values in Equation 9.
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# 4 EXPERIMENTS
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In this section, we set W-A-G-E bits to 2-8-8-8 as default for all layers in a CNN or MLP. The bitwidth $k _ { W }$ is 2 for ternary weights, which implies that there are no multiplications during inference. Constant parameter $\beta$ is 1.5 to make equal probabilities for ternary weights when initialized randomly. Activations and errors should be of the same bitwidth since computation graph of backpropagation is similar to inference and might be applied in the same partition of hardware or memristor array (Sheridan et al., 2017). Although XNOR-Net achieves 1-bit activations, reducing errors to 4 or less bits dramatically degenerates accuracies in our tests, so the bitwidth $k _ { A }$ and $k _ { E }$ are increased to 8 simultaneously. Weights are stored with 8-bit integers during training and ternarized by two constant symmetrical thresholds during inference. We first build the computation graph for a vanilla network, then insert quantization nodes in forward propagation and override gradients in backward propagation for each layer on Tensorflow (Abadi et al., 2016). Our method is evaluated on MNIST, SVHN, CIFAR10 and ILSVRC12 (Russakovsky et al., 2015) and Table 1 shows the comparison results.
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# 4.1 IMPLEMENT DETAILS
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MNIST: A variation of LeNet-5 (LeCun et al., 1998) with 32C5-MP2-64C5-MP2-512FC-10SSE is adopted. The input grayscale images are regarded as activations and quantized by Equation 8 where $\alpha$ equals to 1. The learning rate $\eta$ in WAGE remains as 1 for the whole 100 epochs. We report average accuracy of 10 runs on the test set.
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SVHN & CIFAR10: We use a VGG-like network (Simonyan & Zisserman, 2014) with $2 \times ( 1 2 8 { \bf C } 3 )$ - $\mathbf { M P } 2 . 2 \times ( 2 5 6 \mathbf { C } 3 ) \mathbf { - M P } 2 . 2 \times ( 5 1 2 \mathbf { C } 3 ) .$ -MP2-1024FC-10SSE. For CIFAR10 dataset, we follow the data augmentation in Lee et al. (2015) for training: 4 pixels are padded on each side, and a $3 2 \times 3 2$ patch is randomly cropped from the padded image or its horizontal flip. For testing, only single view of the original $3 2 \times 3 2$ image is evaluated. The model is trained with mini-batch size of 128 and totally 300 epochs. Learning rate $\eta$ is set to 8 and divided by 8 at epoch 200 and epoch 250. The original images are scaled and biased to the range of $[ - 1 , + 1 ]$ for 8-bit integer activation representation. As for SVHN dataset, we leave out randomly flip augmentation and reduce training epochs to 40 since it is a rather big dataset. The error rate is evaluated in the same way as MNIST.
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ImageNet: WAGE framework is evaluated on ILSVRC12 dataset with AlexNet (Krizhevsky et al., 2012) model but removes dropout and local response normalization layers. Images are firstly resized to $2 5 6 \times 2 5 6$ then randomly cropped to $2 2 4 \times 2 2 4$ and horizontally flipped, followed by bias subtraction as CIFAR10. For testing, the single center crop in validation set is evaluated. Since ImageNet task is much difficult than CIFAR10 and has 1000 categories, it is hard to converge when applying SSE or hinge loss criterion in WAGE, so we add Softmax and remove quantizations in the last layer for fear of severe accuracy drop (Tang et al., 2017). The model is trained with mini-batch size of 256 and totally 70 epochs. Learning rate $\eta$ is set to 4 and divided by 8 at epoch 60 and epoch 65.
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Table 1: Test or validation error rates $( \% )$ in previous works and WAGE on multiple datasets. Opt denotes gradient descent optimizer and withM means SGD with momentum, BN represents for batch normalization and 32 bits refers to float32, ImageNet top- $\mathbf { \nabla } \cdot \mathbf { k }$ format: top1/top5.
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<table><tr><td>Method</td><td>kw</td><td>kA</td><td>kG</td><td>kE</td><td>Opt</td><td>BN</td><td>MNIST</td><td>SVHN</td><td>CIFAR10</td><td>ImageNet</td></tr><tr><td>BC</td><td>1</td><td>32</td><td>32</td><td>32</td><td>Adam</td><td>V</td><td>1.29</td><td>2.30</td><td>9.90</td><td>1</td></tr><tr><td>BNN</td><td>1</td><td>1</td><td>32</td><td>32</td><td>Adam</td><td>√</td><td>0.96</td><td>2.53</td><td>10.15</td><td>1</td></tr><tr><td>BWN1</td><td>1</td><td>32</td><td>32</td><td>32</td><td>withM</td><td>√</td><td>1</td><td>1</td><td>1</td><td>43.2/20.6</td></tr><tr><td>XNOR</td><td>1</td><td>1</td><td>32</td><td>32</td><td>Adam</td><td>√</td><td>-</td><td>1</td><td>1</td><td>55.8/30.8</td></tr><tr><td>TWN</td><td>2</td><td>32</td><td>32</td><td>32</td><td>withM</td><td>√</td><td>0.65</td><td>-</td><td>7.44</td><td>34.7/13.8</td></tr><tr><td>TTQ</td><td>2</td><td>32</td><td>32</td><td>32</td><td>Adam</td><td>√</td><td>-</td><td>1</td><td>6.44</td><td>42.5/20.3</td></tr><tr><td>DoReFa²</td><td>8</td><td>8</td><td>32</td><td>8</td><td>Adam</td><td>√</td><td>1</td><td>2.30</td><td>-</td><td>47.0/-</td></tr><tr><td>TernGrad3</td><td>32</td><td>32</td><td>2</td><td>32</td><td>Adam</td><td>√</td><td>1</td><td>1</td><td>14.36</td><td>42.4/19.5</td></tr><tr><td>WAGE</td><td>2</td><td>8</td><td>8</td><td>8</td><td>SGD</td><td>×</td><td>0.40</td><td>1.92</td><td>6.78</td><td>51.6/27.8</td></tr></table>
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# 4.2 TRAINING CURVES AND REGULARIZATION
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We further compare WAGE variations and a vanilla CNN on CIFAR10. The vanilla CNN has the same VGG-like architecture described above except that none quantization of any operand or operation is applied. We add batch normalization in each layer and Softmax for the last layer, replace SSE with cross-entropy criterion, and then use a L2 weight decay of 1e-4 and momentum of 0.9 for training. The learning rate is set to 0.1 and divided by 10 at epoch 200 and epoch 250. For variations of WAGE, pattern 28ff has no quantization nodes in backpropagation. Although the 28ff pattern has the same optimizer and learning rate annealing method as the vanilla pattern, we find that weight updates are decreased by the rescale factor $\alpha$ in Equation 7. Therefore, the learning rate for $2 8 \mathrm { f f }$ is amplified and tuned, which reduces the error rate by $3 \%$ . Figure 3 shows the training curves of three counterparts. It can be seen that the 2888 pattern has comparable convergence rate to the vanilla CNN, better accuracy than those who only discretize weights and activations in inference time, though slightly more volatile. The discretization of backpropagation somehow acts as another type of regularization and have significant error rate drop when decreasing learning rate $\eta$ .
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+
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+
Figure 3: Training curves of WAGE variations and a vanilla CNN on CIFAR10.
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+
|
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+
# 4.3 BITWIDTH OF ERRORS
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+
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+
The bitwidth $k _ { E }$ is set to 8 as default in previous experiments. To further explore a proper bitwidth and its truncated boundary, we firstly export errors from vanilla CNN for CIFAR10 after 100 training epochs. The histogram of errors in the last convolution layer among 128 mini-batch data is shown in Figure 4. It is obvious that errors approximately obey logarithmic normal distribution where values are relatively small and have significantly large range. When quantized with $k _ { E }$ -bit integers, a proper window function should be chosen to truncate the distribution while retaining the approximate orientations for backpropagation. For more details about the layerwise histograms of all W, A, G, E operands, see Figure 5.
|
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+
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+
Firstly, the upper (right) boundary is immobilized to the maximum absolute value among all elements in errors as described in Equation 9. Then the left boundary will be based on the bitwidth $k _ { E }$ . We conduct a series of experiments for $k _ { E }$ ranging from 4 to 15. The boxplot in Figure 4 indicates that 4-8 bits of errors represented by integers are enough for CIFAR10 classification task. Bitwidth 8 is chosen as default to match the 8-bit image color levels and most operands in the micro control unit (MCU). The histogram of errors in the same layer of WAGE-2888 shows that after being shifted and quantized layer by layer, the distribution of errors reshapes and mostly aggregates into truncated window. Thus, most information for orientations is retained. Besides, the smaller values in errors have negligible effects on previous orientations though accumulated layer by layer, which are partially discarded in quantization.
|
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+
|
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+
Since the width of the window has been optimized, we left-shift the window with factor $\gamma$ to explore its horizontal position. The right boundary can be formulated as $m a x \{ | e | \} / \gamma$ . Table 2 shows the effect of shifting errors: although large values are in the minority, they play critical roles for backpropagation training while the majority with small values actually act as noises.
|
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+
|
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+
Table 2: Test error rates $( \% )$ on CIFAR10 when left-shift upper boundary with factor $\gamma$
|
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+
|
| 193 |
+
<table><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>8</td></tr><tr><td rowspan=1 colspan=1>error</td><td rowspan=1 colspan=1>6.78</td><td rowspan=1 colspan=1>7.31</td><td rowspan=1 colspan=1>8.08</td><td rowspan=1 colspan=1>16.92</td></tr></table>
|
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+
|
| 195 |
+
# 4.4 BITWIDTH OF GRADIENTS
|
| 196 |
+
|
| 197 |
+
The bitwidth $k _ { G }$ is set to 8 as default in previous experiments. Although weights are propagated with ternary values in inference and achieve $1 6 \times$ compression rate than float32 weights, they are saved and accumulated in a relatively higher bitwidth (8 bits) for backpropagation training. Therefore, the overall compression rate is only $4 \times$ . The inconsistent bitwidth between weight updates $k _ { G }$ and their effects in inference $k _ { W }$ provides indispensable buffer space. Otherwise, there might be too many weights changing their ternary values in each iteration, making training very slow and unstable. To further explore a proper bitwidth for gradients, we use WAGE 2-8-8-8 in CIFAR10 as baseline and range $k _ { G }$ from 2 to 12, the learning rate $\eta$ is divided by 2 every time the $k _ { G }$ decreases 1 bit to keep approximately equal weights accumulation in large number of iterations. Results from Table 3 show the effect of $k _ { G }$ and indicate the similar bitwidth requirement as previous experiments for $k _ { E }$ .
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+
|
| 199 |
+
|
| 200 |
+
Figure 4: Left are histograms of errors $e$ for same layer in vanilla network and WAGE-2888 network. Upper boundaries are the $m a x \{ | e | \}$ while lower boundaries are determined by the bitwidth $k _ { E }$ . The 10 run accuracies of different $k _ { E }$ are shown on the right.
|
| 201 |
+
|
| 202 |
+
Table 3: Test error rates $( \% )$ on CIFAR10 with different $k _ { G }$
|
| 203 |
+
|
| 204 |
+
<table><tr><td rowspan=1 colspan=1>kG</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>12</td></tr><tr><td rowspan=1 colspan=1>error</td><td rowspan=1 colspan=1>54.22</td><td rowspan=1 colspan=1>51.57</td><td rowspan=1 colspan=1>28.22</td><td rowspan=1 colspan=1>18.01</td><td rowspan=1 colspan=1>11.48</td><td rowspan=1 colspan=1>7.61</td><td rowspan=1 colspan=1>6.78</td><td rowspan=1 colspan=1>6.63</td><td rowspan=1 colspan=1>6.43</td><td rowspan=1 colspan=1>6.55</td><td rowspan=1 colspan=1>6.57</td></tr></table>
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+
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+
For ImageNet implementation, we conduct six patterns to show bitwidth requirements: 2888 from Table 1, 288C for more accurate errors (12 bits), 28C8 for larger buffer space, 28f8 for none quantization of gradients, 28ff for errors and gradients in float32 as unlimited case and its BN counterpart. The accuracy of original AlexNet reproduction is reported as baseline. Learning rate $\eta$ is set to 64 and divided by 8 in 28C8 pattern, 0.01 and divided by 10 in 28f8, 28ff counterparts and vanilla AlexNet. We observe overfitting when increasing $k _ { G }$ thus add L2 weight decay of 1e-4, 1e-4 and 5e-4 for 28f8, 28ff and 28ff-BN patterns, respectively. In table 4, the comparison between pattern 28C8 and 288C reveals that it might be more important to make more buffer space $k _ { G }$ for gradient accumulation than to keep high-resolution orientation $k _ { E }$ . Besides, when it comes to ImageNet dataset, the gradient accumulation, i.e., the bit width of gradients $( k _ { G } )$ and batch normalization become more important (Li et al., 2017) since samples in training set are so variant.
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+
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+
To avoid external memory consumption of full-precision weights during training, Deng et al. (2018) achieved 1-bit weights representation in both training and inference. They use a much larger minibatch size of 1000 and float32 backpropagation dataflow to accumulate more precise weight updates, equally compensating the buffer space in WAGE provided by external bits of $k _ { G }$ . However, large batch size will dramatically increase total training time, counteracting the speed benefits brought by integer arithmetic units. Besides, intermediate variables like feature maps often consume much more memory than weights and linearly correlated with mini-batch size. Therefore, we apply bigger $k _ { G }$ for better convergence rate, accuracy and lower memory usage.
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Table 4: Top-5 error rates $( \% )$ on ImageNet with different $k _ { G }$ and $k _ { E }$
|
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+
|
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+
<table><tr><td rowspan=1 colspan=1>Pattern</td><td rowspan=1 colspan=1>Vanilla</td><td rowspan=1 colspan=1>28ff-BN</td><td rowspan=1 colspan=1>28ff</td><td rowspan=1 colspan=1>28f8</td><td rowspan=1 colspan=1>28C8</td><td rowspan=1 colspan=1>288C</td><td rowspan=1 colspan=1>2888</td></tr><tr><td rowspan=1 colspan=1>error</td><td rowspan=1 colspan=1>19.29</td><td rowspan=1 colspan=1>20.67</td><td rowspan=1 colspan=1>24.14</td><td rowspan=1 colspan=1>23.92</td><td rowspan=1 colspan=1>26.88</td><td rowspan=1 colspan=1>28.06</td><td rowspan=1 colspan=1>27.82</td></tr></table>
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# 5 DISCUSSION AND FUTURE WORK
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The goal of this work is to demonstrate potentials of applying training and inference with lowbitwidth integers in DNNs. Compared with FP16, 8-bit integer operations will not only reduce the energy and area costs for IC design (about $5 \times$ , see Table 5), but also halve the memory accesses costs and memory size requirements during training, which will greatly benefit mobile devices with on-site learning capability. There are some points not involved in this work but yet to be improved or solved in future algorithm developments and hardware deployment.
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MAC Operation: WAGE framework is mainly tested with 2-8-8-8 bitwidth configuration, which means that though there are no multiplications during inference with ternary weights, MACs are still needed to calculate $\textbf { { g } }$ in training. Possible solution is 2-2-8-8 pattern if we do not consider the matching of bitwidths between $\textbf { \em a }$ and $e$ . However, ternary $\textbf { \em a }$ will dramatically slow down convergence and hurt accuracy since $Q ( x , 2 )$ has two relatively high thresholds and clear most outputs of each layer at the beginning of training, this phenomenon is also observed in our BNN replication.
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Non-linear quantization: The linear mapping with uniform distance is adopted in WAGE for its simplicity. However, non-linear quantization method like logarithmic representation (Miyashita et al., 2016; Zhou et al., 2017) might be more efficient because the weights and activations in a trained network naturally have logarithmic normal distributions as shown in Figure 4. Besides, values in logarithmic representation have much larger range with fewer bits than fixed-point representation and are naturally encoded in digital hardware. It is promising to training DNNs with integers encoded with logarithmic representation.
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Normalization: Normalization layers like Softmax and batch normalization are avoided or removed in some WAGE demonstrations. We think normalizations are essential for end-to-end multi-channel perception where sensors with different modalities have different input distributions, as well as cross-model features encoding and cognition where information from different branches gather to form higher-level representations. Therefore, a better way to quantize normalization is of great interest in further studies.
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Table 5: Rough relative costs in $4 5 \mathrm { n m } 0 . 9 \mathrm { V }$ from Sze et al. (2017).
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+
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+
<table><tr><td rowspan=2 colspan=1>Operation</td><td rowspan=1 colspan=1>Energy(pJ)</td><td rowspan=1 colspan=1>Area(um2)</td></tr><tr><td rowspan=1 colspan=1>MUL ADD</td><td rowspan=1 colspan=1>MUL ADD</td></tr><tr><td rowspan=1 colspan=1>8-bit INT</td><td rowspan=1 colspan=1>0.2 pJ 0.03 pJ</td><td rowspan=1 colspan=1>282 36</td></tr><tr><td rowspan=1 colspan=1>16-bit FP</td><td rowspan=1 colspan=1>1.1 pJ 0.40 pJ</td><td rowspan=1 colspan=1>1640 1360</td></tr><tr><td rowspan=1 colspan=1>32-bit FP</td><td rowspan=1 colspan=1>3.7 pJ 0.90 pJ</td><td rowspan=1 colspan=1>7700 4184</td></tr></table>
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+
|
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# 6 CONCLUSION
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+
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WAGE empowers pure low-bitwidth integer dataflow in DNNs for both training and inference. We introduce a new initialization method and a layer-wise constant scaling factor to replace batch normalization, which is a pain spot for network quantization. Many other components for training are also considered or simplified by alternative solutions. In addition, the bitwidth requirements for error computation and gradient accumulation are explored. Experiments reveal that we can quantize relative values of gradients, as well as discard the majority of small values and their orders of magnitude in backpropagation. Although the accumulation for weights updates are indispensable for stable convergence and final accuracy, there still remain works for compression and memory consumption can be further reduced in training. WAGE achieves state-of-art accuracies on multiple datasets with 2-8-8-8 bitwidth configuration. It is promising for incremental works via fine-tuning, more efficient mapping, quantization of batch normalization, etc. Overall, we introduce a framework without floating-point representation and demonstrate the potential to implement both discrete training and inference on integer-based lightweight ASIC or FPGA with on-site learning capability.
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# ACKNOWLEDGMENTS
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This work is partially supported by the Project of NSFC (61327902), the SuZhou-Tsinghua innovation leading program (2016SZ0102), the National Natural Science Foundation of China (61603209) and the Independent Research Plan of Tsinghua University (20151080467). We discuss a lot with Peng Jiao and Lei Deng, gratefully acknowledge for their thoughtful comments.
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| 235 |
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+
# REFERENCES
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Shuchang Zhou, Yuxin Wu, Zekun Ni, Xinyu Zhou, He Wen, and Yuheng Zou. Dorefa-net: Training low bitwidth convolutional neural networks with low bitwidth gradients. arXiv preprint arXiv:1606.06160, 2016.
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Chenzhuo Zhu, Song Han, Huizi Mao, and William J Dally. Trained ternary quantization. arXiv preprint arXiv:1612.01064, 2016.
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| 297 |
+
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| 298 |
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# A ALGORITHM
|
| 299 |
+
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| 300 |
+
We assume that network structures are defined and initialized with Equation 5. The annotations after pseudo code are potential corresponding operations for implementation in a fixed-point dataflow.
|
| 301 |
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| 302 |
+
Algorithm 1 Training an $I$ -layer net with WAGE method on floating-point-based or integer-based device. Weights, activations, gradients and errors are quantized according to Equations 6 - 12.
|
| 303 |
+
|
| 304 |
+
Require: a mini-batch of inputs and targets $( \pmb { a } _ { q } ^ { 0 } , \pmb { a } ^ { * } )$ which are quantized to $k _ { A }$ -bit integers, shiftbased $\alpha$ for each layer, learning rate scheduler $\eta$ , previous weight $W$ saved in $k _ { G }$ bits.
|
| 305 |
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| 306 |
+
Ensure: updated weights $W _ { t + 1 }$
|
| 307 |
+
|
| 308 |
+
1. Forward propagation:
|
| 309 |
+
1: for $i = 1$ to $I$ do
|
| 310 |
+
2: $W _ { q } ^ { i } Q _ { W } ( W ^ { i } )$
|
| 311 |
+
3: $\pmb { a } ^ { i } \gets R e L U ( \pmb { a } _ { q } ^ { i - 1 } \pmb { W } _ { q } ^ { i } )$
|
| 312 |
+
4: $\pmb { a } _ { q } ^ { i } Q _ { A } ( \pmb { a } ^ { i } )$
|
| 313 |
+
|
| 314 |
+
5: end for 2. Back propagation: Compute $\mathbf { \Psi } _ { e ^ { I } } \mathbf { \bar { \Psi } } _ { } \mathbf { \frac { \partial \mathcal { L } } { \partial \mathbf { a } ^ { I } } }$ knowing $\pmb { a } ^ { I }$ and $\mathbf { \delta } \mathbf { \textit { a } } ^ { * }$
|
| 315 |
+
|
| 316 |
+
6: for $i = I$ to 1 do
|
| 317 |
+
7: 0 $e _ { q } ^ { i } Q _ { E } ( e ^ { i } )$
|
| 318 |
+
8: $e ^ { i - 1 } e _ { q } ^ { i } W _ { q } ^ { i }$
|
| 319 |
+
9: g i ← e iq T a i − 1q
|
| 320 |
+
10: $\Delta W ^ { i } Q _ { G } ( \pmb { g } ^ { i } )$
|
| 321 |
+
11: Update and Clip $\dot { W } ^ { i }$ according to Equation 12
|
| 322 |
+
12: end for
|
| 323 |
+
|
| 324 |
+
# B LAYERWISE HISTOGRAM
|
| 325 |
+
|
| 326 |
+
|
| 327 |
+
Figure 5: Layerwise histograms of a trained VGG-like network with bitwidth configuration: 2-8- 8-8 and learning rate $\eta$ equals to 8. The Y-axis represents for probability in W-plots and G-plots, and logarithmic probability in A-plots and E-plots, respectively. In A-plots histograms are one-layer ahead so the first figure shows the quantized input data.
|
parse/train/HJGXzmspb/HJGXzmspb_content_list.json
ADDED
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|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "TRAINING AND INFERENCE WITH INTEGERS IN DEEP NEURAL NETWORKS ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
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"bbox": [
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| 7 |
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| 8 |
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| 9 |
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| 10 |
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| 11 |
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],
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| 12 |
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"page_idx": 0
|
| 13 |
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},
|
| 14 |
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{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Shuang $\\mathbf { W } \\mathbf { u } ^ { 1 }$ , Guoqi $\\mathbf { L i } ^ { 1 }$ , Feng Chen2, Luping Shi1 ",
|
| 17 |
+
"bbox": [
|
| 18 |
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| 19 |
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| 20 |
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| 21 |
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| 22 |
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],
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| 23 |
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"page_idx": 0
|
| 24 |
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},
|
| 25 |
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{
|
| 26 |
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"type": "text",
|
| 27 |
+
"text": "1Department of Precision Instrument \n2Department of Automation \nCenter for Brain Inspired Computing Research \nBeijing Innovation Center for Future Chip \nTsinghua University \n{lpshi,chenfeng}@mail.tsinghua.edu.cn ",
|
| 28 |
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"bbox": [
|
| 29 |
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| 30 |
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| 31 |
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|
| 32 |
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268
|
| 33 |
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],
|
| 34 |
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"page_idx": 0
|
| 35 |
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},
|
| 36 |
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{
|
| 37 |
+
"type": "text",
|
| 38 |
+
"text": "ABSTRACT ",
|
| 39 |
+
"text_level": 1,
|
| 40 |
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"bbox": [
|
| 41 |
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452,
|
| 42 |
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|
| 43 |
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544,
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| 44 |
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320
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| 45 |
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],
|
| 46 |
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"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "Researches on deep neural networks with discrete parameters and their deployment in embedded systems have been active and promising topics. Although previous works have successfully reduced precision in inference, transferring both training and inference processes to low-bitwidth integers has not been demonstrated simultaneously. In this work, we develop a new method termed as “WAGE” to discretize both training and inference, where weights (W), activations (A), gradients (G) and errors (E) among layers are shifted and linearly constrained to low-bitwidth integers. To perform pure discrete dataflow for fixed-point devices, we further replace batch normalization by a constant scaling layer and simplify other components that are arduous for integer implementation. Improved accuracies can be obtained on multiple datasets, which indicates that WAGE somehow acts as a type of regularization. Empirically, we demonstrate the potential to deploy training in hardware systems such as integer-based deep learning accelerators and neuromorphic chips with comparable accuracy and higher energy efficiency, which is crucial to future AI applications in variable scenarios with transfer and continual learning demands. ",
|
| 51 |
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"bbox": [
|
| 52 |
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233,
|
| 53 |
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|
| 54 |
+
764,
|
| 55 |
+
559
|
| 56 |
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],
|
| 57 |
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"page_idx": 0
|
| 58 |
+
},
|
| 59 |
+
{
|
| 60 |
+
"type": "text",
|
| 61 |
+
"text": "1 INTRODUCTION ",
|
| 62 |
+
"text_level": 1,
|
| 63 |
+
"bbox": [
|
| 64 |
+
176,
|
| 65 |
+
585,
|
| 66 |
+
336,
|
| 67 |
+
602
|
| 68 |
+
],
|
| 69 |
+
"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "text",
|
| 73 |
+
"text": "Recently deep neural networks (DNNs) are being widely used for numerous AI applications (Krizhevsky et al., 2012; Hinton et al., 2012; Silver et al., 2016). Depending on the massive tunable parameters, DNNs are considered to have powerful multi-level feature extraction and representation abilities. However, training DNNs needs energy-intensive devices such as GPU and CPU with high precision (float32) processing units and abundant memory, which has greatly challenged their extensive applications for portable devices. In addition, a state-of-art network often has far more weights and effective capacity to shatter all training samples (Zhang et al., 2016), leading to overfitting easily. ",
|
| 74 |
+
"bbox": [
|
| 75 |
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174,
|
| 76 |
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| 77 |
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|
| 78 |
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|
| 79 |
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],
|
| 80 |
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"page_idx": 0
|
| 81 |
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},
|
| 82 |
+
{
|
| 83 |
+
"type": "text",
|
| 84 |
+
"text": "As a result, there is much interest in reducing the size of network during inference (Hubara et al., 2016; Rastegari et al., 2016; Li et al., 2016), as well as dedicated hardware for commercial solutions (Jouppi et al., 2017; Chen et al., 2017; Shi et al., 2015). Due to the accumulation in stochastic gradient descent (SGD) optimization, the precision demand for training is usually higher than inference (Hubara et al., 2016; Li et al., 2017). Therefore, most of the existing techniques only focus on the deployment of a well-trained compressed network, while still keeping high precision and computational complexity during training. In this work, we address this problem as how to process both training and inference with low-bitwidth integers, which is essential for implementing DNNs in dedicated hardware. To this end, two fundamental issues are addressed for discretely training DNNs: i) how to quantize all the operands and operations, and ii) how many bits or states are needed for SGD computation and accumulation. ",
|
| 85 |
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"bbox": [
|
| 86 |
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|
| 87 |
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| 88 |
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|
| 89 |
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| 90 |
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],
|
| 91 |
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"page_idx": 0
|
| 92 |
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},
|
| 93 |
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{
|
| 94 |
+
"type": "text",
|
| 95 |
+
"text": "With respect to the issues, we propose a framework termed as “WAGE” that constrains weights (W), activations (A), gradients (G) and errors (E) among all layers to low-bitwidth integers in both training and inference. Firstly, for operands, linear mapping and orientation-preserved shifting are applied to achieve ternary weights, 8-bit integers for activations and gradients accumulation. Secondly, for operations, batch normalization (Ioffe & Szegedy, 2015) is replaced by a constant scaling factor. Other techniques for fine-tuning such as SGD optimizer with momentum and L2 regularization are simplified or abandoned with little performance degradation. Considering the overall bidirectional propagation, we completely streamline inference into accumulate-compare cycles and training into low-bitwidth multiply-accumulate (MAC) cycles with alignment operations, respectively. ",
|
| 96 |
+
"bbox": [
|
| 97 |
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176,
|
| 98 |
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|
| 99 |
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|
| 100 |
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|
| 101 |
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],
|
| 102 |
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"page_idx": 0
|
| 103 |
+
},
|
| 104 |
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{
|
| 105 |
+
"type": "text",
|
| 106 |
+
"text": "",
|
| 107 |
+
"bbox": [
|
| 108 |
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|
| 109 |
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|
| 110 |
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823,
|
| 111 |
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188
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| 112 |
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],
|
| 113 |
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"page_idx": 1
|
| 114 |
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},
|
| 115 |
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{
|
| 116 |
+
"type": "text",
|
| 117 |
+
"text": "We heuristically explore the bitwidth requirements of integers for error computation and gradient accumulation, which have rarely been discussed in previous works. Experiments indicate that it is the relative values (orientations) rather than absolute values (orders of magnitude) in error that guides previous layers to converge. Moreover, small values have negligible effects on previous orientations though propagated layer by layer, which can be partially discarded in quantization. We leverage these phenomena and use an orientation-preserved shifting operation to constrain errors. As for the gradient accumulation, though weights are quantized to ternary values in inference, a relatively higher bitwidth is indispensable to store and accumulate gradient updates. ",
|
| 118 |
+
"bbox": [
|
| 119 |
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|
| 120 |
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|
| 121 |
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|
| 122 |
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| 123 |
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],
|
| 124 |
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"page_idx": 1
|
| 125 |
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},
|
| 126 |
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{
|
| 127 |
+
"type": "text",
|
| 128 |
+
"text": "The proposed framework is evaluated on MNIST, CIFAR10, SVHN, ImageNet datasets. Comparing to those who only discretize weights and activations at inference time, it has comparable accuracy and can further alleviate overfitting, indicating some type of regularization. WAGE produces pure bidirectional low-precision integer dataflow for DNNs, which can be applied for training and inference in dedicated hardware neatly. We publish the code on GitHub1. ",
|
| 129 |
+
"bbox": [
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| 130 |
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| 131 |
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| 132 |
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| 133 |
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| 134 |
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],
|
| 135 |
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"page_idx": 1
|
| 136 |
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},
|
| 137 |
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{
|
| 138 |
+
"type": "text",
|
| 139 |
+
"text": "2 RELATED WORK ",
|
| 140 |
+
"text_level": 1,
|
| 141 |
+
"bbox": [
|
| 142 |
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|
| 143 |
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| 144 |
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| 145 |
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| 146 |
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],
|
| 147 |
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"page_idx": 1
|
| 148 |
+
},
|
| 149 |
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{
|
| 150 |
+
"type": "text",
|
| 151 |
+
"text": "We mainly focus on reducing precision of operands and operations in both training and inference. Orthogonal and complementary techniques for reducing complexity like network compression, pruning (Han et al., 2015; Zhou et al., 2017) and compact architectures (Howard et al., 2017) are impressively efficient but outside the scope this paper. ",
|
| 152 |
+
"bbox": [
|
| 153 |
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|
| 154 |
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|
| 155 |
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|
| 156 |
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|
| 157 |
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],
|
| 158 |
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"page_idx": 1
|
| 159 |
+
},
|
| 160 |
+
{
|
| 161 |
+
"type": "text",
|
| 162 |
+
"text": "Weight and activation Courbariaux et al. (2015); Hubara et al. (2016) propose methods to train DNNs with binary weights (BC) and activations (BNN) successively. They add noises to weights and activations as a form of regularization but real-valued gradients are accumulated in real-valued variables, suggesting that high precision accumulation is likely required for SGD optimization. XNOR-Net (Rastegari et al., 2016) has a filter-wise scaling factor for weights to improve the performance. Convolutions in XNOR-Net can be implemented efficiently using XNOR logical units and bit-count operations. However, these floating-point factors are calculated simultaneously during training, which generally aggravates the training effort. In TWN (Li et al., 2016) and TTQ (Zhu et al., 2016) two symmetric thresholds are introduced to constrain the weights to be ternary-valued: $\\{ + 1 , 0 , - 1 \\}$ . They claimed a tradeoff between model complexity and expressive ability. ",
|
| 163 |
+
"bbox": [
|
| 164 |
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|
| 165 |
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|
| 166 |
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|
| 167 |
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640
|
| 168 |
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],
|
| 169 |
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"page_idx": 1
|
| 170 |
+
},
|
| 171 |
+
{
|
| 172 |
+
"type": "text",
|
| 173 |
+
"text": "Gradient computation and accumulation DoReFa-Net (Zhou et al., 2016) quantizes gradients to low-bitwidth floating-point numbers with discrete states in the backward pass. TernGrad (Wen et al., 2017) quantizes gradient updates to ternary values to reduce the overhead of gradient synchronization in distributed training. Nevertheless, weights in DoReFa-Net and TernGrad are stored and updated with float32 during training like previous works. Besides, the quantization of batch normalization and its derivative is ignored. Thus, the overall computation graph for the training process is still presented with float32 and more complex with external quantization. Generally, it is difficult to apply DoReFa-Net training in an integer-based hardware directly, but it shows potential for exploring high-dimensional discrete spaces with discrete gradient descent directions. ",
|
| 174 |
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"bbox": [
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| 175 |
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| 176 |
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| 177 |
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| 178 |
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| 179 |
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],
|
| 180 |
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"page_idx": 1
|
| 181 |
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},
|
| 182 |
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{
|
| 183 |
+
"type": "text",
|
| 184 |
+
"text": "3 WAGE QUANTIZATION ",
|
| 185 |
+
"text_level": 1,
|
| 186 |
+
"bbox": [
|
| 187 |
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| 188 |
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| 189 |
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| 190 |
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| 191 |
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],
|
| 192 |
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"page_idx": 1
|
| 193 |
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},
|
| 194 |
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{
|
| 195 |
+
"type": "text",
|
| 196 |
+
"text": "The main idea of WAGE quantization is to constrain four operands to low-bitwidth integers: weight $W$ and activation $^ { a }$ in inference, error $e$ and gradient $\\textbf { { g } }$ in backpropagation training, see Figure 1. We extend the original definition of errors to multi-layer: error $e$ is the gradient of activation $\\textbf { \\em a }$ for the perspective of each convolution or fully-connected layer, while gradient $\\textbf { { g } }$ particularly refers to the gradient accumulation of weight $W$ . Considering the $i$ -th layer of a feed-forward network, we ",
|
| 197 |
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"bbox": [
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| 198 |
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| 199 |
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| 200 |
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| 201 |
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896
|
| 202 |
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],
|
| 203 |
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"page_idx": 1
|
| 204 |
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},
|
| 205 |
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{
|
| 206 |
+
"type": "text",
|
| 207 |
+
"text": "have: ",
|
| 208 |
+
"bbox": [
|
| 209 |
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173,
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| 210 |
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104,
|
| 211 |
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210,
|
| 212 |
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117
|
| 213 |
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],
|
| 214 |
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"page_idx": 2
|
| 215 |
+
},
|
| 216 |
+
{
|
| 217 |
+
"type": "equation",
|
| 218 |
+
"img_path": "images/e52fc3b4198793e13a34e9a3c792af01c6092e0cf52f2fbc93067b970e71cf74.jpg",
|
| 219 |
+
"text": "$$\ne ^ { i } = { \\frac { \\partial { \\mathcal { L } } } { \\partial a ^ { i } } } , \\pmb { g } ^ { i } = { \\frac { \\partial { \\mathcal { L } } } { \\partial W ^ { i } } }\n$$",
|
| 220 |
+
"text_format": "latex",
|
| 221 |
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"bbox": [
|
| 222 |
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| 223 |
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| 224 |
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| 225 |
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|
| 226 |
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],
|
| 227 |
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"page_idx": 2
|
| 228 |
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},
|
| 229 |
+
{
|
| 230 |
+
"type": "text",
|
| 231 |
+
"text": "where $\\mathcal { L }$ is the loss function. We separate these two terms that are mixed up in most existing schemes. The gradient of weight $\\textbf { { g } }$ and the gradient of activation $e$ flow to different paths in each layer, which is a fork both in inference and in backward training and generally acts as node of MAC operations. ",
|
| 232 |
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"bbox": [
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| 233 |
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| 234 |
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| 235 |
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| 236 |
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| 237 |
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],
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| 238 |
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"page_idx": 2
|
| 239 |
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},
|
| 240 |
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{
|
| 241 |
+
"type": "text",
|
| 242 |
+
"text": "For the forward propagation in the $i$ -th layer, assuming that weights are stored and accumulated with $k _ { G }$ -bit integers, then numerous works strive for a better quantization function $Q _ { W } ( \\cdot )$ that maps higher precision weights to their $k _ { W }$ -bit reflections, for example, $[ - 0 . 9 , 0 . 1 , 0 . 7 ]$ to $[ - 1 , 0 , 1 ]$ . Although weights are accumulated with high precision like float32, the deployment of the reflections in dedicated hardware are much more memory efficient after training. Activations are quantized with function $Q _ { A } ( \\cdot )$ to $k _ { A }$ bits to align the increased bitwidth caused by MACs. Weights and activations are discretized to even binary values in previous works, then MACs degrade into logical and bit-count operations that are extremely efficient (Rastegari et al., 2016). ",
|
| 243 |
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"bbox": [
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| 244 |
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| 245 |
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| 246 |
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| 247 |
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| 248 |
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"type": "text",
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"text": "For the backward propagation in the $i$ -th layer, the gradients of activations and weights are calculated by the derivatives of MACs that are generally considered to be in 16-bit floating-point precision at least. As illustrated in Figure 1, the MACs between $k _ { A }$ -bit inputs and $k _ { W }$ -bit weights will increase the bitwidth of outputs to $[ k _ { A } + k _ { W } - 1 ]$ in signed integer representation, and the similar broadening happens to errors $e$ as well. In consideration of training with only low-bitwidth integers, we propose additional functions $Q _ { E } ( \\cdot )$ and $Q _ { G } ( \\cdot )$ to constrain bitwidth of $e$ and $\\textbf { { g } }$ to $k _ { E }$ bits and $k _ { G }$ bits, respectively. In general, where there is a MAC operation, there are quantization operators named $Q _ { W } ( \\cdot ) , \\bar { Q _ { A } ( \\cdot ) } , \\bar { Q _ { G } ( \\cdot ) }$ and $Q _ { E } ( \\cdot )$ in inference and backpropagation. ",
|
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"type": "image",
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"img_path": "images/817241608beb2d22ed2d6644708fd32b8726cbfd6e15f235e1d6beb8e0af7390.jpg",
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"image_caption": [
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| 266 |
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"Figure 1: Four operators $Q _ { W } ( \\cdot )$ , $Q _ { A } ( \\cdot )$ , $Q _ { G } ( \\cdot )$ , $Q _ { E } ( \\cdot )$ added in WAGE computation dataflow to reduce precision, bitwidth of signed integers are below or on the right of arrows, activations are included in MAC for concision. "
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],
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"type": "text",
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"text": "3.1 SHIFT-BASED LINEAR MAPPING AND STOCHASTIC ROUNDING ",
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"text_level": 1,
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"type": "text",
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"text": "In WAGE quantization, we adopt a linear mapping with $k$ -bit integers for simplicity, where continuous and unbounded values are discretized with uniform distance $\\sigma$ : ",
|
| 292 |
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"bbox": [
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"type": "equation",
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"img_path": "images/52e4450db0c76d518b3f9362e5c9324695287d293150c0fc3743a7afb000483b.jpg",
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"text": "$$\n\\sigma ( k ) = 2 ^ { 1 - k } , k \\in \\mathbb { N } _ { + }\n$$",
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"bbox": [
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"type": "text",
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"text": "Then the basic quantization function that converts a floating-point number $x$ to its $k$ -bitwidth signed integer representation can be formulated as: ",
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"img_path": "images/b6dad99583cc722ff4458adf26f4e666167228c59389bb9d28bcac8273efdb0a.jpg",
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"text": "$$\nQ ( x , k ) = C l i p \\left\\{ \\sigma ( k ) \\cdot r o u n d \\left[ \\frac { x } { \\sigma ( k ) } \\right] , - 1 + \\sigma ( k ) , 1 - \\sigma ( k ) \\right\\}\n$$",
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"type": "text",
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"text": "where round approximates continuous values to their nearest discrete states. Clip is the saturation function that clips unbounded values to $[ - 1 + \\sigma , 1 - \\sigma ]$ , where the negative maximum value $- 1$ is removed to maintain symmetry. For example, $Q ( x , 2 )$ quantizes $\\{ - 1 , 0 . 2 , 0 . 6 \\}$ to $\\{ - 0 . 5 , 0 , 0 . 5 \\}$ . Equation 3 is merely used for simulation in floating-point hardware like GPU, whereas in a fixedpoint device, quantization and saturation is satisfied automatically. ",
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"text": "",
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"text": "Before applying linear mapping in some operands (e.g., error), we introduce an additional monolithic scaling factor for shifting values distribution to an appropriate order of magnitude, otherwise values will be all saturated or cleared by Equation 3. The scaling factor is calculated by $S h i f t$ function and then divided in later steps: ",
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"bbox": [
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"type": "equation",
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"img_path": "images/a9bcb3be87a40368961ad981326b05b0c150fbcfb26e73ec18f7ed1157271616.jpg",
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"text": "$$\nS h i f t ( x ) = 2 ^ { r o u n d ( \\log _ { 2 } x ) }\n$$",
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"type": "text",
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"text": "Finally, we propose stochastic rounding to substitute small and real-valued updates for gradient accumulation in training. Section 3.3.4 will detail the implementation of operator $Q _ { G } ( \\cdot )$ , where high bitwidth gradients are constrained to $k _ { G }$ -bit integers stochastically by a 16-bit random number generator. Figure 2 summarizes quantization methods used in WAGE. ",
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{
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| 395 |
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"type": "image",
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"img_path": "images/8125076d734879fbe78137103e8555a0f12b097c5c8576071434819e97b020a4.jpg",
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| 397 |
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"image_caption": [
|
| 398 |
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"Figure 2: Quantization methods used in WAGE. The notation $P , \\pmb { x } , \\lfloor \\cdot \\rfloor$ and $\\lceil \\cdot \\rceil$ denotes probability, vector, floor and ceil, respectively. $S h i f t ( \\cdot )$ refers to Equation 4 with a certain argument. "
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],
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{
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"type": "text",
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| 411 |
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"text": "3.2 WEIGHT INITIALIZATION",
|
| 412 |
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"text_level": 1,
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"type": "text",
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"text": "In previous works, weights are binarized directly by sgn function or ternarized by threshold parameters calculated during training. However, BNN fails to converge without batch normalization because weight values $\\pm 1$ are rather big for a typical DNN. Batch normalization not only efficiently avoids the problem of exploding and vanishing gradients, but also alleviates the demand for proper initialization. However, normalizing outputs for each layer and computing their gradients are quite complex without floating point unit (FPU). Besides, the moving averages of batch outputs occupy external memory. BNN shows a shift-based variation of batch normalization but it is hard to transform all of the elements to the fixed-point representations. As a result, weights should be cautiously initialized in this work where batch normalization is simplified to a constant scaling layer. A modified initialization method based on MSRA (He et al., 2015) can be formulated as: ",
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| 433 |
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"type": "equation",
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| 434 |
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"img_path": "images/ab645a64de455a8e86ef3fc2895e2dd6cbcb050c29f55c93f610992455a4ee52.jpg",
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| 435 |
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"text": "$$\nW \\sim U ( - L , + L ) , L = m a x \\{ \\sqrt { 6 / n _ { i n } } , L _ { m i n } \\} , L _ { m i n } = \\beta \\sigma\n$$",
|
| 436 |
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| 446 |
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"type": "text",
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"text": "where $n _ { i n }$ is the layer fan-in number, and the original limit $\\sqrt { 6 / n _ { i n } }$ in MSRA is calculated to keep same variance between inputs and outputs of the same layer theoretically. The additional limit $L _ { m i n }$ is a minimum value that the uniform distribution $U$ should reach, and $\\beta$ is a constant greater than 1 to create overlaps between minimum step size $\\sigma$ and maximum value $L$ . In case of $k _ { W }$ -bit linear mapping, if weights $W$ are quantized directly with original limits, we will get all-zero tensors when bitwidth $k _ { W }$ is small enough, e.g., 4, or fan-in $n _ { i n }$ is wide enough, where initialized weights may never reach the minimum step $\\sigma$ presented by fixed-point integers. So $L _ { m i n }$ ensures that weights can go beyond $\\sigma$ and quantized to non-zero values after $Q _ { W } ( \\cdot )$ when initialized randomly. ",
|
| 448 |
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"type": "text",
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"text": "3.3 QUANTIZATION DETAILS ",
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"type": "text",
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"text": "3.3.1 WEIGHT $Q _ { W } ( \\cdot )$ ",
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"type": "text",
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"text": "The modified initialization in Equation 5 will amplify weights holistically and guarantee their proper distribution, then $W$ is quantized directly with Equation 3: ",
|
| 483 |
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| 492 |
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"type": "equation",
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| 493 |
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"img_path": "images/d29727bbb0c5cbf73330f7c077997de913a4675f04a76766663cc6e2d995ffbc.jpg",
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"text": "$$\n\\boldsymbol { W _ { q } } = \\boldsymbol { Q _ { W } } ( \\boldsymbol { W } ) = \\boldsymbol { Q } ( \\boldsymbol { W } , \\boldsymbol { k _ { W } } )\n$$",
|
| 495 |
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"type": "text",
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"text": "It should be noted that the variance of weights is scaled compared to the original limit, which will cause exploding of network’s outputs. To alleviate the amplification effect, XNOR-Net proposed a filter-wise scaling factor calculated continuously with full precision. In consideration of integer implementation, we introduce a layer-wise shift-based scaling factor $\\alpha$ to attenuate the amplification effect: ",
|
| 507 |
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"bbox": [
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| 516 |
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"type": "equation",
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| 517 |
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"img_path": "images/af388e26d44d5c9e8bc7520882c9bac02b05c526f7d1955e4ed807597bbb0f58.jpg",
|
| 518 |
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"text": "$$\n\\alpha = m a x \\{ S h i f t ( L _ { m i n } / L ) , 1 \\}\n$$",
|
| 519 |
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|
| 520 |
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"bbox": [
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"type": "text",
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"text": "where $\\alpha$ is a pre-defined constant for each layer determined by the network structure. The modified initialization and attenuation factor $\\alpha$ together approximates floating-point weights to their integer representations, except that $\\alpha$ takes effect after activations to maintain precision of weights presented by $k _ { W }$ -bit integers. ",
|
| 531 |
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"type": "text",
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"text": "3.3.2 ACTIVATION $Q _ { A } ( \\cdot )$ ",
|
| 542 |
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"type": "text",
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| 553 |
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"text": "As stated above, the bitwidth of operands increases after MACs. Then a typical CNN is usually followed with pooling, normalization and activation. Average pooling is avoided because mean operations will increase precision demand. Besides, we hypothesize that batch outputs of each hidden layer approximately have zero-mean, then batch normalization degenerates into to a scaling layer where trainable and batch-calculated scaling parameters are replaced by $\\alpha$ mentioned in Equation 7. If activations are presented in $k _ { A }$ bits, the overall quantization of activations can be formulated as: ",
|
| 554 |
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| 562 |
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| 563 |
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"type": "equation",
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| 564 |
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"img_path": "images/79b3ebcf90ea5f6f38d23449395977e7ebcca740e465da29fc4ba2bd29df538b.jpg",
|
| 565 |
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"text": "$$\n\\pmb { a } _ { q } = Q _ { A } ( \\pmb { a } ) = Q ( \\pmb { a } / \\alpha , k _ { A } )\n$$",
|
| 566 |
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"type": "text",
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"text": "3.3.3 ERROR $Q _ { E } ( \\cdot )$ ",
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| 578 |
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"type": "text",
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"text": "Errors $e$ are calculated layer by layer using the chain rule during training. Although the computation graph of backpropagation is similar to the inference, the inputs are the gradients of $\\mathcal { L }$ , which are relatively small compared to actual inputs for networks. More importantly, the errors are unbounded and might have significantly larger ranges than that of activations, e.g., $[ 1 0 ^ { - 9 } , 1 0 ^ { - 4 } ]$ . DoReFa-Net first applies an affine transform on $e$ to map them into $[ - 1 , 1 ]$ , and then inverts the transform after quantization. Thus, the quantized $e$ are still presented as float32 numbers with discrete states and mostly small values. ",
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| 590 |
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"type": "text",
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| 600 |
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"text": "However, experiments uncover that it is the orientations rather than orders of magnitude in errors that guides previous layers to converge, then the inverse transformation after quantization in DoReFa-Net is no longer needed. The orientation-only preservation prompts us to propagate errors with integer√ √ thoroughly, where error distribution is firstly scaled into $[ - { \\sqrt { 2 } } , + { \\sqrt { 2 } } ]$ by dividing a shift factor as shown in Figure 2 and then quantized by $Q ( e , k _ { E } )$ : ",
|
| 601 |
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"type": "equation",
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"img_path": "images/989403af45a7243dda4ad9ad747e2b01bcb79998479a74f75b8d6d11e5e3e526.jpg",
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| 612 |
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"text": "$$\n\\pmb { e _ { q } } = Q _ { E } ( \\pmb { e } ) = Q ( \\pmb { e } / S h i f t ( m a x \\{ | \\pmb { e } | \\} ) , k _ { E } )\n$$",
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"type": "text",
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| 624 |
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"text": "where $m a x \\{ | e | \\}$ extracts the layer-wise maximum absolute value among all elements in error $e$ , multi-channel for convolution and multi-sample for batch training. The quantization of error discards large proportion of values smaller than $\\sigma$ , we will discuss the influence on accuracy later. ",
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"text": "3.3.4 GRADIENT $Q _ { G } ( \\cdot )$ ",
|
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"text": "Since we only preserve relative values of error after shifting, the gradient updates $\\textbf { { g } }$ derived from MACs between backward errors $e$ and forward activations $^ { a }$ are shifted consequently. We first rescale gradients $\\textbf { { g } }$ with another scaling factor and then bring in shift-based learning rate $\\eta$ : ",
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| 659 |
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"text": "$$\ng _ { s } = \\eta \\cdot g / S h i f t ( m a x \\{ | g | \\} )\n$$",
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"text_format": "latex",
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"bbox": [
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"text": "where $\\eta$ is an integer power of 2. The shifted gradients $\\mathbf { \\nabla } _ { \\mathbf { { \\boldsymbol { g } } } _ { s } }$ represent for minimum step numbers and directions for updating weights. If weights are stored with $k _ { G }$ -bit numbers, the minimum step of modification will be $\\pm 1$ for integers and $\\pm \\sigma ( k _ { G } )$ for floating-point values, respectively. The implement of learning rate $\\eta$ here is quite different from that in a vanilla DNN based on float32. In WAGE, there only remain directions for weights to change and the step sizes are integer multiples of minimum step $\\sigma$ . Shifted gradients $\\mathbf { \\nabla } _ { \\mathbf { { \\boldsymbol { g } } } _ { s } }$ may get greater than 1 if $\\eta$ is 2 or bigger to accelerate training at the beginning, or smaller than 0.5 during latter half of training when learning rate decay is usually applied. As illustrated in Figure 2, to substitute accumulation of small gradients in latter case, we separate $\\mathbf { \\nabla } _ { \\mathbf { { \\boldsymbol { g } } } _ { s } }$ into integer parts and decimal parts, then use a 16-bit random number generator to constrain high bitwidth $\\mathbf { \\nabla } _ { \\mathbf { { \\boldsymbol { g } } } _ { s } }$ to $k _ { G }$ -bit integers stochastically: ",
|
| 672 |
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"type": "text",
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"text": "",
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"img_path": "images/ddf6aaf699c977944e2de35f4afc6962e05e081863d9c7c3628c43ec957459a9.jpg",
|
| 694 |
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"text": "$$\n\\Delta W = Q _ { G } ( g ) = \\sigma ( k _ { G } ) \\cdot s g n ( g _ { s } ) \\cdot \\Big \\{ \\lfloor | g _ { s } | \\rfloor + B e r n o u l l i ( | g _ { s } | - \\lfloor | g _ { s } | \\rfloor ) \\Big \\}\n$$",
|
| 695 |
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"text_format": "latex",
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"bbox": [
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{
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"type": "text",
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"text": "where Bernoulli (Zhou et al., 2016) stochastically samples decimal parts to either 0 or 1. With proper setting of $k _ { G }$ , quantization of gradients will restrict the minimum step size, which may avoid local minimum and overfitting. Furthermore, the gradients will be ternary values when $\\eta$ is not greater than 1, which reduces communication costs for distributed training (Wen et al., 2017). At last, weights $W$ might exceed the range $[ - 1 + \\sigma , 1 - \\sigma ]$ presented by $k _ { G }$ -bit integers after updating with discrete increments $\\Delta \\mathbf { W }$ . So $C l i p$ function is indispensable to saturate and make sure there are only $2 ^ { k _ { G } - 1 } - 1$ states for weights accumulation. In case of the $t$ -th iteration, we have: ",
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"text": "$$\nW _ { t + 1 } = C l i p \\left\\{ { \\cal W } _ { t } - \\Delta { \\cal W } _ { t } , - 1 + \\sigma ( k _ { G } ) , 1 - \\sigma ( k _ { G } ) \\right\\}\n$$",
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"type": "text",
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"text": "3.4 MISCELLANEOUS ",
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| 731 |
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"text_level": 1,
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"type": "text",
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| 742 |
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"text": "From the above, we have illustrated our quantization methods for weights, activations, gradients and errors. See Algorithm 1 for the detailed computation graph. There remain some issues to specify in an overall training process with only integers. ",
|
| 743 |
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"bbox": [
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"type": "text",
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"text": "Gradient descent optimizer like Momentum, RMSProp and Adam contains at least one copy of gradient updates $\\Delta \\mathbf { W }$ or their moving average, doubling memory consumption for weights during training, which is partially equivalent to use bigger $k _ { G }$ . Since the weight updates $\\Delta \\mathbf { W }$ are quantized to integer multiple of $\\sigma$ and scaled by $\\eta$ , we adopt pure mini-batch SGD without any form of momentum or adaptive learning rate to show the potential of reducing storage demands. ",
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| 754 |
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| 764 |
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"text": "Although L2 regularization works quite well for many large-scale DNNs where overfitting occurs commonly, WAGE removes small values in Equation 3 and introduces randomness in Equation 11, acting as certain types of regularization and can get comparable accuracy in later experiments. Thus, we remain L2 weight decay and dropout as supplementary regularization methods. ",
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| 765 |
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| 775 |
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"text": "The Softmax layer and cross-entropy criterion are widely adopted in classification tasks but the calculation of $e ^ { x }$ can hardly be applied in low-bitwidth linear mapping occasions. For tasks with small number of categories, we avoid Softmax layer and apply mean-square-error criterion but omit mean operation to form a sum-square-error (SSE) criterion since shifted errors will get the same values in Equation 9. ",
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| 776 |
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"type": "text",
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"text": "4 EXPERIMENTS ",
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| 787 |
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"text_level": 1,
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"text": "In this section, we set W-A-G-E bits to 2-8-8-8 as default for all layers in a CNN or MLP. The bitwidth $k _ { W }$ is 2 for ternary weights, which implies that there are no multiplications during inference. Constant parameter $\\beta$ is 1.5 to make equal probabilities for ternary weights when initialized randomly. Activations and errors should be of the same bitwidth since computation graph of backpropagation is similar to inference and might be applied in the same partition of hardware or memristor array (Sheridan et al., 2017). Although XNOR-Net achieves 1-bit activations, reducing errors to 4 or less bits dramatically degenerates accuracies in our tests, so the bitwidth $k _ { A }$ and $k _ { E }$ are increased to 8 simultaneously. Weights are stored with 8-bit integers during training and ternarized by two constant symmetrical thresholds during inference. We first build the computation graph for a vanilla network, then insert quantization nodes in forward propagation and override gradients in backward propagation for each layer on Tensorflow (Abadi et al., 2016). Our method is evaluated on MNIST, SVHN, CIFAR10 and ILSVRC12 (Russakovsky et al., 2015) and Table 1 shows the comparison results. ",
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| 799 |
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| 806 |
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| 807 |
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| 808 |
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"type": "text",
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| 809 |
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"text": "4.1 IMPLEMENT DETAILS ",
|
| 810 |
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"text_level": 1,
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| 811 |
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"type": "text",
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| 821 |
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"text": "MNIST: A variation of LeNet-5 (LeCun et al., 1998) with 32C5-MP2-64C5-MP2-512FC-10SSE is adopted. The input grayscale images are regarded as activations and quantized by Equation 8 where $\\alpha$ equals to 1. The learning rate $\\eta$ in WAGE remains as 1 for the whole 100 epochs. We report average accuracy of 10 runs on the test set. ",
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"type": "text",
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"text": "SVHN & CIFAR10: We use a VGG-like network (Simonyan & Zisserman, 2014) with $2 \\times ( 1 2 8 { \\bf C } 3 )$ - $\\mathbf { M P } 2 . 2 \\times ( 2 5 6 \\mathbf { C } 3 ) \\mathbf { - M P } 2 . 2 \\times ( 5 1 2 \\mathbf { C } 3 ) .$ -MP2-1024FC-10SSE. For CIFAR10 dataset, we follow the data augmentation in Lee et al. (2015) for training: 4 pixels are padded on each side, and a $3 2 \\times 3 2$ patch is randomly cropped from the padded image or its horizontal flip. For testing, only single view of the original $3 2 \\times 3 2$ image is evaluated. The model is trained with mini-batch size of 128 and totally 300 epochs. Learning rate $\\eta$ is set to 8 and divided by 8 at epoch 200 and epoch 250. The original images are scaled and biased to the range of $[ - 1 , + 1 ]$ for 8-bit integer activation representation. As for SVHN dataset, we leave out randomly flip augmentation and reduce training epochs to 40 since it is a rather big dataset. The error rate is evaluated in the same way as MNIST. ",
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| 841 |
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"type": "text",
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"text": "ImageNet: WAGE framework is evaluated on ILSVRC12 dataset with AlexNet (Krizhevsky et al., 2012) model but removes dropout and local response normalization layers. Images are firstly resized to $2 5 6 \\times 2 5 6$ then randomly cropped to $2 2 4 \\times 2 2 4$ and horizontally flipped, followed by bias subtraction as CIFAR10. For testing, the single center crop in validation set is evaluated. Since ImageNet task is much difficult than CIFAR10 and has 1000 categories, it is hard to converge when applying SSE or hinge loss criterion in WAGE, so we add Softmax and remove quantizations in the last layer for fear of severe accuracy drop (Tang et al., 2017). The model is trained with mini-batch size of 256 and totally 70 epochs. Learning rate $\\eta$ is set to 4 and divided by 8 at epoch 60 and epoch 65. ",
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"type": "table",
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"img_path": "images/ff6d53cdabfd701becfc22b066472d69515744f47750f0b6b747608fee706dc9.jpg",
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| 855 |
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"table_caption": [
|
| 856 |
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"Table 1: Test or validation error rates $( \\% )$ in previous works and WAGE on multiple datasets. Opt denotes gradient descent optimizer and withM means SGD with momentum, BN represents for batch normalization and 32 bits refers to float32, ImageNet top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ format: top1/top5. "
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],
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"table_footnote": [],
|
| 859 |
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"table_body": "<table><tr><td>Method</td><td>kw</td><td>kA</td><td>kG</td><td>kE</td><td>Opt</td><td>BN</td><td>MNIST</td><td>SVHN</td><td>CIFAR10</td><td>ImageNet</td></tr><tr><td>BC</td><td>1</td><td>32</td><td>32</td><td>32</td><td>Adam</td><td>V</td><td>1.29</td><td>2.30</td><td>9.90</td><td>1</td></tr><tr><td>BNN</td><td>1</td><td>1</td><td>32</td><td>32</td><td>Adam</td><td>√</td><td>0.96</td><td>2.53</td><td>10.15</td><td>1</td></tr><tr><td>BWN1</td><td>1</td><td>32</td><td>32</td><td>32</td><td>withM</td><td>√</td><td>1</td><td>1</td><td>1</td><td>43.2/20.6</td></tr><tr><td>XNOR</td><td>1</td><td>1</td><td>32</td><td>32</td><td>Adam</td><td>√</td><td>-</td><td>1</td><td>1</td><td>55.8/30.8</td></tr><tr><td>TWN</td><td>2</td><td>32</td><td>32</td><td>32</td><td>withM</td><td>√</td><td>0.65</td><td>-</td><td>7.44</td><td>34.7/13.8</td></tr><tr><td>TTQ</td><td>2</td><td>32</td><td>32</td><td>32</td><td>Adam</td><td>√</td><td>-</td><td>1</td><td>6.44</td><td>42.5/20.3</td></tr><tr><td>DoReFa²</td><td>8</td><td>8</td><td>32</td><td>8</td><td>Adam</td><td>√</td><td>1</td><td>2.30</td><td>-</td><td>47.0/-</td></tr><tr><td>TernGrad3</td><td>32</td><td>32</td><td>2</td><td>32</td><td>Adam</td><td>√</td><td>1</td><td>1</td><td>14.36</td><td>42.4/19.5</td></tr><tr><td>WAGE</td><td>2</td><td>8</td><td>8</td><td>8</td><td>SGD</td><td>×</td><td>0.40</td><td>1.92</td><td>6.78</td><td>51.6/27.8</td></tr></table>",
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"type": "text",
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"text": "4.2 TRAINING CURVES AND REGULARIZATION ",
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| 882 |
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"text": "We further compare WAGE variations and a vanilla CNN on CIFAR10. The vanilla CNN has the same VGG-like architecture described above except that none quantization of any operand or operation is applied. We add batch normalization in each layer and Softmax for the last layer, replace SSE with cross-entropy criterion, and then use a L2 weight decay of 1e-4 and momentum of 0.9 for training. The learning rate is set to 0.1 and divided by 10 at epoch 200 and epoch 250. For variations of WAGE, pattern 28ff has no quantization nodes in backpropagation. Although the 28ff pattern has the same optimizer and learning rate annealing method as the vanilla pattern, we find that weight updates are decreased by the rescale factor $\\alpha$ in Equation 7. Therefore, the learning rate for $2 8 \\mathrm { f f }$ is amplified and tuned, which reduces the error rate by $3 \\%$ . Figure 3 shows the training curves of three counterparts. It can be seen that the 2888 pattern has comparable convergence rate to the vanilla CNN, better accuracy than those who only discretize weights and activations in inference time, though slightly more volatile. The discretization of backpropagation somehow acts as another type of regularization and have significant error rate drop when decreasing learning rate $\\eta$ . ",
|
| 883 |
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| 888 |
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|
| 889 |
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"page_idx": 6
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| 890 |
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|
| 891 |
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{
|
| 892 |
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"type": "image",
|
| 893 |
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"img_path": "images/4a436c0768c09e887e192b39a7ea39f460e8366485af64c63c70957efd0b1595.jpg",
|
| 894 |
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"image_caption": [
|
| 895 |
+
"Figure 3: Training curves of WAGE variations and a vanilla CNN on CIFAR10. "
|
| 896 |
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],
|
| 897 |
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|
| 898 |
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| 906 |
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{
|
| 907 |
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"type": "text",
|
| 908 |
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"text": "4.3 BITWIDTH OF ERRORS",
|
| 909 |
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"text_level": 1,
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| 910 |
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| 918 |
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|
| 919 |
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"type": "text",
|
| 920 |
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"text": "The bitwidth $k _ { E }$ is set to 8 as default in previous experiments. To further explore a proper bitwidth and its truncated boundary, we firstly export errors from vanilla CNN for CIFAR10 after 100 training epochs. The histogram of errors in the last convolution layer among 128 mini-batch data is shown in Figure 4. It is obvious that errors approximately obey logarithmic normal distribution where values are relatively small and have significantly large range. When quantized with $k _ { E }$ -bit integers, a proper window function should be chosen to truncate the distribution while retaining the approximate orientations for backpropagation. For more details about the layerwise histograms of all W, A, G, E operands, see Figure 5. ",
|
| 921 |
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"bbox": [
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| 927 |
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| 928 |
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| 929 |
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|
| 930 |
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"type": "text",
|
| 931 |
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"text": "Firstly, the upper (right) boundary is immobilized to the maximum absolute value among all elements in errors as described in Equation 9. Then the left boundary will be based on the bitwidth $k _ { E }$ . We conduct a series of experiments for $k _ { E }$ ranging from 4 to 15. The boxplot in Figure 4 indicates that 4-8 bits of errors represented by integers are enough for CIFAR10 classification task. Bitwidth 8 is chosen as default to match the 8-bit image color levels and most operands in the micro control unit (MCU). The histogram of errors in the same layer of WAGE-2888 shows that after being shifted and quantized layer by layer, the distribution of errors reshapes and mostly aggregates into truncated window. Thus, most information for orientations is retained. Besides, the smaller values in errors have negligible effects on previous orientations though accumulated layer by layer, which are partially discarded in quantization. ",
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| 932 |
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{
|
| 941 |
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"type": "text",
|
| 942 |
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"text": "Since the width of the window has been optimized, we left-shift the window with factor $\\gamma$ to explore its horizontal position. The right boundary can be formulated as $m a x \\{ | e | \\} / \\gamma$ . Table 2 shows the effect of shifting errors: although large values are in the minority, they play critical roles for backpropagation training while the majority with small values actually act as noises. ",
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| 949 |
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"page_idx": 7
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| 950 |
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| 951 |
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{
|
| 952 |
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"type": "table",
|
| 953 |
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"img_path": "images/63d633a8f70184e5e4c8b5e939242461580d275c708695a19924f9d1c1a2f33f.jpg",
|
| 954 |
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"table_caption": [
|
| 955 |
+
"Table 2: Test error rates $( \\% )$ on CIFAR10 when left-shift upper boundary with factor $\\gamma$ "
|
| 956 |
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],
|
| 957 |
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"table_footnote": [],
|
| 958 |
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"table_body": "<table><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>8</td></tr><tr><td rowspan=1 colspan=1>error</td><td rowspan=1 colspan=1>6.78</td><td rowspan=1 colspan=1>7.31</td><td rowspan=1 colspan=1>8.08</td><td rowspan=1 colspan=1>16.92</td></tr></table>",
|
| 959 |
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|
| 966 |
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|
| 967 |
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{
|
| 968 |
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"type": "text",
|
| 969 |
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"text": "4.4 BITWIDTH OF GRADIENTS",
|
| 970 |
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"text_level": 1,
|
| 971 |
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"bbox": [
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| 979 |
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{
|
| 980 |
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"type": "text",
|
| 981 |
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"text": "The bitwidth $k _ { G }$ is set to 8 as default in previous experiments. Although weights are propagated with ternary values in inference and achieve $1 6 \\times$ compression rate than float32 weights, they are saved and accumulated in a relatively higher bitwidth (8 bits) for backpropagation training. Therefore, the overall compression rate is only $4 \\times$ . The inconsistent bitwidth between weight updates $k _ { G }$ and their effects in inference $k _ { W }$ provides indispensable buffer space. Otherwise, there might be too many weights changing their ternary values in each iteration, making training very slow and unstable. To further explore a proper bitwidth for gradients, we use WAGE 2-8-8-8 in CIFAR10 as baseline and range $k _ { G }$ from 2 to 12, the learning rate $\\eta$ is divided by 2 every time the $k _ { G }$ decreases 1 bit to keep approximately equal weights accumulation in large number of iterations. Results from Table 3 show the effect of $k _ { G }$ and indicate the similar bitwidth requirement as previous experiments for $k _ { E }$ . ",
|
| 982 |
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"bbox": [
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|
| 988 |
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"page_idx": 7
|
| 989 |
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},
|
| 990 |
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{
|
| 991 |
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"type": "image",
|
| 992 |
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"img_path": "images/6656974d908cf6666067d8870d249173758fdec06426277d8625030220173579.jpg",
|
| 993 |
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"image_caption": [
|
| 994 |
+
"Figure 4: Left are histograms of errors $e$ for same layer in vanilla network and WAGE-2888 network. Upper boundaries are the $m a x \\{ | e | \\}$ while lower boundaries are determined by the bitwidth $k _ { E }$ . The 10 run accuracies of different $k _ { E }$ are shown on the right. "
|
| 995 |
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],
|
| 996 |
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"image_footnote": [],
|
| 997 |
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"bbox": [
|
| 998 |
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| 1003 |
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| 1004 |
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|
| 1005 |
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|
| 1006 |
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"type": "text",
|
| 1007 |
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"text": "",
|
| 1008 |
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"bbox": [
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| 1009 |
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| 1014 |
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| 1015 |
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| 1016 |
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{
|
| 1017 |
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"type": "table",
|
| 1018 |
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"img_path": "images/d644021159411c3a7902c6a98feaabc9a59d6c1e874997743d932543e72a5d38.jpg",
|
| 1019 |
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"table_caption": [
|
| 1020 |
+
"Table 3: Test error rates $( \\% )$ on CIFAR10 with different $k _ { G }$ "
|
| 1021 |
+
],
|
| 1022 |
+
"table_footnote": [],
|
| 1023 |
+
"table_body": "<table><tr><td rowspan=1 colspan=1>kG</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>12</td></tr><tr><td rowspan=1 colspan=1>error</td><td rowspan=1 colspan=1>54.22</td><td rowspan=1 colspan=1>51.57</td><td rowspan=1 colspan=1>28.22</td><td rowspan=1 colspan=1>18.01</td><td rowspan=1 colspan=1>11.48</td><td rowspan=1 colspan=1>7.61</td><td rowspan=1 colspan=1>6.78</td><td rowspan=1 colspan=1>6.63</td><td rowspan=1 colspan=1>6.43</td><td rowspan=1 colspan=1>6.55</td><td rowspan=1 colspan=1>6.57</td></tr></table>",
|
| 1024 |
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|
| 1031 |
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|
| 1032 |
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{
|
| 1033 |
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"type": "text",
|
| 1034 |
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"text": "For ImageNet implementation, we conduct six patterns to show bitwidth requirements: 2888 from Table 1, 288C for more accurate errors (12 bits), 28C8 for larger buffer space, 28f8 for none quantization of gradients, 28ff for errors and gradients in float32 as unlimited case and its BN counterpart. The accuracy of original AlexNet reproduction is reported as baseline. Learning rate $\\eta$ is set to 64 and divided by 8 in 28C8 pattern, 0.01 and divided by 10 in 28f8, 28ff counterparts and vanilla AlexNet. We observe overfitting when increasing $k _ { G }$ thus add L2 weight decay of 1e-4, 1e-4 and 5e-4 for 28f8, 28ff and 28ff-BN patterns, respectively. In table 4, the comparison between pattern 28C8 and 288C reveals that it might be more important to make more buffer space $k _ { G }$ for gradient accumulation than to keep high-resolution orientation $k _ { E }$ . Besides, when it comes to ImageNet dataset, the gradient accumulation, i.e., the bit width of gradients $( k _ { G } )$ and batch normalization become more important (Li et al., 2017) since samples in training set are so variant. ",
|
| 1035 |
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"bbox": [
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| 1041 |
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| 1042 |
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|
| 1043 |
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|
| 1044 |
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"type": "text",
|
| 1045 |
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"text": "To avoid external memory consumption of full-precision weights during training, Deng et al. (2018) achieved 1-bit weights representation in both training and inference. They use a much larger minibatch size of 1000 and float32 backpropagation dataflow to accumulate more precise weight updates, equally compensating the buffer space in WAGE provided by external bits of $k _ { G }$ . However, large batch size will dramatically increase total training time, counteracting the speed benefits brought by integer arithmetic units. Besides, intermediate variables like feature maps often consume much more memory than weights and linearly correlated with mini-batch size. Therefore, we apply bigger $k _ { G }$ for better convergence rate, accuracy and lower memory usage. ",
|
| 1046 |
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"bbox": [
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| 1052 |
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| 1053 |
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| 1054 |
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{
|
| 1055 |
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"type": "table",
|
| 1056 |
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"img_path": "images/0f764fd3a049484afdcc1adf780762091f36c805741b82f6ea13f33bab2836ad.jpg",
|
| 1057 |
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"table_caption": [
|
| 1058 |
+
"Table 4: Top-5 error rates $( \\% )$ on ImageNet with different $k _ { G }$ and $k _ { E }$ "
|
| 1059 |
+
],
|
| 1060 |
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"table_footnote": [],
|
| 1061 |
+
"table_body": "<table><tr><td rowspan=1 colspan=1>Pattern</td><td rowspan=1 colspan=1>Vanilla</td><td rowspan=1 colspan=1>28ff-BN</td><td rowspan=1 colspan=1>28ff</td><td rowspan=1 colspan=1>28f8</td><td rowspan=1 colspan=1>28C8</td><td rowspan=1 colspan=1>288C</td><td rowspan=1 colspan=1>2888</td></tr><tr><td rowspan=1 colspan=1>error</td><td rowspan=1 colspan=1>19.29</td><td rowspan=1 colspan=1>20.67</td><td rowspan=1 colspan=1>24.14</td><td rowspan=1 colspan=1>23.92</td><td rowspan=1 colspan=1>26.88</td><td rowspan=1 colspan=1>28.06</td><td rowspan=1 colspan=1>27.82</td></tr></table>",
|
| 1062 |
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|
| 1063 |
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|
| 1068 |
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|
| 1069 |
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},
|
| 1070 |
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{
|
| 1071 |
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"type": "text",
|
| 1072 |
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"text": "5 DISCUSSION AND FUTURE WORK ",
|
| 1073 |
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"text_level": 1,
|
| 1074 |
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|
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|
| 1080 |
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|
| 1081 |
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|
| 1082 |
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|
| 1083 |
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"type": "text",
|
| 1084 |
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"text": "The goal of this work is to demonstrate potentials of applying training and inference with lowbitwidth integers in DNNs. Compared with FP16, 8-bit integer operations will not only reduce the energy and area costs for IC design (about $5 \\times$ , see Table 5), but also halve the memory accesses costs and memory size requirements during training, which will greatly benefit mobile devices with on-site learning capability. There are some points not involved in this work but yet to be improved or solved in future algorithm developments and hardware deployment. ",
|
| 1085 |
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|
| 1092 |
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|
| 1093 |
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|
| 1094 |
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"type": "text",
|
| 1095 |
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"text": "MAC Operation: WAGE framework is mainly tested with 2-8-8-8 bitwidth configuration, which means that though there are no multiplications during inference with ternary weights, MACs are still needed to calculate $\\textbf { { g } }$ in training. Possible solution is 2-2-8-8 pattern if we do not consider the matching of bitwidths between $\\textbf { \\em a }$ and $e$ . However, ternary $\\textbf { \\em a }$ will dramatically slow down convergence and hurt accuracy since $Q ( x , 2 )$ has two relatively high thresholds and clear most outputs of each layer at the beginning of training, this phenomenon is also observed in our BNN replication. ",
|
| 1096 |
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"bbox": [
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| 1099 |
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|
| 1102 |
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| 1103 |
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|
| 1104 |
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|
| 1105 |
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"type": "text",
|
| 1106 |
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"text": "Non-linear quantization: The linear mapping with uniform distance is adopted in WAGE for its simplicity. However, non-linear quantization method like logarithmic representation (Miyashita et al., 2016; Zhou et al., 2017) might be more efficient because the weights and activations in a trained network naturally have logarithmic normal distributions as shown in Figure 4. Besides, values in logarithmic representation have much larger range with fewer bits than fixed-point representation and are naturally encoded in digital hardware. It is promising to training DNNs with integers encoded with logarithmic representation. ",
|
| 1107 |
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| 1110 |
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| 1111 |
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507
|
| 1112 |
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],
|
| 1113 |
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"page_idx": 9
|
| 1114 |
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},
|
| 1115 |
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{
|
| 1116 |
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"type": "text",
|
| 1117 |
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"text": "Normalization: Normalization layers like Softmax and batch normalization are avoided or removed in some WAGE demonstrations. We think normalizations are essential for end-to-end multi-channel perception where sensors with different modalities have different input distributions, as well as cross-model features encoding and cognition where information from different branches gather to form higher-level representations. Therefore, a better way to quantize normalization is of great interest in further studies. ",
|
| 1118 |
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| 1122 |
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598
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|
| 1124 |
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"page_idx": 9
|
| 1125 |
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},
|
| 1126 |
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{
|
| 1127 |
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"type": "table",
|
| 1128 |
+
"img_path": "images/b7116e8382caa9e6307f608a14085c1b09b912611afff679c2a62fda50cd9559.jpg",
|
| 1129 |
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"table_caption": [
|
| 1130 |
+
"Table 5: Rough relative costs in $4 5 \\mathrm { n m } 0 . 9 \\mathrm { V }$ from Sze et al. (2017). "
|
| 1131 |
+
],
|
| 1132 |
+
"table_footnote": [],
|
| 1133 |
+
"table_body": "<table><tr><td rowspan=2 colspan=1>Operation</td><td rowspan=1 colspan=1>Energy(pJ)</td><td rowspan=1 colspan=1>Area(um2)</td></tr><tr><td rowspan=1 colspan=1>MUL ADD</td><td rowspan=1 colspan=1>MUL ADD</td></tr><tr><td rowspan=1 colspan=1>8-bit INT</td><td rowspan=1 colspan=1>0.2 pJ 0.03 pJ</td><td rowspan=1 colspan=1>282 36</td></tr><tr><td rowspan=1 colspan=1>16-bit FP</td><td rowspan=1 colspan=1>1.1 pJ 0.40 pJ</td><td rowspan=1 colspan=1>1640 1360</td></tr><tr><td rowspan=1 colspan=1>32-bit FP</td><td rowspan=1 colspan=1>3.7 pJ 0.90 pJ</td><td rowspan=1 colspan=1>7700 4184</td></tr></table>",
|
| 1134 |
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"bbox": [
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| 1140 |
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| 1141 |
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},
|
| 1142 |
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{
|
| 1143 |
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"type": "text",
|
| 1144 |
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"text": "6 CONCLUSION ",
|
| 1145 |
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"text_level": 1,
|
| 1146 |
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| 1153 |
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| 1154 |
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|
| 1155 |
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"type": "text",
|
| 1156 |
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"text": "WAGE empowers pure low-bitwidth integer dataflow in DNNs for both training and inference. We introduce a new initialization method and a layer-wise constant scaling factor to replace batch normalization, which is a pain spot for network quantization. Many other components for training are also considered or simplified by alternative solutions. In addition, the bitwidth requirements for error computation and gradient accumulation are explored. Experiments reveal that we can quantize relative values of gradients, as well as discard the majority of small values and their orders of magnitude in backpropagation. Although the accumulation for weights updates are indispensable for stable convergence and final accuracy, there still remain works for compression and memory consumption can be further reduced in training. WAGE achieves state-of-art accuracies on multiple datasets with 2-8-8-8 bitwidth configuration. It is promising for incremental works via fine-tuning, more efficient mapping, quantization of batch normalization, etc. Overall, we introduce a framework without floating-point representation and demonstrate the potential to implement both discrete training and inference on integer-based lightweight ASIC or FPGA with on-site learning capability. ",
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"text": "ACKNOWLEDGMENTS ",
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"text_level": 1,
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"text": "This work is partially supported by the Project of NSFC (61327902), the SuZhou-Tsinghua innovation leading program (2016SZ0102), the National Natural Science Foundation of China (61603209) and the Independent Research Plan of Tsinghua University (20151080467). We discuss a lot with Peng Jiao and Lei Deng, gratefully acknowledge for their thoughtful comments. ",
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"text": "Shuchang Zhou, Yuxin Wu, Zekun Ni, Xinyu Zhou, He Wen, and Yuheng Zou. Dorefa-net: Training low bitwidth convolutional neural networks with low bitwidth gradients. arXiv preprint arXiv:1606.06160, 2016. ",
|
| 1522 |
+
"bbox": [
|
| 1523 |
+
174,
|
| 1524 |
+
790,
|
| 1525 |
+
823,
|
| 1526 |
+
833
|
| 1527 |
+
],
|
| 1528 |
+
"page_idx": 11
|
| 1529 |
+
},
|
| 1530 |
+
{
|
| 1531 |
+
"type": "text",
|
| 1532 |
+
"text": "Chenzhuo Zhu, Song Han, Huizi Mao, and William J Dally. Trained ternary quantization. arXiv preprint arXiv:1612.01064, 2016. ",
|
| 1533 |
+
"bbox": [
|
| 1534 |
+
169,
|
| 1535 |
+
842,
|
| 1536 |
+
825,
|
| 1537 |
+
871
|
| 1538 |
+
],
|
| 1539 |
+
"page_idx": 11
|
| 1540 |
+
},
|
| 1541 |
+
{
|
| 1542 |
+
"type": "text",
|
| 1543 |
+
"text": "A ALGORITHM ",
|
| 1544 |
+
"text_level": 1,
|
| 1545 |
+
"bbox": [
|
| 1546 |
+
176,
|
| 1547 |
+
102,
|
| 1548 |
+
316,
|
| 1549 |
+
117
|
| 1550 |
+
],
|
| 1551 |
+
"page_idx": 12
|
| 1552 |
+
},
|
| 1553 |
+
{
|
| 1554 |
+
"type": "text",
|
| 1555 |
+
"text": "We assume that network structures are defined and initialized with Equation 5. The annotations after pseudo code are potential corresponding operations for implementation in a fixed-point dataflow. ",
|
| 1556 |
+
"bbox": [
|
| 1557 |
+
174,
|
| 1558 |
+
133,
|
| 1559 |
+
823,
|
| 1560 |
+
162
|
| 1561 |
+
],
|
| 1562 |
+
"page_idx": 12
|
| 1563 |
+
},
|
| 1564 |
+
{
|
| 1565 |
+
"type": "text",
|
| 1566 |
+
"text": "Algorithm 1 Training an $I$ -layer net with WAGE method on floating-point-based or integer-based device. Weights, activations, gradients and errors are quantized according to Equations 6 - 12. ",
|
| 1567 |
+
"bbox": [
|
| 1568 |
+
173,
|
| 1569 |
+
176,
|
| 1570 |
+
823,
|
| 1571 |
+
207
|
| 1572 |
+
],
|
| 1573 |
+
"page_idx": 12
|
| 1574 |
+
},
|
| 1575 |
+
{
|
| 1576 |
+
"type": "text",
|
| 1577 |
+
"text": "Require: a mini-batch of inputs and targets $( \\pmb { a } _ { q } ^ { 0 } , \\pmb { a } ^ { * } )$ which are quantized to $k _ { A }$ -bit integers, shiftbased $\\alpha$ for each layer, learning rate scheduler $\\eta$ , previous weight $W$ saved in $k _ { G }$ bits. ",
|
| 1578 |
+
"bbox": [
|
| 1579 |
+
174,
|
| 1580 |
+
209,
|
| 1581 |
+
821,
|
| 1582 |
+
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|
| 1583 |
+
],
|
| 1584 |
+
"page_idx": 12
|
| 1585 |
+
},
|
| 1586 |
+
{
|
| 1587 |
+
"type": "text",
|
| 1588 |
+
"text": "Ensure: updated weights $W _ { t + 1 }$ ",
|
| 1589 |
+
"bbox": [
|
| 1590 |
+
176,
|
| 1591 |
+
238,
|
| 1592 |
+
388,
|
| 1593 |
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],
|
| 1595 |
+
"page_idx": 12
|
| 1596 |
+
},
|
| 1597 |
+
{
|
| 1598 |
+
"type": "text",
|
| 1599 |
+
"text": "1. Forward propagation: \n1: for $i = 1$ to $I$ do \n2: $W _ { q } ^ { i } Q _ { W } ( W ^ { i } )$ \n3: $\\pmb { a } ^ { i } \\gets R e L U ( \\pmb { a } _ { q } ^ { i - 1 } \\pmb { W } _ { q } ^ { i } )$ \n4: $\\pmb { a } _ { q } ^ { i } Q _ { A } ( \\pmb { a } ^ { i } )$ ",
|
| 1600 |
+
"bbox": [
|
| 1601 |
+
179,
|
| 1602 |
+
252,
|
| 1603 |
+
377,
|
| 1604 |
+
327
|
| 1605 |
+
],
|
| 1606 |
+
"page_idx": 12
|
| 1607 |
+
},
|
| 1608 |
+
{
|
| 1609 |
+
"type": "text",
|
| 1610 |
+
"text": "5: end for 2. Back propagation: Compute $\\mathbf { \\Psi } _ { e ^ { I } } \\mathbf { \\bar { \\Psi } } _ { } \\mathbf { \\frac { \\partial \\mathcal { L } } { \\partial \\mathbf { a } ^ { I } } }$ knowing $\\pmb { a } ^ { I }$ and $\\mathbf { \\delta } \\mathbf { \\textit { a } } ^ { * }$ ",
|
| 1611 |
+
"bbox": [
|
| 1612 |
+
181,
|
| 1613 |
+
329,
|
| 1614 |
+
467,
|
| 1615 |
+
369
|
| 1616 |
+
],
|
| 1617 |
+
"page_idx": 12
|
| 1618 |
+
},
|
| 1619 |
+
{
|
| 1620 |
+
"type": "text",
|
| 1621 |
+
"text": "6: for $i = I$ to 1 do \n7: 0 $e _ { q } ^ { i } Q _ { E } ( e ^ { i } )$ \n8: $e ^ { i - 1 } e _ { q } ^ { i } W _ { q } ^ { i }$ \n9: g i ← e iq T a i − 1q \n10: $\\Delta W ^ { i } Q _ { G } ( \\pmb { g } ^ { i } )$ \n11: Update and Clip $\\dot { W } ^ { i }$ according to Equation 12 \n12: end for ",
|
| 1622 |
+
"bbox": [
|
| 1623 |
+
178,
|
| 1624 |
+
369,
|
| 1625 |
+
526,
|
| 1626 |
+
476
|
| 1627 |
+
],
|
| 1628 |
+
"page_idx": 12
|
| 1629 |
+
},
|
| 1630 |
+
{
|
| 1631 |
+
"type": "text",
|
| 1632 |
+
"text": "B LAYERWISE HISTOGRAM ",
|
| 1633 |
+
"text_level": 1,
|
| 1634 |
+
"bbox": [
|
| 1635 |
+
174,
|
| 1636 |
+
102,
|
| 1637 |
+
418,
|
| 1638 |
+
118
|
| 1639 |
+
],
|
| 1640 |
+
"page_idx": 13
|
| 1641 |
+
},
|
| 1642 |
+
{
|
| 1643 |
+
"type": "image",
|
| 1644 |
+
"img_path": "images/43901665d2658825e17c80f353f099762fdcf082af6320b8c868442672e31a6e.jpg",
|
| 1645 |
+
"image_caption": [
|
| 1646 |
+
"Figure 5: Layerwise histograms of a trained VGG-like network with bitwidth configuration: 2-8- 8-8 and learning rate $\\eta$ equals to 8. The Y-axis represents for probability in W-plots and G-plots, and logarithmic probability in A-plots and E-plots, respectively. In A-plots histograms are one-layer ahead so the first figure shows the quantized input data. "
|
| 1647 |
+
],
|
| 1648 |
+
"image_footnote": [],
|
| 1649 |
+
"bbox": [
|
| 1650 |
+
171,
|
| 1651 |
+
132,
|
| 1652 |
+
825,
|
| 1653 |
+
666
|
| 1654 |
+
],
|
| 1655 |
+
"page_idx": 13
|
| 1656 |
+
}
|
| 1657 |
+
]
|
parse/train/HJGXzmspb/HJGXzmspb_middle.json
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parse/train/HJGXzmspb/HJGXzmspb_model.json
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parse/train/HkxBJT4YvB/HkxBJT4YvB.md
ADDED
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|
| 1 |
+
# LEARNING DISENTANGLED REPRESENTATIONS FORCOUNTERFACTUAL REGRESSION
|
| 2 |
+
|
| 3 |
+
Negar Hassanpour & Russell Greiner
|
| 4 |
+
|
| 5 |
+
Department of Computing Science University of Alberta Edmonton, Alberta, T6G 2E8, CANADA {hassanpo,rgreiner}@ualberta.ca
|
| 6 |
+
|
| 7 |
+
# ABSTRACT
|
| 8 |
+
|
| 9 |
+
We consider the challenge of estimating treatment effects from observational data; and point out that, in general, only some factors based on the observed covariates $X$ contribute to selection of the treatment $T$ , and only some to determining the outcomes $Y$ . We model this by considering three underlying sources of $\{ X , T , Y \}$ and show that explicitly modeling these sources offers great insight to guide designing models that better handle selection bias in observational datasets. This paper is an attempt to conceptualize this line of thought and provide a path to explore it further.
|
| 10 |
+
|
| 11 |
+
In this work, we propose an algorithm to (1) identify disentangled representations of the above-mentioned underlying factors from any given observational dataset $\mathcal { D }$ and (2) leverage this knowledge to reduce, as well as account for, the negative impact of selection bias on estimating the treatment effects from $\mathcal { D }$ . Our empirical results show that the proposed method achieves state-of-the-art performance in both individual and population based evaluation measures.
|
| 12 |
+
|
| 13 |
+
# 1 INTRODUCTION
|
| 14 |
+
|
| 15 |
+
As we rely more and more on artificial intelligence (AI) to automate the decision making processes, accurately estimating the causal effects of taking different actions gains an essential role. A prominent example is precision medicine $- i . e .$ , the customization of health-care tailored to each individual patient – which attempts to identify which medical procedure $t \in \tau$ will benefit a certain patient $x$ the most, in terms of the treatment outcome $y \in \mathbb { R }$ . Learning such models requires answering counterfactual questions (Rubin, 1974; Pearl, 2009) such as: “Would this patient have lived longer [and by how much], had she received an alternative treatment?”.
|
| 16 |
+
|
| 17 |
+
For notation: a dataset $\mathcal { D } = \{ [ x _ { i } , t _ { i } , y _ { i } ] \} _ { i = 1 } ^ { N }$ used for treatment effect estimation has the following format: for the $i ^ { t h }$ instance (e.g., patient), we have some context information $x _ { i } \in \mathcal { X } \subseteq \mathbb { R } ^ { K }$ (e.g., age, BMI, blood work, etc.), the administered treatment $t _ { i }$ chosen from a set of treatment options $\tau$ (e.g., {0: medication, 1: surgery}), and the respective observed outcome $y _ { i } \in \mathcal { V }$ (e.g., survival time; $\mathcal { V } \subseteq \mathbb { R } ^ { + }$ ) as a result of receiving treatment $t _ { i }$ . Note that $\mathcal { D }$ only contains the outcome of the administered treatment (aka observed outcome: $y _ { i }$ ), but not the outcome(s) of the alternative treatment(s) (aka counterfactual outcome(s): $y _ { i } ^ { t }$ for $t \in \mathcal { T } \backslash \{ t _ { i } \} )$ , which are inherently unobservable. For the binary-treatment case, we denote the alternative treatment as $\neg t _ { i } = 1 - t _ { i }$ .
|
| 18 |
+
|
| 19 |
+
Pearl (2009) demonstrates that, in general, causal relationships can only be learned by experimentation (on-line exploration), or running a Randomized Controlled Trial (RCT), where the treatment assignment does not depend on the individual $X - { \mathsf { s e e } }$ Figure 1(a). In many cases, however, this is expensive, unethical, or even infeasible. Here, we are forced to approximate treatment effects from off-line datasets collected through Observational Studies. In such datasets, the administered treatment $T$ depends on some or all attributes of individual $X - { \mathsf { s e e } }$ Figure 1(b). Here, as $\operatorname* { P r } ( T | X ) \neq \operatorname* { P r } ( T )$ , we say these datasets exhibit selection bias (Imbens & Rubin, 2015). Figure 2 illustrates selection bias in an example (synthetic) observational dataset.
|
| 20 |
+
|
| 21 |
+
|
| 22 |
+
Figure 1: Belief net structure for randomized controlled trials and observational studies. Here, ${ Y ^ { \overline { { 0 } } } } ( { Y ^ { 1 } } )$ is the outcome of applying $T =$ treatment#0 (#1) to the individual represented by $X$ .
|
| 23 |
+
|
| 24 |
+
Here, we want to accurately estimate the Individual Treatment Effect (ITE) for each instance $i - i . e .$ , to estimate $\mathbf { e } _ { i } = y _ { i } ^ { 1 } - y _ { i } ^ { 0 }$ . We frame the solution as learning the function $f : \mathcal { X } \times \mathcal { T } \mathcal { Y }$ that can accurately predict the outcomes (both observed $\hat { y _ { i } } ^ { t _ { i } }$ as well as counterfactuals $\hat { y _ { i } } ^ { \lnot t _ { i } }$ ) given the context information $x _ { i }$ for each individual. As mentioned earlier, there are two challenges associated with estimating treatment effects:
|
| 25 |
+
|
| 26 |
+
(i) The fact that counterfactual outcomes are unobservable (i.e., not present in any training data) makes estimating treatment effects more difficult than the generalization problem in the supervised learning paradigm. This is an inherent characteristic of this task.
|
| 27 |
+
(ii) Selection bias in observational datasets implies having fewer instances within each treatment arm at specific regions of the domain. This sparsity, in turn, would decrease the accuracy and confidence of predicting counterfactuals at those regions.
|
| 28 |
+
|
| 29 |
+
This paper addresses the second challenge by investigating the root causes of selection bias, by dissecting and identifying the underlying factors that can generate an observational dataset $\mathcal { D }$ , and leveraging this knowledge to reduce, as well as account for, the negative impact of selection bias on estimating the treatment effects from $\mathcal { D }$ . In this work, we borrow ideas from the representation learning literature (Bengio et al., 2013) in order to reduce selection bias and from the domain adaptation literature (Shimodaira, 2000) in order to account for the remainder selection bias that (might) still exist after its reduction.
|
| 30 |
+
|
| 31 |
+
Our analysis relies on the following assumptions: Assumption 1: Unconfoundedness (Rosenbaum & Rubin, 1983) – There are no unobserved confounders (i.e., covariates that contribute to both treatment selection procedure as well as determination of outcomes). Formally, $\{ Y ^ { t } \} _ { t \in { \mathcal { T } } } \bot T \mid X$ .
|
| 32 |
+
|
| 33 |
+
Assumption 2: Overlap (Imbens, 2004) – Every individual $x$ should have a non-zero chance of being assigned to any treatment arm. That is,
|
| 34 |
+
|
| 35 |
+
|
| 36 |
+
Figure 2: An example observational dataset. Here, to treat heart disease, a doctor typically prescribes surgery $( t = 1 )$ to younger patients (•) and medication ${ \bf \boldsymbol { t } } = 0$ ) to older ones ${ \bf \Xi } ( { \bf \Lambda } )$ . Note that instances with larger (resp., smaller) $x$ values have a higher chance to be assigned to the $t = 0$ (resp., 1) treatment arm; hence we have selection bias. The counterfactual outcomes (only used for evaluation purpose) are illustrated by small $\bullet ( \mathbf { \alpha } )$ for $\neg t = 1$ (0).
|
| 37 |
+
|
| 38 |
+
These two assumptions together are called strong ignorability (Rosenbaum & Rubin, 1983). Imbens & Wooldridge (2009) showed that strong ignorability is sufficient for ITE to be identifiable.
|
| 39 |
+
|
| 40 |
+
Without loss of generality, we assume that the random variable $X$ follows a(n unknown) joint probability distribution $\operatorname* { P r } ( X | \Gamma , \Delta , \Upsilon )$ , treatment $T$ follows $\mathrm { P r } ( T | \Gamma , \Delta )$ , and outcome $\boldsymbol { Y } ^ { T }$ follows $\mathrm { P r } _ { { \cal T } } ( { \cal Y } ^ { T } | \Delta , \Upsilon )$ , where $\Gamma , \Delta$ , and $\Upsilon$ represent the three underlying factors1 that generate an observational dataset $\mathcal { D }$ . The respective graphical model is illustrated in Figure 3. Conforming with the statements above, note that the graphical model also suggests that selection bias is induced by factors $\Gamma$ and $\Delta$ , where $\Delta$ represents the confounding factors between $T$ and $Y$ .
|
| 41 |
+
|
| 42 |
+
Main contribution: We argue that explicit identification of the underlying factors $\{ \Gamma , \Delta , \Upsilon \}$ in observational datasets offers great insight to guide designing models that better handle selection bias and consequently achieve better performance in terms of estimating ITEs. In this paper, we propose a model, named Disentangled Representations for CounterFactual Regression (DR-CFR), that is optimized to do exactly that. We also present experiments that demonstrate the advantages of this perspective; and show empirically that the proposed method outperforms state-of-the-art models in a variety of data generation scenarios with different dimensionality of factors; see below.
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Figure 3: Underlying factors of $X$ ; $\Gamma$ $( \Upsilon )$ are factors that partially determine only $T$ $( Y )$ but not the other random variable; and $\Delta$ are confounders; Selection bias is induced by factors $\Gamma$ and $\Delta$ .
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# 2 RELATED WORKS
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Selection bias in observational datasets is equivalent to a domain adaptation scenario where a model is trained on a “source” (observed) data distribution, but should perform well on a “target” (counterfactual) one. Learning treatment effects from observational datasets is closely related to “off-policy learning from logged bandit feedback” – cf., (Swaminathan & Joachims, 2015a), whose goal is learning an optimal policy that selects the best personalized treatment for each individual. A common statistical solution is re-weighting certain data instances to balance the source and target distributions. The majority of re-weighting approaches belong to the Inverse Propensity Weighting (IPW) family of methods – cf., (Austin, 2011; Bottou et al., 2013; Swaminathan & Joachims, 2015c). While IPW methods are unbiased, they suffer from high variance. Swaminathan & Joachims (2015b) proposed the Counterfactual Risk Minimization (CRM) principle to alleviate this issue. In summary, re-weighting is an attempt to account for the selection bias.
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Johansson et al. (2016) is among the pioneer works that explored ways to use techniques from representation learning (Bengio et al., 2013) to reduce the selection bias. Shalit et al. (2017) present a refined version of (Johansson et al., 2016)’s method that learns a common representation space $\Phi ( x ) = \phi$ by minimizing the discrepancy (Mansour et al., 2009) (hereinafter “disc”) between the conditional distributions of $\phi$ given $t = 0$ versus $\phi$ , given $t = 1$ . That is,
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$$
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\mathsf { d i s c } \Big ( \big \{ \Phi ( x _ { i } ) \big \} _ { i : t _ { i } = 0 } , \big \{ \Phi ( x _ { i } ) \big \} _ { i : t _ { i } = 1 } \Big )
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$$
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which is (effectively) a regularization term that attempts to reduce selection bias in the learned representation. On top of this representation learning network, they trained two regression networks $h ^ { t } ( \phi )$ – one for each treatment arm $( t \in \{ 0 , 1 \} )$ ) – that predict the outcomes.
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Hassanpour & Greiner (2019) argued that the learned representation cannot and should not remove all the selection bias, as the confounders not only contribute to choosing a treatment but also to determining the respective outcomes.2 As a result, where there are confounders (which is a common situation), even $\phi$ would exhibit some selection bias, although less than that in the original domain $x$ They built on the work of (Shalit et al., 2017) by introducing context-aware importance sampling weights, that attempt to account for the above-mentioned remainder selection bias. These weights
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$$
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\omega _ { i } = 1 + { \frac { \operatorname* { P r } ( \phi _ { i } \mid \neg t _ { i } ) } { \operatorname* { P r } ( \phi _ { i } \mid t _ { i } ) } } = 1 + { \frac { \operatorname* { P r } ( t _ { i } ) } { 1 - \operatorname* { P r } ( t _ { i } ) } } \cdot { \frac { 1 - \pi { \bigl ( } t _ { i } \mid \phi _ { i } { \bigr ) } } { \pi { \bigl ( } t _ { i } \mid \phi _ { i } { \bigr ) } } }
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$$
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are designed to enhance performance of estimating both factual as well as counterfacual outcomes (by the 1 and $\frac { \operatorname* { P r } ( \phi \mid \lnot t ) } { \operatorname* { P r } ( \phi \mid t ) }$ terms, respectively), where $\pi ( t _ { i } | \phi _ { i } )$ is the probability of assigning the observed $t _ { i }$ conditioned on the learned context $\phi _ { i }$ .
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Note that both (Shalit et al., 2017) and (Hassanpour & Greiner, 2019) use $\Phi$ to model the concatenation of factors $\Delta$ and $\Upsilon$ (see Figure 3). Although it does make sense that there should be no discrepancy between conditional distributions of $\Upsilon$ , the $\Delta$ factor should model the confounding factors, which by definition, must embed some information about treatment assignment. This would result in a positive discrepancy between conditional distributions of $\Delta$ that should not be minimized. Thus, minimizing Equation (1) with respect to $\Phi$ can lead to problematic results as it discards some of the confounders.
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Yao et al. (2018) proposed the Similarity preserved Individual Treatment Effect (SITE) method, which extends Shalit et al. (2017)’s framework by adding a local similarity preserving component. This component acts as a regularization term, that attempts to retain the same neighbourhood relationships in the learned representation space as exhibited in the original space, by matching the propensity scores $\operatorname* { P r } ( t = 1 | \bar { \boldsymbol { x } } )$ and $\textstyle \operatorname* { P r } ( t = { \bar { 1 } } | \phi )$ . This, however, results in learning sub-optimal representations when $\Gamma \neq \emptyset$ as SITE tries to keep instances whose $\Gamma \mathrm { s }$ are far apart, also far apart in $\phi$ . In other words, this component penalizes reducing selection bias in $\phi$ by not discarding the irrelevant information present in $\Gamma$ even when it does not hurt the outcome estimation at all.
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Our work has many similarities to (Kuang et al., 2017), who decomposed $X$ into two subsets: confounding and adjustment variables, which are similar to our $\Delta$ and $\Upsilon$ factors respectively. They then used an optimization algorithm for identifying these variables, to ultimately find an unbiased estimate of the Average Treatment Effect (ATE). We extend their work in three ways: (i) In addition to confounders and adjustment variables, we also identify the factors that determine the treatment and have no effect on the outcome (i.e., Γ). (ii) Unlike (Kuang et al., 2017) that take a linear approach by tagging the raw features as either confounders or adjustment variables, our proposed method has the capacity to learn [non-linear] representations of the underlying factors. (iii) Our method facilitates estimating both ATE as well ITE, whereas (Kuang et al., 2017) cannot provide estimates of ITEs.
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# 3 LEARNING DISENTANGLED REPRESENTATIONS
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We assume, without loss of generality, that any dataset of the form $\{ X , T , Y \}$ is generated from three underlying factors $\{ \Gamma , \bar { \Delta } , \Upsilon \}$ , as illustrated in Figure 3. 3 Observe that the factor $\Gamma$ (resp., $\Upsilon$ ) partially determines only $T$ (resp., $Y$ ), but not the other variables; and $\Delta$ includes the confounding factors between $T$ and $Y$ . This graphical model suggests that selection bias is induced by factors $\Gamma$ and $\Delta$ . It also shows that the outcome depends on the factors $\Delta$ and $\Upsilon$ . Inspired by this graphical model, our model architecture incorporates the following components:
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• Three representation learning networks; one for each underlying factor: $\Gamma ( x ) , \Delta ( x )$ , and $\Upsilon ( x )$ . • Two regression networks; one for each treatment arm: $h ^ { 0 } ( \Delta ( x ) , \Upsilon ( x ) )$ and $h ^ { 1 } ( \Delta ( x ) , \Upsilon ( x ) )$ . • Two logistic networks: $\pi _ { 0 } ( t | \Gamma ( x ) , \Delta ( x ) )$ to model the logging policy – aka behaviour policy in Reinforcement Learning; $c f .$ , (Sutton & Barto, 1998) – and $\pi ( t | \Delta ( x ) )$ to design weights that account for the confounders’ impact.
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We therefore try to minimize the following objective function:
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$$
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\begin{array} { l } { \displaystyle J ( \Gamma , \Delta , \Upsilon , h ^ { 0 } , h ^ { 1 } , \pi _ { 0 } ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \omega \big ( t _ { i } , \Delta ( x _ { i } ) \big ) \cdot \mathcal { L } \big [ y _ { i } , h ^ { t _ { i } } \big ( \Delta ( x _ { i } ) , \Upsilon ( x _ { i } ) \big ) \big ] } \\ { \displaystyle \quad \quad + \alpha \cdot { \mathsf { d i s c } } \big ( \{ \Upsilon ( x _ { i } ) \} _ { i : t _ { i } = 0 } , \{ \Upsilon ( x _ { i } ) \} _ { i : t _ { i } = 1 } \big ) } \\ { \displaystyle \quad \quad + \beta \cdot \frac { 1 } { N } \sum _ { i = 1 } ^ { N } - \log \big [ \pi _ { 0 } \big ( t _ { i } | \Gamma ( x _ { i } ) , \Delta ( x _ { i } ) \big ) \big ] } \\ { \displaystyle \quad \quad + \lambda \cdot \mathfrak { R e g } ( \Gamma , \Delta , \Upsilon , h ^ { 0 } , h ^ { 1 } , \pi _ { 0 } ) } \end{array}
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$$
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where $\omega \left( t _ { i } , \Delta ( x _ { i } ) \right)$ is the re-weighting function; $\mathcal { L } \big [ y _ { i } , h ^ { t _ { i } } \big ( \Delta ( x _ { i } ) , \Upsilon ( x _ { i } ) \big ) \big ]$ is the prediction loss for observed outcomes (aka factual loss); $\mathsf { d i s c } \big ( \{ \Upsilon ( x ) \} _ { i : t _ { i } = 0 } , \{ \Upsilon ( x ) \} _ { i : t _ { i } = 1 } \big )$ calculates the discrepancy between conditional distributions of $\Upsilon$ given $t = 0$ versus given $t = 1 ; - \log \pi _ { 0 } ( \cdot )$ is the cross entropy loss of predicting the assigned treatments given the learned context; and $\mathfrak { R e g } ( \cdot )$ is the regularization term for penalizing model complexity. The following sections elaborate on each of these terms.
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# 3.1 FACTUAL LOSS: $\mathcal { L } \big [ \boldsymbol { y } , h ^ { t } \big ( \Delta ( \boldsymbol { x } ) , \Upsilon ( \boldsymbol { x } ) \big ) \big ]$
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Similar to (Johansson et al., 2016; Shalit et al., 2017; Hassanpour & Greiner, 2019; Yao et al., 2018), we train two regression networks $h ^ { 0 }$ and $h ^ { 1 }$ , one for each treatment arm. As guided by the graphical model in Figure 3, the inputs to these regression networks are the outputs of the $\Delta ( x )$ and $\Upsilon ( x )$ representation networks and their outputs are the predicted outcomes for their respective treatments.
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Note that the prediction loss $\mathcal { L }$ can only be calculated on the observed outcomes (hence the name factual loss), as counterfactual outcomes are not available in any training set. This would be an L2-loss for real-valued outcomes and a log-loss for binary outcomes. By minimizing the factual loss, we ensure that the union of the learned representations $\dot { \Delta } ( x )$ and $\Upsilon ( x )$ retain enough information needed for accurate estimation of the observed outcomes.
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# 3.2 RE-WEIGHTING FUNCTION: $\omega ( t , \Delta ( x ) )$
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We follow (Hassanpour & Greiner, 2019)’s design for weights as re-stated in Equation (2), with the modification that we employ $\Delta$ to calculate the weights instead of $\Phi$ . Although following the same design, we anticipate our weights should perform better in practice than those in (Hassanpour & Greiner, 2019) as: (i) no confounders are discarded due to minimizing the imbalance loss (because our disc is defined based on $\Upsilon$ , not $\Phi$ ); and (ii) only the legitimate confounders are used to derive the weights (i.e., $\Delta )$ , not the ones that have not contributed to treatment selection (i.e., Υ).
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Notably, the weights design in Equation (2) is different from the common practice in re-weighting techniques (e.g., IPW) in that the weights are calculated based on all factors that determine $T$ (i.e., $\Gamma$ as well as $\Delta$ ). However, we argue that incorporation of $\Gamma$ in the weights might result in emphasizing the wrong instances. In other words, since the factual loss $\mathcal { L }$ is only sensitive to factors $\Delta$ and $\Upsilon$ , and not $\Gamma$ , re-weighting $\mathcal { L }$ according to $\Gamma$ would yield a wrong objective function to be optimized.
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$$
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\mathsf { d i s c } \big ( \{ \Upsilon ( x _ { i } ) \} _ { i : t _ { i } = 0 } , \{ \Upsilon ( x _ { i } ) \} _ { i : t _ { i } = 1 } \big )
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$$
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According to Figure 3, $\Upsilon$ should be independent of $T$ due to the collider structure at $Y$ . Therefore,
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$$
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\Upsilon \ \perp \ T \quad \implies \quad \operatorname* { P r } ( \Upsilon \mid T ) = \operatorname* { P r } ( \Upsilon ) \quad \implies \quad \operatorname* { P r } ( \Upsilon \mid T = 0 ) = \operatorname* { P r } ( \Upsilon \mid T = 1 )
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$$
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We used Maximum Mean Discrepancy (MMD) (Gretton et al., 2012) to calculate dissimilarity between the two conditional distributions of $\Upsilon$ given $t = 0$ versus $t = 1$ .
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By minimizing the imbalance loss, we ensure that the learned factor $\Upsilon$ embeds no information about $T$ and all the confounding factors are retained in $\Delta$ . Capturing all the confounders in $\Delta$ and only in $\Delta$ is the hallmark of the proposed method, as we will use it for optimal re-weighting of the factual loss term (next section). Note that this differs from Shalit et al. (2017)’s approach in that they do not distinguish between the independent factors $\Delta$ and $\Upsilon$ ; and minimizing the loss defined on only one factor $\Phi$ which might erroneously suggest discarding some of the confounders in $\Delta$ .
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# 3.4 CROSS ENTROPY LOSS: $- \log \left[ \pi _ { 0 } \big ( t | \Gamma ( x ) , \Delta ( x ) \big ) \right]$
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We model the logging policy as a logistic regression network parameterized by $\big [ W _ { 0 } , b _ { 0 } \big ]$ as follows: $\pi _ { 0 } \bigl ( t | \psi \bigr ) = \Bigl [ 1 + e ^ { - \bigl ( 2 t - 1 \bigr ) ( \psi \cdot W _ { 0 } + b _ { 0 } ) } \Bigr ] ^ { - 1 }$ , where $\psi$ is the concatenation of matrices $\Gamma$ and $\Delta$ Minimizing the cross entropy loss enforces learning $\Gamma$ and $\Delta$ in a way that allows $\pi _ { 0 } ( \cdot )$ to predict the assigned treatments. In other words, the union of the learned representations of $\Gamma$ and $\Delta$ retain enough information to recover the logging policy that guided the treatment assignments.
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# 4 EXPERIMENTS
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# 4.1 BENCHMARKS
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Evaluating treatment effect estimation methods is problematic on real-world datasets since, as mentioned earlier, their counterfactual outcomes are inherently unobservable. A common solution is to synthesize datasets where the outcomes of all possible treatments are available, then discard some outcomes to create a proper observational dataset with characteristics (such as selection bias) similar to a real-world one – cf., (Beygelzimer & Langford, 2009; Hassanpour & Greiner, 2018). In this work, we use two such benchmarks: our synthetic series of datasets as well as a publicly available benchmark: the Infant Health and Development Program (IHDP) (Hill, 2011).
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# 4.1.1 SYNTHETIC DATASETS
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We generated our synthetic datasets according to the following process, which takes as input the sample size $N$ ; dimensionalities $[ m _ { \Gamma } , m _ { \Delta } , m _ { \Upsilon } ] \in \mathcal { Z } ^ { + ( 3 ) }$ ; for each factor $L \in \{ \Gamma , \Delta , \Upsilon \}$ , the means and covariance matrices $\left( \mu _ { L } , \Sigma _ { L } \right)$ ; and a scalar $\zeta$ that determines the slope of the logistic curve.
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• For each latent factor $L \in \{ \Gamma , \Delta , \Upsilon \}$
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– Form $L$ by drawing $N$ instances (each of size $m _ { L }$ ) from $\mathcal { N } ( \mu _ { L } , \Sigma _ { L } )$ ,
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– Concatenate $\Gamma , \Delta$ , and $\Upsilon$ to make the covariates matrix $X$ [of size $N \times ( m _ { \Gamma } + m _ { \Delta } + m _ { \Upsilon } ) ]$
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– Concatenate $\Gamma$ and $\Delta$ to make $\Psi$ [of size $N \times \left( m _ { \Gamma } + m _ { \Delta } \right) ]$
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– Concatenate $\Delta$ and $\Upsilon$ to make $\Phi$ [of size $N \times ( m _ { \Delta } + m _ { \Upsilon } ) ]$
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• For treatment $T$
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– Sample $m _ { \Gamma } + m _ { \Delta }$ tuple of coefficients $\theta$ from $\mathcal { N } ( 0 , 1 ) ^ { m _ { \Gamma } + m _ { \Delta } }$
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– Define the logging policy as $\begin{array} { r } { \pi _ { 0 } ( t = 1 | z ) = \frac { 1 } { 1 + \exp ( - \zeta z ) } } \end{array}$ , where $z = \Psi \cdot \theta$
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– For each instance $x _ { i }$ , sample treatment $t _ { i }$ from the Bernoulli distribution with parameter $\pi _ { 0 } ( t = 1 | z _ { i } )$
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• For outcomes $Y ^ { 0 }$ and $Y ^ { 1 }$ :
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– Sample $m _ { \Delta } + m _ { \Upsilon }$ tuple of coefficients $\vartheta ^ { 0 }$ and $\vartheta ^ { 1 }$ from $\mathcal { N } ( 0 , 1 ) ^ { m _ { \Delta } + m _ { \Upsilon } }$ – Define $y ^ { 0 } = ( \Phi \circ \Phi \circ \Phi + 0 . 5 ) \cdot \vartheta ^ { 0 } / ( m _ { \Delta } + m _ { \Upsilon } ) + \varepsilon$ and $y ^ { 1 } = ( \Phi \circ \Phi ) \cdot \vartheta ^ { 1 } / ( m _ { \Delta } + m \Upsilon ) + \varepsilon ,$ , where $\varepsilon$ is a white noise sampled from $\mathcal { N } ( 0 , 0 . 1 )$ and $\circ$ is the symbol for element-wise (Hadamard/Schur) product.
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We considered all the viable datasets in a mesh generated by $m _ { \Gamma } , m _ { \Delta } , m _ { \Upsilon } \in \{ 0 , 4 , 8 \}$ . This creates 24 scenarios4 that consider all possible situations in terms of the relative sizes of the factors $\Gamma , \Delta$ , and $\Upsilon$ . For each scenario, we synthesized five datasets with various initial random seeds.
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# 4.1.2 INFANT HEALTH AND DEVELOPMENT PROGRAM (IHDP)
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The original RCT data was designed to evaluate the effect of specialist home visits on future cognitive test scores of premature infants. Hill (2011) induced selection bias by removing a non-random subset of the treated population to create a realistic observational dataset. The resulting dataset contains 747 instances (608 control, 139 treated) with 25 covariates. We run our experiments on the same benchmark (100 realizations of outcomes) provided by and used in (Johansson et al., 2016; Shalit
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(a) Slice of the weights matrix that connects {the variables in $X$ belonging to $\Gamma$ } to {the first layer of the representation network that attempts to identify $\Gamma \}$ . The size of this slice is $m _ { \Gamma } \times K$ .
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(b) Slice of the weights matrix that connects {the variables in $X$ not belonging to $\Gamma \}$ to {the first layer of the representation network that attempts to identify $\Gamma$ }. The size of this slice is $( m _ { \Delta } + m _ { \Upsilon } ) \times K$ .
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Figure 4: Visualization of slicing the learned weights matrix in the first layer of the representation network (number of neurons: $K$ ) for identifying $\Gamma$ (best viewed in color).
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Figure 5: Radar charts that visualize the capability of DR-CFR in identifying the underlying factors $\Gamma , \Delta$ , and $\Upsilon$ . Each vertex on the polygons is identified with the factors’ dimension sequence $( m _ { \Gamma } \underline { { { m } } } _ { \Delta \underline { { { - } } } } m \underline { { { \Upsilon } } } )$ of the associated synthetic dataset. The polygons’ radii are scaled between $0 { : } 0 . 0 9$ and quantify the average weights of the first slice (in dotted magenta) and the second slice (in cyan).
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et al., 2017). Outcomes of this semi-synthetic benchmark were simulated according to response surfaces provided in the Non-Parametric Causal Inference (NPCI) package (Dorie, 2016).
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# 4.2 RESULTS AND DISCUSSIONS
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# 4.2.1 EVALUATING IDENTIFICATION OF FACTORS $\{ \Gamma , \Delta , \Upsilon \}$
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First, we want to determine if the proposed method is able to identify the variables that belong to each underlying factor. To do so, we look at the weight matrix in the first layer of each representation network, which is of size $( m _ { \Gamma } + m _ { \Delta } + m _ { \Upsilon } ) \times K$ , where $K$ is the number of neurons in the first hidden layer of the respective representation network. For example, to check if $\Gamma$ is identified properly, we partition the weights matrix into two slices, as shown in Figure 4, and calculate the average of each slice. The first slice [referred to as $\mathbf { S } _ { \Gamma }$ ; highlighted in Figure 4(a)] pertains to “ Γ’s ground truth variables in $X ^ { \dag }$ and the second slice $[ S _ { \neg \Gamma }$ ; Figure 4(b)] pertains to “variables in $X$ that do not belong to $\Gamma ^ { \ast }$ . Constructing $\mathbf { S } _ { \Delta }$ , $\mathbf { S } _ { \lnot \Delta }$ , $\mathtt { S } _ { \mathtt { Y } }$ , and $\mathbf { S } _ { \neg \Upsilon }$ follow a similar procedure.
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If the proposed method achieves a good identification, then we expect the average of the absolute values of weights in $\mathrm { \bf S _ { \mathrm { { T } } } }$ should be higher than that of $\mathbf { S } _ { \lnot \Gamma }$ ; this same claim should hold for $( \mathsf { S } _ { \Delta } , \mathsf { S } _ { \neg \Delta } )$ and $( \mathsf { S r } , \mathsf { S } _ { \neg \Upsilon } )$ as well. Note that only the relative relationships between the average weights in either of the slices matter; since this analysis is aimed at checking whether, for example, for identifying $\Gamma$ , its respective representation network has indeed learned to emphasize on “Γ’s ground truth variables in $X$ ” more than the other variables in $X$ . Figure 5 illustrates the identification performance of DR-CFR according to this analysis; showing empirically that the proposed method successfully identifies all the three underlying factors, for all synthetic datasets.
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Figure 6: Radar charts for visualizing the PEHE performance results on the synthetic datasets. Training sample size on the left chart is 2,500 and on the right chart is 10,000. Each vertex on the polygons is identified with the factors’ dimension sequence $( m _ { \Gamma _ { - } } m _ { \Delta _ { - } } m _ { \Upsilon } )$ of the associated group of datasets. The polygons’ radii are scaled between $0 : 0 . 8$ to quantify the PEHE values (i.e., the closer to the centre, the smaller the PEHE). The dashed purple curve illustrates the results of the proposed method.
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# 4.2.2 EVALUATING ESTIMATION OF TREATMENT EFFECTS
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Given a synthetic dataset (that include both factual as well as counterfactual outcomes), one can evaluate treatment effect estimation methods with two types of performance measures:
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• Individual-based: “Precision in Estimation of Heterogeneous Effect” $\begin{array} { r } { \mathrm { P E H E } { = } \sqrt { \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \left( \hat { \mathbf { e } } _ { i } - \mathbf { e } _ { i } \right) ^ { 2 } } } \end{array}$ where $\hat { \mathbf { e } } _ { i } = \hat { y } _ { i } ^ { 1 } - \hat { y } _ { i } ^ { 0 }$ is the predicted effect and $\mathbf { e } _ { i } = y _ { i } ^ { 1 } - y _ { i } ^ { 0 }$ is the true effect. • Population-based: “Bias of the Average Treatment Effect” $\epsilon _ { \mathrm { A T E } } = \left| { \mathrm { A T E } } - { \widehat { \mathrm { A T E } } } \right|$ where $\mathrm { A T E = }$ $\begin{array} { r } { \frac { 1 } { N } \sum _ { i = 1 } ^ { N } y _ { i } ^ { 1 } - \frac { 1 } { N } \sum _ { j = 1 } ^ { N } y _ { j } ^ { 0 } } \end{array}$ in which $y _ { i } ^ { 1 }$ and $y _ { j } ^ { 0 }$ are the true outcomes for the respective treatments and $\widehat { \mathrm { A T E } }$ is calculated based on the estimated outcomes.
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In this paper, we compare performances of the following treatment effect estimation methods: 5
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• CFR: CounterFactual Regression (Shalit et al., 2017).
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• CFR-ISW: CFR with Importance Sampling Weights (Hassanpour & Greiner, 2019).
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• SITE: Similarity preserved Individual Treatment Effect (Yao et al., 2018).
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• DR-CFR: Disentangled Representations for CFR – our proposed method.
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| 187 |
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Figure 6 visualizes the PEHE measures in radar charts for these four methods, trained with datasets of size $N = 2 { , } 5 0 0$ (left) and $N { = } 1 0 { , } 0 0 0$ (right). As expected, all methods perform better with observing more training data; however, DR-CFR took the most advantage by reducing PEHE the most (by 0.15, going down from 0.60 to 0.45), while CFR, CFR-ISW, and SITE reduced PEHE by 0.07, 0.08, and 0.08 respectively.
|
| 188 |
+
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| 189 |
+
Table 1 summarizes the PEHE and $\epsilon _ { \mathrm { A T E } }$ measures (lower is better) for all scenarios, in terms of mean and standard deviation of all the $2 4 \times 5$ datasets, in order to give a unified view on the performance.
|
| 190 |
+
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| 191 |
+
Table 2: IHDP datasets (100 with $N { = } 7 4 7$ )
|
| 192 |
+
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| 193 |
+
<table><tr><td>Methods</td><td>PEHE</td><td>EATE</td></tr><tr><td>CFR</td><td>0.81 (0.30)</td><td>0.13 (0.12)</td></tr><tr><td>CFR-ISW</td><td>0.73 (0.28)</td><td>0.11 (0.10)</td></tr><tr><td>SITE</td><td>0.73 (0.33)</td><td>0.10 (0.09)</td></tr><tr><td>DR-CFR</td><td>0.65 (0.37)</td><td>0.03 (0.04)</td></tr></table>
|
| 194 |
+
|
| 195 |
+
Table 1: Synthetic datasets $2 4 \times 5$ with $N = 1 0 , 0 0 0 )$
|
| 196 |
+
|
| 197 |
+
<table><tr><td>Methods</td><td>PEHE</td><td>EATE</td></tr><tr><td>CFR</td><td>0.61 (0.05)</td><td>0.021 (0.018)</td></tr><tr><td>CFR-ISW</td><td>0.58 (0.06)</td><td>0.017 (0.009)</td></tr><tr><td>SITE</td><td>0.63 (0.05)</td><td>0.035 (0.039)</td></tr><tr><td>DR-CFR</td><td>0.45 (0.11)</td><td>0.013 (0.006)</td></tr></table>
|
| 198 |
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| 199 |
+
PEHE and $\epsilon _ { \mathrm { A T E } }$ measures (lower is better) represented in the form of “mean (standard deviation)”.
|
| 200 |
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| 201 |
+
DR-CFR achieves the best performance among the contending methods. These results are statistically significant based on the Welch’s unpaired t-test with $\alpha { = } 0 . 0 5$ . Table 2 summarizes the PEHE and $\epsilon _ { \mathrm { A T E } }$ measures on the IHDP benchmark. The results are reported in terms of mean and standard deviation over the 100 datasets with various realizations of outcomes. Again, DR-CFR achieves the best performance (statistically significant for $\epsilon _ { \mathrm { A T E } }$ ) among the contending methods.
|
| 202 |
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| 203 |
+
# 5 FUTURE WORKS AND CONCLUSION
|
| 204 |
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| 205 |
+
The majority of methods proposed to estimate treatment effects – including this work – fall under the category of discriminative approaches. A promising direction is to consider developing generative models, in an attempt to shed light on the true underlying data generating mechanism. Perhaps this could also facilitate generating new, virtual, yet realistic data instances – similar to what is done in computer vision. Louizos et al. (2017)’s method is a notable generative approach, which uses Variational Auto-Encoder (VAE) to extract latent confounders from their observed proxies. While that work is an interesting step in that direction, it is not yet capable of addressing the problem of selection bias. We believe that our proposed perspective on the problem can be helpful to solve this open question. This is left to future work.
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| 206 |
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+
In this paper, we studied the problem of estimating treatment effect from observational studies. We argued that not all factors in the observed covariates $X$ might contribute to the procedure of selecting treatment $T$ , or more importantly, determining the outcomes $Y$ . We modeled this using three underlying sources of $X$ , $T$ , and $Y$ , and showed that explicit identification of these sources offers great insight to help us design models that better handle selection bias in observational datasets. We proposed an algorithm, Disentangled Representations for CounterFactual Regression (DR-CFR), that can (1) identify disentangled representations of the above-mentioned underlying sources and (2) leverage this knowledge to reduce as well as account for the negative impact of selection bias on estimating the treatment effects from observational data. Our empirical results showed that the proposed method achieves state-of-the-art performance in both individual and population based evaluation measures.
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# ACKNOWLEDGEMENTS
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The authors gratefully acknowledge financial support from Natural Sciences and Engineering Research Council of Canada (NSERC) and Alberta Machine Intelligence Institute (Amii). We wish to thank Dr. Pouria Ramazi and Shivam Raj for fruitful conversations, and Dr. Fredrik Johansson for publishing/maintaining the code-base for the CFR method online. We also would like to thank the ICLR 2020 anonymous reviewers, as well as Dr. Kun Kuang and Tianle Liu, for their valuable reviews, which helped improve this paper.
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| 1 |
+
[
|
| 2 |
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{
|
| 3 |
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"type": "text",
|
| 4 |
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"text": "LEARNING DISENTANGLED REPRESENTATIONS FORCOUNTERFACTUAL REGRESSION",
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| 5 |
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"text_level": 1,
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| 6 |
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"bbox": [
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},
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| 14 |
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{
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| 15 |
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"type": "text",
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| 16 |
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"text": "Negar Hassanpour & Russell Greiner ",
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| 17 |
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"bbox": [
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},
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| 25 |
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{
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| 26 |
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"type": "text",
|
| 27 |
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"text": "Department of Computing Science University of Alberta Edmonton, Alberta, T6G 2E8, CANADA {hassanpo,rgreiner}@ualberta.ca ",
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| 28 |
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"bbox": [
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"type": "text",
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"text": "ABSTRACT ",
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"text": "We consider the challenge of estimating treatment effects from observational data; and point out that, in general, only some factors based on the observed covariates $X$ contribute to selection of the treatment $T$ , and only some to determining the outcomes $Y$ . We model this by considering three underlying sources of $\\{ X , T , Y \\}$ and show that explicitly modeling these sources offers great insight to guide designing models that better handle selection bias in observational datasets. This paper is an attempt to conceptualize this line of thought and provide a path to explore it further. ",
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"text": "In this work, we propose an algorithm to (1) identify disentangled representations of the above-mentioned underlying factors from any given observational dataset $\\mathcal { D }$ and (2) leverage this knowledge to reduce, as well as account for, the negative impact of selection bias on estimating the treatment effects from $\\mathcal { D }$ . Our empirical results show that the proposed method achieves state-of-the-art performance in both individual and population based evaluation measures. ",
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"text": "1 INTRODUCTION ",
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"text": "As we rely more and more on artificial intelligence (AI) to automate the decision making processes, accurately estimating the causal effects of taking different actions gains an essential role. A prominent example is precision medicine $- i . e .$ , the customization of health-care tailored to each individual patient – which attempts to identify which medical procedure $t \\in \\tau$ will benefit a certain patient $x$ the most, in terms of the treatment outcome $y \\in \\mathbb { R }$ . Learning such models requires answering counterfactual questions (Rubin, 1974; Pearl, 2009) such as: “Would this patient have lived longer [and by how much], had she received an alternative treatment?”. ",
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"text": "For notation: a dataset $\\mathcal { D } = \\{ [ x _ { i } , t _ { i } , y _ { i } ] \\} _ { i = 1 } ^ { N }$ used for treatment effect estimation has the following format: for the $i ^ { t h }$ instance (e.g., patient), we have some context information $x _ { i } \\in \\mathcal { X } \\subseteq \\mathbb { R } ^ { K }$ (e.g., age, BMI, blood work, etc.), the administered treatment $t _ { i }$ chosen from a set of treatment options $\\tau$ (e.g., {0: medication, 1: surgery}), and the respective observed outcome $y _ { i } \\in \\mathcal { V }$ (e.g., survival time; $\\mathcal { V } \\subseteq \\mathbb { R } ^ { + }$ ) as a result of receiving treatment $t _ { i }$ . Note that $\\mathcal { D }$ only contains the outcome of the administered treatment (aka observed outcome: $y _ { i }$ ), but not the outcome(s) of the alternative treatment(s) (aka counterfactual outcome(s): $y _ { i } ^ { t }$ for $t \\in \\mathcal { T } \\backslash \\{ t _ { i } \\} )$ , which are inherently unobservable. For the binary-treatment case, we denote the alternative treatment as $\\neg t _ { i } = 1 - t _ { i }$ . ",
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"text": "Pearl (2009) demonstrates that, in general, causal relationships can only be learned by experimentation (on-line exploration), or running a Randomized Controlled Trial (RCT), where the treatment assignment does not depend on the individual $X - { \\mathsf { s e e } }$ Figure 1(a). In many cases, however, this is expensive, unethical, or even infeasible. Here, we are forced to approximate treatment effects from off-line datasets collected through Observational Studies. In such datasets, the administered treatment $T$ depends on some or all attributes of individual $X - { \\mathsf { s e e } }$ Figure 1(b). Here, as $\\operatorname* { P r } ( T | X ) \\neq \\operatorname* { P r } ( T )$ , we say these datasets exhibit selection bias (Imbens & Rubin, 2015). Figure 2 illustrates selection bias in an example (synthetic) observational dataset. ",
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"Figure 1: Belief net structure for randomized controlled trials and observational studies. Here, ${ Y ^ { \\overline { { 0 } } } } ( { Y ^ { 1 } } )$ is the outcome of applying $T =$ treatment#0 (#1) to the individual represented by $X$ . "
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"text": "Here, we want to accurately estimate the Individual Treatment Effect (ITE) for each instance $i - i . e .$ , to estimate $\\mathbf { e } _ { i } = y _ { i } ^ { 1 } - y _ { i } ^ { 0 }$ . We frame the solution as learning the function $f : \\mathcal { X } \\times \\mathcal { T } \\mathcal { Y }$ that can accurately predict the outcomes (both observed $\\hat { y _ { i } } ^ { t _ { i } }$ as well as counterfactuals $\\hat { y _ { i } } ^ { \\lnot t _ { i } }$ ) given the context information $x _ { i }$ for each individual. As mentioned earlier, there are two challenges associated with estimating treatment effects: ",
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"text": "(i) The fact that counterfactual outcomes are unobservable (i.e., not present in any training data) makes estimating treatment effects more difficult than the generalization problem in the supervised learning paradigm. This is an inherent characteristic of this task. \n(ii) Selection bias in observational datasets implies having fewer instances within each treatment arm at specific regions of the domain. This sparsity, in turn, would decrease the accuracy and confidence of predicting counterfactuals at those regions. ",
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"text": "This paper addresses the second challenge by investigating the root causes of selection bias, by dissecting and identifying the underlying factors that can generate an observational dataset $\\mathcal { D }$ , and leveraging this knowledge to reduce, as well as account for, the negative impact of selection bias on estimating the treatment effects from $\\mathcal { D }$ . In this work, we borrow ideas from the representation learning literature (Bengio et al., 2013) in order to reduce selection bias and from the domain adaptation literature (Shimodaira, 2000) in order to account for the remainder selection bias that (might) still exist after its reduction. ",
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"text": "Our analysis relies on the following assumptions: Assumption 1: Unconfoundedness (Rosenbaum & Rubin, 1983) – There are no unobserved confounders (i.e., covariates that contribute to both treatment selection procedure as well as determination of outcomes). Formally, $\\{ Y ^ { t } \\} _ { t \\in { \\mathcal { T } } } \\bot T \\mid X$ . ",
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"text": "Assumption 2: Overlap (Imbens, 2004) – Every individual $x$ should have a non-zero chance of being assigned to any treatment arm. That is, ",
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"Figure 2: An example observational dataset. Here, to treat heart disease, a doctor typically prescribes surgery $( t = 1 )$ to younger patients (•) and medication ${ \\bf \\boldsymbol { t } } = 0$ ) to older ones ${ \\bf \\Xi } ( { \\bf \\Lambda } )$ . Note that instances with larger (resp., smaller) $x$ values have a higher chance to be assigned to the $t = 0$ (resp., 1) treatment arm; hence we have selection bias. The counterfactual outcomes (only used for evaluation purpose) are illustrated by small $\\bullet ( \\mathbf { \\alpha } )$ for $\\neg t = 1$ (0). "
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"text": "These two assumptions together are called strong ignorability (Rosenbaum & Rubin, 1983). Imbens & Wooldridge (2009) showed that strong ignorability is sufficient for ITE to be identifiable. ",
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"text": "Without loss of generality, we assume that the random variable $X$ follows a(n unknown) joint probability distribution $\\operatorname* { P r } ( X | \\Gamma , \\Delta , \\Upsilon )$ , treatment $T$ follows $\\mathrm { P r } ( T | \\Gamma , \\Delta )$ , and outcome $\\boldsymbol { Y } ^ { T }$ follows $\\mathrm { P r } _ { { \\cal T } } ( { \\cal Y } ^ { T } | \\Delta , \\Upsilon )$ , where $\\Gamma , \\Delta$ , and $\\Upsilon$ represent the three underlying factors1 that generate an observational dataset $\\mathcal { D }$ . The respective graphical model is illustrated in Figure 3. Conforming with the statements above, note that the graphical model also suggests that selection bias is induced by factors $\\Gamma$ and $\\Delta$ , where $\\Delta$ represents the confounding factors between $T$ and $Y$ . ",
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"text": "",
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"text": "Main contribution: We argue that explicit identification of the underlying factors $\\{ \\Gamma , \\Delta , \\Upsilon \\}$ in observational datasets offers great insight to guide designing models that better handle selection bias and consequently achieve better performance in terms of estimating ITEs. In this paper, we propose a model, named Disentangled Representations for CounterFactual Regression (DR-CFR), that is optimized to do exactly that. We also present experiments that demonstrate the advantages of this perspective; and show empirically that the proposed method outperforms state-of-the-art models in a variety of data generation scenarios with different dimensionality of factors; see below. ",
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"Figure 3: Underlying factors of $X$ ; $\\Gamma$ $( \\Upsilon )$ are factors that partially determine only $T$ $( Y )$ but not the other random variable; and $\\Delta$ are confounders; Selection bias is induced by factors $\\Gamma$ and $\\Delta$ . "
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"text": "2 RELATED WORKS ",
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"text": "Selection bias in observational datasets is equivalent to a domain adaptation scenario where a model is trained on a “source” (observed) data distribution, but should perform well on a “target” (counterfactual) one. Learning treatment effects from observational datasets is closely related to “off-policy learning from logged bandit feedback” – cf., (Swaminathan & Joachims, 2015a), whose goal is learning an optimal policy that selects the best personalized treatment for each individual. A common statistical solution is re-weighting certain data instances to balance the source and target distributions. The majority of re-weighting approaches belong to the Inverse Propensity Weighting (IPW) family of methods – cf., (Austin, 2011; Bottou et al., 2013; Swaminathan & Joachims, 2015c). While IPW methods are unbiased, they suffer from high variance. Swaminathan & Joachims (2015b) proposed the Counterfactual Risk Minimization (CRM) principle to alleviate this issue. In summary, re-weighting is an attempt to account for the selection bias. ",
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"text": "Johansson et al. (2016) is among the pioneer works that explored ways to use techniques from representation learning (Bengio et al., 2013) to reduce the selection bias. Shalit et al. (2017) present a refined version of (Johansson et al., 2016)’s method that learns a common representation space $\\Phi ( x ) = \\phi$ by minimizing the discrepancy (Mansour et al., 2009) (hereinafter “disc”) between the conditional distributions of $\\phi$ given $t = 0$ versus $\\phi$ , given $t = 1$ . That is, ",
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"text": "$$\n\\mathsf { d i s c } \\Big ( \\big \\{ \\Phi ( x _ { i } ) \\big \\} _ { i : t _ { i } = 0 } , \\big \\{ \\Phi ( x _ { i } ) \\big \\} _ { i : t _ { i } = 1 } \\Big )\n$$",
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"text": "which is (effectively) a regularization term that attempts to reduce selection bias in the learned representation. On top of this representation learning network, they trained two regression networks $h ^ { t } ( \\phi )$ – one for each treatment arm $( t \\in \\{ 0 , 1 \\} )$ ) – that predict the outcomes. ",
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"text": "Hassanpour & Greiner (2019) argued that the learned representation cannot and should not remove all the selection bias, as the confounders not only contribute to choosing a treatment but also to determining the respective outcomes.2 As a result, where there are confounders (which is a common situation), even $\\phi$ would exhibit some selection bias, although less than that in the original domain $x$ They built on the work of (Shalit et al., 2017) by introducing context-aware importance sampling weights, that attempt to account for the above-mentioned remainder selection bias. These weights ",
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"text": "$$\n\\omega _ { i } = 1 + { \\frac { \\operatorname* { P r } ( \\phi _ { i } \\mid \\neg t _ { i } ) } { \\operatorname* { P r } ( \\phi _ { i } \\mid t _ { i } ) } } = 1 + { \\frac { \\operatorname* { P r } ( t _ { i } ) } { 1 - \\operatorname* { P r } ( t _ { i } ) } } \\cdot { \\frac { 1 - \\pi { \\bigl ( } t _ { i } \\mid \\phi _ { i } { \\bigr ) } } { \\pi { \\bigl ( } t _ { i } \\mid \\phi _ { i } { \\bigr ) } } }\n$$",
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{
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| 353 |
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"type": "text",
|
| 354 |
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"text": "are designed to enhance performance of estimating both factual as well as counterfacual outcomes (by the 1 and $\\frac { \\operatorname* { P r } ( \\phi \\mid \\lnot t ) } { \\operatorname* { P r } ( \\phi \\mid t ) }$ terms, respectively), where $\\pi ( t _ { i } | \\phi _ { i } )$ is the probability of assigning the observed $t _ { i }$ conditioned on the learned context $\\phi _ { i }$ . ",
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"text": "Note that both (Shalit et al., 2017) and (Hassanpour & Greiner, 2019) use $\\Phi$ to model the concatenation of factors $\\Delta$ and $\\Upsilon$ (see Figure 3). Although it does make sense that there should be no discrepancy between conditional distributions of $\\Upsilon$ , the $\\Delta$ factor should model the confounding factors, which by definition, must embed some information about treatment assignment. This would result in a positive discrepancy between conditional distributions of $\\Delta$ that should not be minimized. Thus, minimizing Equation (1) with respect to $\\Phi$ can lead to problematic results as it discards some of the confounders. ",
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"text": "Yao et al. (2018) proposed the Similarity preserved Individual Treatment Effect (SITE) method, which extends Shalit et al. (2017)’s framework by adding a local similarity preserving component. This component acts as a regularization term, that attempts to retain the same neighbourhood relationships in the learned representation space as exhibited in the original space, by matching the propensity scores $\\operatorname* { P r } ( t = 1 | \\bar { \\boldsymbol { x } } )$ and $\\textstyle \\operatorname* { P r } ( t = { \\bar { 1 } } | \\phi )$ . This, however, results in learning sub-optimal representations when $\\Gamma \\neq \\emptyset$ as SITE tries to keep instances whose $\\Gamma \\mathrm { s }$ are far apart, also far apart in $\\phi$ . In other words, this component penalizes reducing selection bias in $\\phi$ by not discarding the irrelevant information present in $\\Gamma$ even when it does not hurt the outcome estimation at all. ",
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"type": "text",
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"text": "Our work has many similarities to (Kuang et al., 2017), who decomposed $X$ into two subsets: confounding and adjustment variables, which are similar to our $\\Delta$ and $\\Upsilon$ factors respectively. They then used an optimization algorithm for identifying these variables, to ultimately find an unbiased estimate of the Average Treatment Effect (ATE). We extend their work in three ways: (i) In addition to confounders and adjustment variables, we also identify the factors that determine the treatment and have no effect on the outcome (i.e., Γ). (ii) Unlike (Kuang et al., 2017) that take a linear approach by tagging the raw features as either confounders or adjustment variables, our proposed method has the capacity to learn [non-linear] representations of the underlying factors. (iii) Our method facilitates estimating both ATE as well ITE, whereas (Kuang et al., 2017) cannot provide estimates of ITEs. ",
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"type": "text",
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"text": "3 LEARNING DISENTANGLED REPRESENTATIONS",
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"type": "text",
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"text": "We assume, without loss of generality, that any dataset of the form $\\{ X , T , Y \\}$ is generated from three underlying factors $\\{ \\Gamma , \\bar { \\Delta } , \\Upsilon \\}$ , as illustrated in Figure 3. 3 Observe that the factor $\\Gamma$ (resp., $\\Upsilon$ ) partially determines only $T$ (resp., $Y$ ), but not the other variables; and $\\Delta$ includes the confounding factors between $T$ and $Y$ . This graphical model suggests that selection bias is induced by factors $\\Gamma$ and $\\Delta$ . It also shows that the outcome depends on the factors $\\Delta$ and $\\Upsilon$ . Inspired by this graphical model, our model architecture incorporates the following components: ",
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"text": "• Three representation learning networks; one for each underlying factor: $\\Gamma ( x ) , \\Delta ( x )$ , and $\\Upsilon ( x )$ . • Two regression networks; one for each treatment arm: $h ^ { 0 } ( \\Delta ( x ) , \\Upsilon ( x ) )$ and $h ^ { 1 } ( \\Delta ( x ) , \\Upsilon ( x ) )$ . • Two logistic networks: $\\pi _ { 0 } ( t | \\Gamma ( x ) , \\Delta ( x ) )$ to model the logging policy – aka behaviour policy in Reinforcement Learning; $c f .$ , (Sutton & Barto, 1998) – and $\\pi ( t | \\Delta ( x ) )$ to design weights that account for the confounders’ impact. ",
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"text": "We therefore try to minimize the following objective function: ",
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"img_path": "images/b583978c0447127039b0390a4b4c19e30d6f2b374ae8a233e5d8116d7d5414c5.jpg",
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"text": "$$\n\\begin{array} { l } { \\displaystyle J ( \\Gamma , \\Delta , \\Upsilon , h ^ { 0 } , h ^ { 1 } , \\pi _ { 0 } ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\omega \\big ( t _ { i } , \\Delta ( x _ { i } ) \\big ) \\cdot \\mathcal { L } \\big [ y _ { i } , h ^ { t _ { i } } \\big ( \\Delta ( x _ { i } ) , \\Upsilon ( x _ { i } ) \\big ) \\big ] } \\\\ { \\displaystyle \\quad \\quad + \\alpha \\cdot { \\mathsf { d i s c } } \\big ( \\{ \\Upsilon ( x _ { i } ) \\} _ { i : t _ { i } = 0 } , \\{ \\Upsilon ( x _ { i } ) \\} _ { i : t _ { i } = 1 } \\big ) } \\\\ { \\displaystyle \\quad \\quad + \\beta \\cdot \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } - \\log \\big [ \\pi _ { 0 } \\big ( t _ { i } | \\Gamma ( x _ { i } ) , \\Delta ( x _ { i } ) \\big ) \\big ] } \\\\ { \\displaystyle \\quad \\quad + \\lambda \\cdot \\mathfrak { R e g } ( \\Gamma , \\Delta , \\Upsilon , h ^ { 0 } , h ^ { 1 } , \\pi _ { 0 } ) } \\end{array}\n$$",
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"text": "where $\\omega \\left( t _ { i } , \\Delta ( x _ { i } ) \\right)$ is the re-weighting function; $\\mathcal { L } \\big [ y _ { i } , h ^ { t _ { i } } \\big ( \\Delta ( x _ { i } ) , \\Upsilon ( x _ { i } ) \\big ) \\big ]$ is the prediction loss for observed outcomes (aka factual loss); $\\mathsf { d i s c } \\big ( \\{ \\Upsilon ( x ) \\} _ { i : t _ { i } = 0 } , \\{ \\Upsilon ( x ) \\} _ { i : t _ { i } = 1 } \\big )$ calculates the discrepancy between conditional distributions of $\\Upsilon$ given $t = 0$ versus given $t = 1 ; - \\log \\pi _ { 0 } ( \\cdot )$ is the cross entropy loss of predicting the assigned treatments given the learned context; and $\\mathfrak { R e g } ( \\cdot )$ is the regularization term for penalizing model complexity. The following sections elaborate on each of these terms. ",
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"text": "3.1 FACTUAL LOSS: $\\mathcal { L } \\big [ \\boldsymbol { y } , h ^ { t } \\big ( \\Delta ( \\boldsymbol { x } ) , \\Upsilon ( \\boldsymbol { x } ) \\big ) \\big ]$ ",
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"type": "text",
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"text": "Similar to (Johansson et al., 2016; Shalit et al., 2017; Hassanpour & Greiner, 2019; Yao et al., 2018), we train two regression networks $h ^ { 0 }$ and $h ^ { 1 }$ , one for each treatment arm. As guided by the graphical model in Figure 3, the inputs to these regression networks are the outputs of the $\\Delta ( x )$ and $\\Upsilon ( x )$ representation networks and their outputs are the predicted outcomes for their respective treatments. ",
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"text": "Note that the prediction loss $\\mathcal { L }$ can only be calculated on the observed outcomes (hence the name factual loss), as counterfactual outcomes are not available in any training set. This would be an L2-loss for real-valued outcomes and a log-loss for binary outcomes. By minimizing the factual loss, we ensure that the union of the learned representations $\\dot { \\Delta } ( x )$ and $\\Upsilon ( x )$ retain enough information needed for accurate estimation of the observed outcomes. ",
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"text": "3.2 RE-WEIGHTING FUNCTION: $\\omega ( t , \\Delta ( x ) )$ ",
|
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"text": "We follow (Hassanpour & Greiner, 2019)’s design for weights as re-stated in Equation (2), with the modification that we employ $\\Delta$ to calculate the weights instead of $\\Phi$ . Although following the same design, we anticipate our weights should perform better in practice than those in (Hassanpour & Greiner, 2019) as: (i) no confounders are discarded due to minimizing the imbalance loss (because our disc is defined based on $\\Upsilon$ , not $\\Phi$ ); and (ii) only the legitimate confounders are used to derive the weights (i.e., $\\Delta )$ , not the ones that have not contributed to treatment selection (i.e., Υ). ",
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"text": "Notably, the weights design in Equation (2) is different from the common practice in re-weighting techniques (e.g., IPW) in that the weights are calculated based on all factors that determine $T$ (i.e., $\\Gamma$ as well as $\\Delta$ ). However, we argue that incorporation of $\\Gamma$ in the weights might result in emphasizing the wrong instances. In other words, since the factual loss $\\mathcal { L }$ is only sensitive to factors $\\Delta$ and $\\Upsilon$ , and not $\\Gamma$ , re-weighting $\\mathcal { L }$ according to $\\Gamma$ would yield a wrong objective function to be optimized. ",
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"type": "equation",
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"text": "$$\n\\mathsf { d i s c } \\big ( \\{ \\Upsilon ( x _ { i } ) \\} _ { i : t _ { i } = 0 } , \\{ \\Upsilon ( x _ { i } ) \\} _ { i : t _ { i } = 1 } \\big )\n$$",
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"type": "text",
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"text": "According to Figure 3, $\\Upsilon$ should be independent of $T$ due to the collider structure at $Y$ . Therefore, ",
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"text": "$$\n\\Upsilon \\ \\perp \\ T \\quad \\implies \\quad \\operatorname* { P r } ( \\Upsilon \\mid T ) = \\operatorname* { P r } ( \\Upsilon ) \\quad \\implies \\quad \\operatorname* { P r } ( \\Upsilon \\mid T = 0 ) = \\operatorname* { P r } ( \\Upsilon \\mid T = 1 )\n$$",
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"text": "We used Maximum Mean Discrepancy (MMD) (Gretton et al., 2012) to calculate dissimilarity between the two conditional distributions of $\\Upsilon$ given $t = 0$ versus $t = 1$ . ",
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"text": "By minimizing the imbalance loss, we ensure that the learned factor $\\Upsilon$ embeds no information about $T$ and all the confounding factors are retained in $\\Delta$ . Capturing all the confounders in $\\Delta$ and only in $\\Delta$ is the hallmark of the proposed method, as we will use it for optimal re-weighting of the factual loss term (next section). Note that this differs from Shalit et al. (2017)’s approach in that they do not distinguish between the independent factors $\\Delta$ and $\\Upsilon$ ; and minimizing the loss defined on only one factor $\\Phi$ which might erroneously suggest discarding some of the confounders in $\\Delta$ . ",
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"text": "3.4 CROSS ENTROPY LOSS: $- \\log \\left[ \\pi _ { 0 } \\big ( t | \\Gamma ( x ) , \\Delta ( x ) \\big ) \\right]$ ",
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"text": "We model the logging policy as a logistic regression network parameterized by $\\big [ W _ { 0 } , b _ { 0 } \\big ]$ as follows: $\\pi _ { 0 } \\bigl ( t | \\psi \\bigr ) = \\Bigl [ 1 + e ^ { - \\bigl ( 2 t - 1 \\bigr ) ( \\psi \\cdot W _ { 0 } + b _ { 0 } ) } \\Bigr ] ^ { - 1 }$ , where $\\psi$ is the concatenation of matrices $\\Gamma$ and $\\Delta$ Minimizing the cross entropy loss enforces learning $\\Gamma$ and $\\Delta$ in a way that allows $\\pi _ { 0 } ( \\cdot )$ to predict the assigned treatments. In other words, the union of the learned representations of $\\Gamma$ and $\\Delta$ retain enough information to recover the logging policy that guided the treatment assignments. ",
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"type": "text",
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"text": "4 EXPERIMENTS ",
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| 618 |
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"text": "4.1 BENCHMARKS ",
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"text": "Evaluating treatment effect estimation methods is problematic on real-world datasets since, as mentioned earlier, their counterfactual outcomes are inherently unobservable. A common solution is to synthesize datasets where the outcomes of all possible treatments are available, then discard some outcomes to create a proper observational dataset with characteristics (such as selection bias) similar to a real-world one – cf., (Beygelzimer & Langford, 2009; Hassanpour & Greiner, 2018). In this work, we use two such benchmarks: our synthetic series of datasets as well as a publicly available benchmark: the Infant Health and Development Program (IHDP) (Hill, 2011). ",
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"text": "4.1.1 SYNTHETIC DATASETS ",
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| 663 |
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"type": "text",
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| 664 |
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"text": "We generated our synthetic datasets according to the following process, which takes as input the sample size $N$ ; dimensionalities $[ m _ { \\Gamma } , m _ { \\Delta } , m _ { \\Upsilon } ] \\in \\mathcal { Z } ^ { + ( 3 ) }$ ; for each factor $L \\in \\{ \\Gamma , \\Delta , \\Upsilon \\}$ , the means and covariance matrices $\\left( \\mu _ { L } , \\Sigma _ { L } \\right)$ ; and a scalar $\\zeta$ that determines the slope of the logistic curve. ",
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"bbox": [
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"type": "text",
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"text": "• For each latent factor $L \\in \\{ \\Gamma , \\Delta , \\Upsilon \\}$ ",
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"type": "text",
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"text": "– Form $L$ by drawing $N$ instances (each of size $m _ { L }$ ) from $\\mathcal { N } ( \\mu _ { L } , \\Sigma _ { L } )$ , \n– Concatenate $\\Gamma , \\Delta$ , and $\\Upsilon$ to make the covariates matrix $X$ [of size $N \\times ( m _ { \\Gamma } + m _ { \\Delta } + m _ { \\Upsilon } ) ]$ \n– Concatenate $\\Gamma$ and $\\Delta$ to make $\\Psi$ [of size $N \\times \\left( m _ { \\Gamma } + m _ { \\Delta } \\right) ]$ \n– Concatenate $\\Delta$ and $\\Upsilon$ to make $\\Phi$ [of size $N \\times ( m _ { \\Delta } + m _ { \\Upsilon } ) ]$ ",
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"bbox": [
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"type": "text",
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"text": "• For treatment $T$ ",
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"bbox": [
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"type": "text",
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"text": "– Sample $m _ { \\Gamma } + m _ { \\Delta }$ tuple of coefficients $\\theta$ from $\\mathcal { N } ( 0 , 1 ) ^ { m _ { \\Gamma } + m _ { \\Delta } }$ \n– Define the logging policy as $\\begin{array} { r } { \\pi _ { 0 } ( t = 1 | z ) = \\frac { 1 } { 1 + \\exp ( - \\zeta z ) } } \\end{array}$ , where $z = \\Psi \\cdot \\theta$ \n– For each instance $x _ { i }$ , sample treatment $t _ { i }$ from the Bernoulli distribution with parameter $\\pi _ { 0 } ( t = 1 | z _ { i } )$ ",
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"type": "text",
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"text": "• For outcomes $Y ^ { 0 }$ and $Y ^ { 1 }$ : ",
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"text": "– Sample $m _ { \\Delta } + m _ { \\Upsilon }$ tuple of coefficients $\\vartheta ^ { 0 }$ and $\\vartheta ^ { 1 }$ from $\\mathcal { N } ( 0 , 1 ) ^ { m _ { \\Delta } + m _ { \\Upsilon } }$ – Define $y ^ { 0 } = ( \\Phi \\circ \\Phi \\circ \\Phi + 0 . 5 ) \\cdot \\vartheta ^ { 0 } / ( m _ { \\Delta } + m _ { \\Upsilon } ) + \\varepsilon$ and $y ^ { 1 } = ( \\Phi \\circ \\Phi ) \\cdot \\vartheta ^ { 1 } / ( m _ { \\Delta } + m \\Upsilon ) + \\varepsilon ,$ , where $\\varepsilon$ is a white noise sampled from $\\mathcal { N } ( 0 , 0 . 1 )$ and $\\circ$ is the symbol for element-wise (Hadamard/Schur) product. ",
|
| 731 |
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"bbox": [
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|
| 739 |
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|
| 740 |
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"type": "text",
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"text": "We considered all the viable datasets in a mesh generated by $m _ { \\Gamma } , m _ { \\Delta } , m _ { \\Upsilon } \\in \\{ 0 , 4 , 8 \\}$ . This creates 24 scenarios4 that consider all possible situations in terms of the relative sizes of the factors $\\Gamma , \\Delta$ , and $\\Upsilon$ . For each scenario, we synthesized five datasets with various initial random seeds. ",
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"type": "text",
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"text": "4.1.2 INFANT HEALTH AND DEVELOPMENT PROGRAM (IHDP) ",
|
| 753 |
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"text_level": 1,
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"text": "The original RCT data was designed to evaluate the effect of specialist home visits on future cognitive test scores of premature infants. Hill (2011) induced selection bias by removing a non-random subset of the treated population to create a realistic observational dataset. The resulting dataset contains 747 instances (608 control, 139 treated) with 25 covariates. We run our experiments on the same benchmark (100 realizations of outcomes) provided by and used in (Johansson et al., 2016; Shalit ",
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"img_path": "images/be8c139c8de9971e467b02283551bfb07a320d6664c0fc239d7f7db750501290.jpg",
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"image_caption": [],
|
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"text": "(a) Slice of the weights matrix that connects {the variables in $X$ belonging to $\\Gamma$ } to {the first layer of the representation network that attempts to identify $\\Gamma \\}$ . The size of this slice is $m _ { \\Gamma } \\times K$ . ",
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"type": "image",
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"img_path": "images/ccdfad925bc7e1fbf3b2aa5162f7860b2859be53c1aea4078e98c72a31c0b480.jpg",
|
| 800 |
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"image_caption": [
|
| 801 |
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"(b) Slice of the weights matrix that connects {the variables in $X$ not belonging to $\\Gamma \\}$ to {the first layer of the representation network that attempts to identify $\\Gamma$ }. The size of this slice is $( m _ { \\Delta } + m _ { \\Upsilon } ) \\times K$ . "
|
| 802 |
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],
|
| 803 |
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"image_footnote": [],
|
| 804 |
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"bbox": [
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{
|
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"type": "image",
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"img_path": "images/bfd1a1a067a00361b4e48b570e7735c213ca23a53bb3a01a6c4410b87ac95c0f.jpg",
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| 815 |
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"image_caption": [
|
| 816 |
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"Figure 4: Visualization of slicing the learned weights matrix in the first layer of the representation network (number of neurons: $K$ ) for identifying $\\Gamma$ (best viewed in color). ",
|
| 817 |
+
"Figure 5: Radar charts that visualize the capability of DR-CFR in identifying the underlying factors $\\Gamma , \\Delta$ , and $\\Upsilon$ . Each vertex on the polygons is identified with the factors’ dimension sequence $( m _ { \\Gamma } \\underline { { { m } } } _ { \\Delta \\underline { { { - } } } } m \\underline { { { \\Upsilon } } } )$ of the associated synthetic dataset. The polygons’ radii are scaled between $0 { : } 0 . 0 9$ and quantify the average weights of the first slice (in dotted magenta) and the second slice (in cyan). "
|
| 818 |
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],
|
| 819 |
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"image_footnote": [],
|
| 820 |
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"bbox": [
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| 829 |
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"type": "text",
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| 830 |
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"text": "et al., 2017). Outcomes of this semi-synthetic benchmark were simulated according to response surfaces provided in the Non-Parametric Causal Inference (NPCI) package (Dorie, 2016). ",
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| 831 |
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"bbox": [
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"type": "text",
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| 841 |
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"text": "4.2 RESULTS AND DISCUSSIONS ",
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| 842 |
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"text_level": 1,
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|
| 852 |
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"type": "text",
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| 853 |
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"text": "4.2.1 EVALUATING IDENTIFICATION OF FACTORS $\\{ \\Gamma , \\Delta , \\Upsilon \\}$ ",
|
| 854 |
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"text": "First, we want to determine if the proposed method is able to identify the variables that belong to each underlying factor. To do so, we look at the weight matrix in the first layer of each representation network, which is of size $( m _ { \\Gamma } + m _ { \\Delta } + m _ { \\Upsilon } ) \\times K$ , where $K$ is the number of neurons in the first hidden layer of the respective representation network. For example, to check if $\\Gamma$ is identified properly, we partition the weights matrix into two slices, as shown in Figure 4, and calculate the average of each slice. The first slice [referred to as $\\mathbf { S } _ { \\Gamma }$ ; highlighted in Figure 4(a)] pertains to “ Γ’s ground truth variables in $X ^ { \\dag }$ and the second slice $[ S _ { \\neg \\Gamma }$ ; Figure 4(b)] pertains to “variables in $X$ that do not belong to $\\Gamma ^ { \\ast }$ . Constructing $\\mathbf { S } _ { \\Delta }$ , $\\mathbf { S } _ { \\lnot \\Delta }$ , $\\mathtt { S } _ { \\mathtt { Y } }$ , and $\\mathbf { S } _ { \\neg \\Upsilon }$ follow a similar procedure. ",
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| 866 |
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"bbox": [
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|
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"type": "text",
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"text": "If the proposed method achieves a good identification, then we expect the average of the absolute values of weights in $\\mathrm { \\bf S _ { \\mathrm { { T } } } }$ should be higher than that of $\\mathbf { S } _ { \\lnot \\Gamma }$ ; this same claim should hold for $( \\mathsf { S } _ { \\Delta } , \\mathsf { S } _ { \\neg \\Delta } )$ and $( \\mathsf { S r } , \\mathsf { S } _ { \\neg \\Upsilon } )$ as well. Note that only the relative relationships between the average weights in either of the slices matter; since this analysis is aimed at checking whether, for example, for identifying $\\Gamma$ , its respective representation network has indeed learned to emphasize on “Γ’s ground truth variables in $X$ ” more than the other variables in $X$ . Figure 5 illustrates the identification performance of DR-CFR according to this analysis; showing empirically that the proposed method successfully identifies all the three underlying factors, for all synthetic datasets. ",
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| 877 |
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"bbox": [
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},
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| 885 |
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{
|
| 886 |
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"type": "image",
|
| 887 |
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"img_path": "images/40463570b2bcee413d0b671e3a2c4447d89f03f0c4b6d664b2b31356dcb79f68.jpg",
|
| 888 |
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"image_caption": [
|
| 889 |
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"Figure 6: Radar charts for visualizing the PEHE performance results on the synthetic datasets. Training sample size on the left chart is 2,500 and on the right chart is 10,000. Each vertex on the polygons is identified with the factors’ dimension sequence $( m _ { \\Gamma _ { - } } m _ { \\Delta _ { - } } m _ { \\Upsilon } )$ of the associated group of datasets. The polygons’ radii are scaled between $0 : 0 . 8$ to quantify the PEHE values (i.e., the closer to the centre, the smaller the PEHE). The dashed purple curve illustrates the results of the proposed method. "
|
| 890 |
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|
| 891 |
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"image_footnote": [],
|
| 892 |
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"bbox": [
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| 899 |
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|
| 900 |
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|
| 901 |
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"type": "text",
|
| 902 |
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"text": "",
|
| 903 |
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| 910 |
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|
| 911 |
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{
|
| 912 |
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"type": "text",
|
| 913 |
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"text": "4.2.2 EVALUATING ESTIMATION OF TREATMENT EFFECTS ",
|
| 914 |
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| 915 |
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| 922 |
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},
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| 923 |
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|
| 924 |
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"type": "text",
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| 925 |
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"text": "Given a synthetic dataset (that include both factual as well as counterfactual outcomes), one can evaluate treatment effect estimation methods with two types of performance measures: ",
|
| 926 |
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"bbox": [
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| 935 |
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|
| 936 |
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"text": "• Individual-based: “Precision in Estimation of Heterogeneous Effect” $\\begin{array} { r } { \\mathrm { P E H E } { = } \\sqrt { \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\left( \\hat { \\mathbf { e } } _ { i } - \\mathbf { e } _ { i } \\right) ^ { 2 } } } \\end{array}$ where $\\hat { \\mathbf { e } } _ { i } = \\hat { y } _ { i } ^ { 1 } - \\hat { y } _ { i } ^ { 0 }$ is the predicted effect and $\\mathbf { e } _ { i } = y _ { i } ^ { 1 } - y _ { i } ^ { 0 }$ is the true effect. • Population-based: “Bias of the Average Treatment Effect” $\\epsilon _ { \\mathrm { A T E } } = \\left| { \\mathrm { A T E } } - { \\widehat { \\mathrm { A T E } } } \\right|$ where $\\mathrm { A T E = }$ $\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } y _ { i } ^ { 1 } - \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } y _ { j } ^ { 0 } } \\end{array}$ in which $y _ { i } ^ { 1 }$ and $y _ { j } ^ { 0 }$ are the true outcomes for the respective treatments and $\\widehat { \\mathrm { A T E } }$ is calculated based on the estimated outcomes. ",
|
| 937 |
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},
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| 945 |
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{
|
| 946 |
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"type": "text",
|
| 947 |
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"text": "In this paper, we compare performances of the following treatment effect estimation methods: 5 ",
|
| 948 |
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"bbox": [
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},
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{
|
| 957 |
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"type": "text",
|
| 958 |
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"text": "• CFR: CounterFactual Regression (Shalit et al., 2017). \n• CFR-ISW: CFR with Importance Sampling Weights (Hassanpour & Greiner, 2019). \n• SITE: Similarity preserved Individual Treatment Effect (Yao et al., 2018). \n• DR-CFR: Disentangled Representations for CFR – our proposed method. ",
|
| 959 |
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| 968 |
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"type": "text",
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| 969 |
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"text": "Figure 6 visualizes the PEHE measures in radar charts for these four methods, trained with datasets of size $N = 2 { , } 5 0 0$ (left) and $N { = } 1 0 { , } 0 0 0$ (right). As expected, all methods perform better with observing more training data; however, DR-CFR took the most advantage by reducing PEHE the most (by 0.15, going down from 0.60 to 0.45), while CFR, CFR-ISW, and SITE reduced PEHE by 0.07, 0.08, and 0.08 respectively. ",
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| 970 |
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},
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| 978 |
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{
|
| 979 |
+
"type": "text",
|
| 980 |
+
"text": "Table 1 summarizes the PEHE and $\\epsilon _ { \\mathrm { A T E } }$ measures (lower is better) for all scenarios, in terms of mean and standard deviation of all the $2 4 \\times 5$ datasets, in order to give a unified view on the performance. ",
|
| 981 |
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"bbox": [
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| 982 |
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176,
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| 983 |
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847,
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| 984 |
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823,
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| 985 |
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876
|
| 986 |
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],
|
| 987 |
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"page_idx": 7
|
| 988 |
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},
|
| 989 |
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{
|
| 990 |
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"type": "table",
|
| 991 |
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"img_path": "images/fe56202948fc249a90925377949232db56aa889337e1db995df4579f2a295049.jpg",
|
| 992 |
+
"table_caption": [
|
| 993 |
+
"Table 2: IHDP datasets (100 with $N { = } 7 4 7$ ) "
|
| 994 |
+
],
|
| 995 |
+
"table_footnote": [],
|
| 996 |
+
"table_body": "<table><tr><td>Methods</td><td>PEHE</td><td>EATE</td></tr><tr><td>CFR</td><td>0.81 (0.30)</td><td>0.13 (0.12)</td></tr><tr><td>CFR-ISW</td><td>0.73 (0.28)</td><td>0.11 (0.10)</td></tr><tr><td>SITE</td><td>0.73 (0.33)</td><td>0.10 (0.09)</td></tr><tr><td>DR-CFR</td><td>0.65 (0.37)</td><td>0.03 (0.04)</td></tr></table>",
|
| 997 |
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"bbox": [
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| 998 |
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540,
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| 999 |
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141,
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| 1000 |
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808,
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| 1001 |
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233
|
| 1002 |
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],
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| 1003 |
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"page_idx": 8
|
| 1004 |
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},
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| 1005 |
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{
|
| 1006 |
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"type": "table",
|
| 1007 |
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"img_path": "images/1d9874fd5746ce65422fe71ac402c3d4e031cde364bc64692de7cc70b56086c7.jpg",
|
| 1008 |
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"table_caption": [
|
| 1009 |
+
"Table 1: Synthetic datasets $2 4 \\times 5$ with $N = 1 0 , 0 0 0 )$ "
|
| 1010 |
+
],
|
| 1011 |
+
"table_footnote": [
|
| 1012 |
+
"PEHE and $\\epsilon _ { \\mathrm { A T E } }$ measures (lower is better) represented in the form of “mean (standard deviation)”. "
|
| 1013 |
+
],
|
| 1014 |
+
"table_body": "<table><tr><td>Methods</td><td>PEHE</td><td>EATE</td></tr><tr><td>CFR</td><td>0.61 (0.05)</td><td>0.021 (0.018)</td></tr><tr><td>CFR-ISW</td><td>0.58 (0.06)</td><td>0.017 (0.009)</td></tr><tr><td>SITE</td><td>0.63 (0.05)</td><td>0.035 (0.039)</td></tr><tr><td>DR-CFR</td><td>0.45 (0.11)</td><td>0.013 (0.006)</td></tr></table>",
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| 1015 |
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"bbox": [
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| 1016 |
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|
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|
| 1021 |
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"page_idx": 8
|
| 1022 |
+
},
|
| 1023 |
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{
|
| 1024 |
+
"type": "text",
|
| 1025 |
+
"text": "DR-CFR achieves the best performance among the contending methods. These results are statistically significant based on the Welch’s unpaired t-test with $\\alpha { = } 0 . 0 5$ . Table 2 summarizes the PEHE and $\\epsilon _ { \\mathrm { A T E } }$ measures on the IHDP benchmark. The results are reported in terms of mean and standard deviation over the 100 datasets with various realizations of outcomes. Again, DR-CFR achieves the best performance (statistically significant for $\\epsilon _ { \\mathrm { A T E } }$ ) among the contending methods. ",
|
| 1026 |
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|
| 1027 |
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| 1033 |
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},
|
| 1034 |
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{
|
| 1035 |
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"type": "text",
|
| 1036 |
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"text": "5 FUTURE WORKS AND CONCLUSION ",
|
| 1037 |
+
"text_level": 1,
|
| 1038 |
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"bbox": [
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|
| 1045 |
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|
| 1046 |
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{
|
| 1047 |
+
"type": "text",
|
| 1048 |
+
"text": "The majority of methods proposed to estimate treatment effects – including this work – fall under the category of discriminative approaches. A promising direction is to consider developing generative models, in an attempt to shed light on the true underlying data generating mechanism. Perhaps this could also facilitate generating new, virtual, yet realistic data instances – similar to what is done in computer vision. Louizos et al. (2017)’s method is a notable generative approach, which uses Variational Auto-Encoder (VAE) to extract latent confounders from their observed proxies. While that work is an interesting step in that direction, it is not yet capable of addressing the problem of selection bias. We believe that our proposed perspective on the problem can be helpful to solve this open question. This is left to future work. ",
|
| 1049 |
+
"bbox": [
|
| 1050 |
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|
| 1051 |
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|
| 1052 |
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| 1053 |
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],
|
| 1055 |
+
"page_idx": 8
|
| 1056 |
+
},
|
| 1057 |
+
{
|
| 1058 |
+
"type": "text",
|
| 1059 |
+
"text": "In this paper, we studied the problem of estimating treatment effect from observational studies. We argued that not all factors in the observed covariates $X$ might contribute to the procedure of selecting treatment $T$ , or more importantly, determining the outcomes $Y$ . We modeled this using three underlying sources of $X$ , $T$ , and $Y$ , and showed that explicit identification of these sources offers great insight to help us design models that better handle selection bias in observational datasets. We proposed an algorithm, Disentangled Representations for CounterFactual Regression (DR-CFR), that can (1) identify disentangled representations of the above-mentioned underlying sources and (2) leverage this knowledge to reduce as well as account for the negative impact of selection bias on estimating the treatment effects from observational data. Our empirical results showed that the proposed method achieves state-of-the-art performance in both individual and population based evaluation measures. ",
|
| 1060 |
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|
| 1061 |
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|
| 1066 |
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| 1067 |
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},
|
| 1068 |
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{
|
| 1069 |
+
"type": "text",
|
| 1070 |
+
"text": "ACKNOWLEDGEMENTS ",
|
| 1071 |
+
"text_level": 1,
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| 1072 |
+
"bbox": [
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+
176,
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},
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{
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| 1081 |
+
"type": "text",
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| 1082 |
+
"text": "The authors gratefully acknowledge financial support from Natural Sciences and Engineering Research Council of Canada (NSERC) and Alberta Machine Intelligence Institute (Amii). We wish to thank Dr. Pouria Ramazi and Shivam Raj for fruitful conversations, and Dr. Fredrik Johansson for publishing/maintaining the code-base for the CFR method online. We also would like to thank the ICLR 2020 anonymous reviewers, as well as Dr. Kun Kuang and Tianle Liu, for their valuable reviews, which helped improve this paper. ",
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| 1340 |
+
821,
|
| 1341 |
+
849
|
| 1342 |
+
],
|
| 1343 |
+
"page_idx": 9
|
| 1344 |
+
},
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| 1345 |
+
{
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| 1346 |
+
"type": "text",
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| 1347 |
+
"text": "Adith Swaminathan and Thorsten Joachims. Batch learning from logged bandit feedback through counterfactual risk minimization. JMLR, 16, 2015a. ",
|
| 1348 |
+
"bbox": [
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173,
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| 1350 |
+
857,
|
| 1351 |
+
820,
|
| 1352 |
+
886
|
| 1353 |
+
],
|
| 1354 |
+
"page_idx": 9
|
| 1355 |
+
},
|
| 1356 |
+
{
|
| 1357 |
+
"type": "text",
|
| 1358 |
+
"text": "Adith Swaminathan and Thorsten Joachims. Counterfactual risk minimization: Learning from logged bandit feedback. In ICML, 2015b. ",
|
| 1359 |
+
"bbox": [
|
| 1360 |
+
174,
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| 1361 |
+
895,
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| 1362 |
+
821,
|
| 1363 |
+
924
|
| 1364 |
+
],
|
| 1365 |
+
"page_idx": 9
|
| 1366 |
+
},
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| 1367 |
+
{
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| 1368 |
+
"type": "text",
|
| 1369 |
+
"text": "Adith Swaminathan and Thorsten Joachims. The self-normalized estimator for counterfactual learning. In NeurIPS, 2015c. ",
|
| 1370 |
+
"bbox": [
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+
173,
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| 1372 |
+
103,
|
| 1373 |
+
825,
|
| 1374 |
+
132
|
| 1375 |
+
],
|
| 1376 |
+
"page_idx": 10
|
| 1377 |
+
},
|
| 1378 |
+
{
|
| 1379 |
+
"type": "text",
|
| 1380 |
+
"text": "Liuyi Yao, Sheng Li, Yaliang Li, Mengdi Huai, Jing Gao, and Aidong Zhang. Representation learning for treatment effect estimation from observational data. In NeurIPS, pp. 2633–2643, 2018. ",
|
| 1381 |
+
"bbox": [
|
| 1382 |
+
174,
|
| 1383 |
+
141,
|
| 1384 |
+
825,
|
| 1385 |
+
170
|
| 1386 |
+
],
|
| 1387 |
+
"page_idx": 10
|
| 1388 |
+
}
|
| 1389 |
+
]
|
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| 1 |
+
# ON CONVERGENCE AND STABILITY OF GANS
|
| 2 |
+
|
| 3 |
+
Anonymous authors Paper under double-blind review
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
We propose studying GAN training dynamics as regret minimization, which is in contrast to the popular view that there is consistent minimization of a divergence between real and generated distributions. We analyze the convergence of GAN training from this new point of view to understand why mode collapse happens. We hypothesize the existence of undesirable local equilibria in this non-convex game to be responsible for mode collapse. We observe that these local equilibria often exhibit sharp gradients of the discriminator function around some real data points. We demonstrate that these degenerate local equilibria can be avoided with a gradient penalty scheme called DRAGAN. We show that DRAGAN enables faster training, achieves improved stability with fewer mode collapses, and leads to generator networks with better modeling performance across a variety of architectures and objective functions.
|
| 8 |
+
|
| 9 |
+
# 1 INTRODUCTION
|
| 10 |
+
|
| 11 |
+
Generative modeling involves taking a set of samples drawn from an unknown data generating distribution $P _ { r e a l }$ and finding an estimate $P _ { m o d e l }$ that closely resembles it. Generative adversarial networks (GAN) (Goodfellow et al., 2014) is a powerful framework used for fitting implicit generative models. The basic setup consists of two networks, the generator and the discriminator, playing against each other in a repeated zero-sum game setting. The goal here is to reach an equilibrium where $P _ { r e a l }$ , $P _ { m o d e l }$ are close, and the alternating gradient updates procedure (AGD) is used to achieve this. However, this process is highly unstable and often results in mode collapse (Goodfellow, 2017). This calls for an deeper investigation into training dynamics of GANs.
|
| 12 |
+
|
| 13 |
+
In this paper, we propose studying GAN training dynamics as a repeated game in which both the players are using no-regret algorithms (Cesa-Bianchi & Lugosi, 2006) and discuss how AGD 1 falls under this paradigm. In contrast, much of the theory (Goodfellow et al., 2014; Arjovsky & Bottou, 2017) and recent developments (Nowozin et al., 2016; Arjovsky et al., 2017; Gulrajani et al., 2017) are based on the unrealistic assumption that the discriminator is playing optimally (in the function space) at each step and as a result, there is consistent minimization of a divergence between real and generated distributions. This corresponds to at least one player using the best-response algorithm (in the function space), and the resulting game dynamics can be completely different in both these cases (Nisan et al., 2007). Thus, there is a clear disconnect between theoretical arguments used as motivation in recent literature and what actually happens in practice.
|
| 14 |
+
|
| 15 |
+
We would like to point out that the latter view can still be useful for reasoning about the asymptotic equilibrium situation but we argue that regret minimization is the more appropriate way to think about GAN training dynamics. So, we analyze the convergence of GAN training from this new point of view to understand why mode collapse happens. We start with a short analysis of the artificial convex-concave case of the GAN game in section 2.2. This setting has a unique solution and guaranteed convergence (of averaged iterates) using no-regret algorithms can be shown with standard arguments from game theory literature. Here, we make explicit, the critical (previously not widely known) connection between AGD used in GAN training and regret minimization. This immediately yields a novel proof for the asymptotic convergence of GAN training, in the non-parametric limit. Prior to our work, such a result (Goodfellow et al., 2014) required a strong assumption that the discriminator is optimal at each step.
|
| 16 |
+
|
| 17 |
+
However, these convergence results do not hold when the game objective function is non-convex, which is the practical case when deep neural networks are used. In non-convex games, global regret minimization and equilibrium computation are computationally hard in general. Recent gametheoretic literature indicates that AGD can end up cycling (Mertikopoulos et al., 2017) or converging to a (potentially bad) local equilibrium, under some conditions (Hazan et al., 2017). We hypothesize these to be the reasons for cycling and mode collapse observed during GAN training, respectively (section 2.3). In this work, we do not explore the cycling issue but focus our attention on the mode collapse problem. In contrast to our hypothesis, the prevalent view of mode collapse and instability (Arjovsky & Bottou, 2017) is that it results from attempting to minimize a strong divergence during training. However, as we argued earlier, GAN training with AGD does not consistently minimize a divergence and therefore, such a theory is not suitable to discuss convergence or to address the stability issue.
|
| 18 |
+
|
| 19 |
+
Next, if mode collapse is indeed the result of an undesirable local equilibrium, a natural question then is how we can avoid it? We make a simple observation that, in the GAN game, mode collapse situations are often accompanied by sharp gradients of the discriminator function around some real data points (section 2.4). Therefore, a simple strategy to mitigate mode collapse is to regularize the discriminator so as to constrain its gradients in the ambient data space. We demonstrate that this improves the stability using a toy experiment with one hidden layer neural networks. This gives rise to a new explanation for why WGAN and gradient penalties might be improving the stability of GAN training – they are mitigating the mode collapse problem by keeping the gradients of the discriminator function small in data space. From this motivation, we propose a training algorithm involving a novel gradient penalty scheme called DRAGAN (Deep Regret Analytic Generative Adversarial Networks) which enables faster training, achieves improved stability and modeling performance (over WGAN-GP (Gulrajani et al., 2017) which is the state-of-the-art stable training procedure) across a variety of architectures and objective functions.
|
| 20 |
+
|
| 21 |
+
Below, we provide a short literature review. Several recent works focus on stabilizing the training of GANs. While some solutions (Radford et al., 2015; Salimans et al., 2016) require the usage of specific architectures (or) modeling objectives, some (Che et al., 2016; Zhao et al., 2016) significantly deviate from the original GAN framework. Other promising works in this direction (Metz et al., 2016; Arjovsky et al., 2017; Qi, 2017; Gulrajani et al., 2017) impose a significant computational overhead. Thus, a fast and versatile method for consistent stable training of GANs is still missing in the literature. Our work is aimed at addressing this.
|
| 22 |
+
|
| 23 |
+
To summarize, our contributions are as follows:
|
| 24 |
+
|
| 25 |
+
• We propose a new way of reasoning about the GAN training dynamics - by viewing AGD as regret minimization.
|
| 26 |
+
• We provide a novel proof for the asymptotic convergence of GAN training in the nonparametric limit and it does not require the discriminator to be optimal at each step.
|
| 27 |
+
• We discuss how AGD can converge to a potentially bad local equilibrium in non-convex games and hypothesize this to be responsible for mode collapse during GAN training.
|
| 28 |
+
• We characterize mode collapse situations with sharp gradients of the discriminator function around some real data points.
|
| 29 |
+
• A novel gradient penalty scheme called DRAGAN is introduced based on this observation and we demonstrate that it mitigates the mode collapse issue.
|
| 30 |
+
|
| 31 |
+
# 2 THEORETICAL ANALYSIS OF GAN TRAINING DYNAMICS
|
| 32 |
+
|
| 33 |
+
We start with a brief description of the GAN framework (section 2.1). We discuss guaranteed convergence in the artificial convex-concave case using no-regret algorithms, and make a critical connection between GAN training process (AGD) and regret minimization (section 2.2). This immediately yields a novel proof for the asymptotic convergence of GAN training in the nonparametric limit. Then, we consider the practical non-convex case and discuss how AGD can converge to a potentially bad local equilibrium here (section 2.3). We characterize mode collapse situations with sharp gradients of the discriminator function around real samples and this provides an effective strategy to avoid them. This naturally leads to the introduction of our gradient penalty scheme DRAGAN (section 2.4). We end with a discussion and comparison with other gradient penalties in the literature (section 2.5).
|
| 34 |
+
|
| 35 |
+
# 2.1 BACKGROUND
|
| 36 |
+
|
| 37 |
+
The GAN framework can be viewed as a repeated zero-sum game, consisting of two players - the generator, which produces synthetic data given some noise source and the discriminator, which is trained to distinguish generator’s samples from the real data. The generator model G is parameterized by $\phi$ , takes a noise vector $\mathbf { z }$ as input, and produces a synthetic sample $G _ { \phi } ( \mathbf { z } )$ . The discriminator model $\mathrm { D }$ is parameterized by $\theta$ , takes a sample $\mathbf { x }$ as input and computes $D _ { \theta } ( \mathbf { x } )$ , which can be interpreted as the probability that $\mathbf { x }$ is real.
|
| 38 |
+
|
| 39 |
+
The models $\mathbf { G }$ , D can be selected from any arbitrary class of functions – in practice, GANs typical rely on deep networks for both. Their cost functions are defined as
|
| 40 |
+
|
| 41 |
+
$$
|
| 42 |
+
\begin{array} { r l } & { J ^ { ( D ) } ( \phi , \theta ) : = - \mathbb { E } _ { x \sim p _ { r e a l } } \log D _ { \theta } ( x ) - \mathbb { E } _ { \mathbf { z } } \log ( 1 - D _ { \theta } ( G _ { \phi } ( z ) ) ) , \mathrm { ~ a r ~ } } \\ & { J ^ { ( G ) } ( \phi , \theta ) : = - J ^ { ( D ) } ( \phi , \theta ) } \end{array}
|
| 43 |
+
$$
|
| 44 |
+
|
| 45 |
+
And the complete game can be specified as -
|
| 46 |
+
|
| 47 |
+
$$
|
| 48 |
+
\underset { \phi } { \operatorname* { m i n } } \underset { \theta } { \operatorname* { m a x } } \left\{ J ( \phi , \theta ) = \mathbb { E } _ { x \sim p _ { r e a l } } \log D _ { \theta } ( x ) + \mathbb { E } _ { \mathbf { z } } \log ( 1 - D _ { \theta } ( G _ { \phi } ( z ) ) ) \right\}
|
| 49 |
+
$$
|
| 50 |
+
|
| 51 |
+
The generator distribution $P _ { m o d e l }$ asymptotically converges to the real distribution $P _ { r e a l }$ if updates are made in the function space and the discriminator is optimal at each step (Goodfellow et al., 2014).
|
| 52 |
+
|
| 53 |
+
2.2 CONVEX-CONCAVE CASE AND NO-REGRET ALGORITHMS
|
| 54 |
+
|
| 55 |
+
According to Sion’s theorem (Sion, 1958), if $\Phi \subset \mathbb { R } ^ { m }$ , $\Theta \subset \mathbb { R } ^ { n }$ such that they are compact and convex sets, and the function $J : \Phi \times \Theta \mathbb { R }$ is convex in its first argument and concave in its second, then we have -
|
| 56 |
+
|
| 57 |
+
$$
|
| 58 |
+
\operatorname* { m i n } _ { \phi \in \Phi } \operatorname* { m a x } _ { \theta \in \Theta } J ( \phi , \theta ) = \operatorname* { m a x } _ { \theta \in \Theta } \operatorname* { m i n } _ { \phi \in \Phi } J ( \phi , \theta )
|
| 59 |
+
$$
|
| 60 |
+
|
| 61 |
+
That is, an equilibrium is guaranteed to exist in this setting where players’ payoffs correspond to the unique value of the game (Neumann, 1928).
|
| 62 |
+
|
| 63 |
+
A natural question then is how we can find such an equilibrium. A simple procedure that players can use is best-response algorithms (BRD). In each round, best-responding players play their optimal strategy given their opponent’s current strategy. Despite its simplicity, BRD are often computationally intractable and they don’t lead to convergence even in simple games. In contrast, a technique that is both efficient and provably works is regret minimization. If both players update their parameters using no-regret algorithms, then it is easy to show that their averaged iterates will converge to an equilibrium pair (Nisan et al., 2007). Let us first define no-regret algorithms.
|
| 64 |
+
|
| 65 |
+
Definition 2.1 (No-regret algorithm). Given a sequence of convex loss functions $L _ { 1 } , L _ { 2 } , \dots :$ $K \mathbb { R }$ , an algorithm that selects a sequence of $k _ { t }$ ’s, each of which may only depend on previously observed $L _ { 1 } , \dots , L _ { t - 1 }$ , is said to have no regret if $\begin{array} { r } { \frac { R ( T ) } { T } = o ( 1 ) } \end{array}$ , where we define
|
| 66 |
+
|
| 67 |
+
$$
|
| 68 |
+
\begin{array} { r } { R ( T ) : = \sum _ { t = 1 } ^ { T } L _ { t } ( k _ { t } ) - \operatorname* { m i n } _ { k \in K } \sum _ { t = 1 } ^ { T } L _ { t } ( k ) } \end{array}
|
| 69 |
+
$$
|
| 70 |
+
|
| 71 |
+
We can apply no-regret learning to our problem of equilibrium finding in the GAN game $J ( \cdot , \cdot )$ as follows. The generator imagines the function $J ( \cdot , \theta _ { t } )$ as its loss function on round $t$ , and similarly the discriminator imagines computes the average itera $- J ( \phi _ { t } , \cdot )$ unctiand $t$ $T$ oun. If f play, each playeris the equilibrium $\begin{array} { r } { \stackrel { \cdot } { \phi } _ { T } : = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \phi _ { \underline { { t } } } } \end{array}$ $\begin{array} { r } { \bar { \theta } _ { T } : = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \theta _ { t } } \end{array}$ $V ^ { * }$ value of the game, and the players suffer regret $R _ { 1 } ( T )$ and $\bar { R _ { 2 } ( T ) }$ respectively, then one can show using standard arguments (Freund & Schapire, 1999) that -
|
| 72 |
+
|
| 73 |
+
$$
|
| 74 |
+
\begin{array} { r } { V ^ { * } - \frac { R _ { 2 } ( T ) } { T } \le \operatorname* { m a x } _ { \theta \in \Theta } J \big ( \bar { \phi } _ { T } , \theta \big ) - \frac { R _ { 2 } ( T ) } { T } \le \operatorname* { m i n } _ { \phi \in \Phi } J \big ( \phi , \bar { \theta } _ { T } \big ) + \frac { R _ { 1 } ( T ) } { T } \le V ^ { * } + \frac { R _ { 1 } ( T ) } { T } . } \end{array}
|
| 75 |
+
$$
|
| 76 |
+
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| 77 |
+
In other words, $\bar { \theta } _ { T }$ and $\bar { \phi } _ { T }$ are "almost optimal" solutions to the game, where the "almost" approximation factor is given by the average regret terms R1(T )+R2(T ) . Under the no-regret condition, the former will vanish, and hence we can guarantee convergence in the limit. Next, we define a popular family of no-regret algorithms.
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+
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| 79 |
+
Definition 2.2 (Follow The Regularized Leader). FTRL (Hazan et al., 2016) selects $k _ { t }$ on round $t$ by solving for arg $\begin{array} { r } { \operatorname* { m i n } _ { k \in \boldsymbol { K } } \{ \bar { \sum } _ { s = 1 } ^ { t - 1 } L _ { s } ( k ) + \frac { 1 } { \eta } \Omega ( k ) \} } \end{array}$ , where $\Omega ( \cdot )$ is some convex regularization function and $\eta$ is a learning rate.
|
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+
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+
Remark: Roughly speaking, if you select the regularization as $\begin{array} { r } { \Omega ( \cdot ) = \frac { 1 } { 2 } \| \cdot \| ^ { 2 } } \end{array}$ , then FTRL becomes the well-known online gradient descent or OGD (Zinkevich, 2003). Ignoring the case of constraint violations, OGD can be written in a simple iterative form: $k _ { t } = k _ { t - 1 } - \eta \nabla L _ { t - 1 } ( k _ { t - 1 } )$ .
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+
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+
The typical GAN training procedure using alternating gradient updates (or simultaneous gradient updates) is almost this - both the players applying online gradient descent. Notice that the $\operatorname* { m i n } / \operatorname* { m a x }$ objective function in GANs involves a stochastic component, with two randomized inputs given on each round, $x$ and $z$ which are sampled from the data distribution and a standard multivariate normal, respectively. Let us write $J _ { x , z } ( \phi , \theta \bar ) : = \log { D _ { \theta } ( x ) } + \log ( 1 - D _ { \theta } ( G _ { \phi } ( z ) ) )$ . Taking expectations with respect to $\mathbf { x }$ and $\mathbf { z }$ , we define the full (non-stochastic) game as $J ( \phi , \theta ) = \mathbb { E } _ { \mathbf { x } , \mathbf { z } } \left[ J _ { x , z } ( \phi , \theta ) \right]$ . But the above online training procedure is still valid with stochastic inputs. That is, the equilibrium computation would proceed similarly, where on each round we sample $x _ { t }$ and $z _ { t }$ , and follow the updates
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+
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+
$$
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+
\phi _ { t + 1 } \phi _ { t } - \eta \nabla _ { \phi } J _ { x _ { t } , z _ { t } } ( \phi _ { t } , \theta _ { t } ) . \quad \mathrm { a n d } \quad \theta _ { t + 1 } \theta _ { t } + \eta ^ { ' } \nabla _ { \theta } J _ { x _ { t } , z _ { t } } ( \phi _ { t } , \theta _ { t } )
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+
$$
|
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+
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+
On a side note, a benefit of this stochastic perspective is that we can get a generalization bound on the mean parameters $\bar { \phi } _ { T }$ after $T$ rounds of optimization. The celebrated "online-to-batch conversion" (Cesa-Bianchi et al., 2004) implies that $\mathbb { E } _ { \mathbf { x } , \mathbf { z } } \big [ J _ { x , z } ( \bar { \phi } _ { T } , \theta ) \big ]$ , for any $\theta$ , is no more than the optimal value $\mathbb { E } _ { \mathbf { x } , \mathbf { z } } [ J _ { x , z } ( \phi ^ { * } , \theta ) ]$ plus an "estimation error" bounded by $\mathbb { E } \left[ { \frac { R _ { 1 } ( T ) + R _ { 2 } ( T ) } { T } } \right]$ , where the expectation is taken with respect to the sequence of samples observed along the way, and any randomness in the algorithm. Analogously, this applies to $\bar { \theta } _ { T } ^ { \star }$ as well. A limitation of this result, however, is that it requires a fresh sample $x _ { t }$ to be used on every round.
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+
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+
To summarize, we discussed in this subsection about how the artificial convex-concave case is easy to solve through regret minimization. While this is a standard result in game theory and online learning literature, it is not widely known in the GAN literature. For instance, Salimans et al. (2016) and Goodfellow (2017) discuss a toy game which is convex-concave and show cycling behavior. But, the simple solution in that case is to just average the iterates. Further, we made explicit, the critical connection between regret minimization and alternating gradient updates procedure used for GAN training. Now, Goodfellow et al. (2014) argue that, if $G$ and $D$ have enough capacity (in the non-parametric limit) and updates are made in the function space, then the GAN game can be considered convex-concave. Thus, our analysis based on regret minimization immediately yields a novel proof for the asymptotic convergence of GANs, without requiring that the discriminator be optimal at each step.
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+
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+
Moreover, the connection between regret minimization and GAN training process gives a novel way to reason about its dynamics. In contrast, the popular view of GAN training as consistently minimizing a divergence arises if the discriminator uses BRD (in the function space) and thus, it has little to do with the actual training process of GANs. As a result, this calls into question the motivation behind many recent developments like WGAN and gradient penalties among others, which improve the training stability of GANs. In the next subsection, we discuss the practical non-convex case and why training instability arises. This provides the necessary ideas to investigate mode collapse from our new perspective.
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# 2.3 NON-CONVEX CASE AND LOCAL EQUILIBRIA
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In practice, we choose $G$ , $D$ to be deep neural networks and the function $J ( \phi , \theta )$ need not be convexconcave anymore. The nice properties we had in the convex-concave case like the existence of a unique solution and guaranteed convergence through regret minimization no longer hold. In fact, regret minimization and equilibrium computation are computationally hard in general non-convex settings. However, analogous to the case of non-convex optimization (also intractable) where we focus on finding local minima, we can look for tractable solution concepts in non-convex games.
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+
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Recent work by Hazan et al. (2017) introduces the notion of local regret and shows that if both the players use a smoothed variant of OGD to minimize this quantity, then the non-convex game converges to some form of local equilibrium, under mild assumptions. The usual training procedure of GANs (AGD) corresponds to using a window size of 1 in their formulation. Thus, GAN training will eventually converge (approximately) to a local equilibrium which is described below or the updates will cycle. We leave it to future works to explore the equally important cycling issue and focus here on the former case.
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+
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Definition 2.3 (Local Equilibrium). A pair $( \phi ^ { * } , \theta ^ { * } )$ is called an $\epsilon$ -approximate local equilibrium if it holds that
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+
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+
$$
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+
\begin{array} { l } { { \forall \phi ^ { ' } , \vert \vert \phi ^ { ' } - \phi ^ { * } \vert \vert \le \eta : J ( \phi ^ { * } , \theta ^ { * } ) \le J ( \phi ^ { ' } , \theta ^ { * } ) + \epsilon } } \\ { { \forall \theta ^ { ' } , \vert \vert \theta ^ { ' } - \theta ^ { * } \vert \vert \le \eta : J ( \phi ^ { * } , \theta ^ { * } ) \ge J ( \phi ^ { * } , \theta ^ { ' } ) - \epsilon } } \end{array}
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+
$$
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+
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+
That is, in a local equilibrium, both the players do not have much of an incentive to switch to any other strategy within a small neighborhood of their current strategies. Now, we turn our attention to the mode collapse issue which poses a significant challenge to the GAN training process. The training is said to have resulted in mode collapse if the generator ends up mapping multiple z vectors to the same output $\mathbf { x }$ , which is assigned a high probability of being real by the discriminator (Goodfellow, 2017). We hypothesize this to be the result of the game converging to bad local equilibria.
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The prevalent view of mode collapse and instability in GAN training (Arjovsky & Bottou, 2017) is that it is caused due to the supports of real and model distributions being disjoint or lying on low-dimensional manifolds. The argument is that this would result in strong distance measures like KL-divergence or JS-divergence getting maxed out, and the generator cannot get useful gradients to learn. In fact, this is the motivation for the introduction of WGAN (Arjovsky et al., 2017). But, as we argued earlier, GAN training does not consistently minimize a divergence as that would require using intractable best-response algorithms. Hence, such a theory is not suitable to discuss convergence or to address the instability of GAN training. Our new view of GAN training process as regret minimization is closer to what is used in practice and provides an alternate explanation for mode collapse - the existence of undesirable local equilibria. The natural question now is how we can avoid them?
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# 2.4 MODE COLLAPSE AND GRADIENT PENALTIES
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The problem of dealing with multiple equilibria in games and how to avoid undesirable ones is an important question in algorithmic game theory (Nisan et al., 2007). In this work, we constrain ourselves to the GAN game and aim to characterize the undesirable local equilibria (mode collapse) in an effort to avoid them. In this direction, after empirically studying multiple mode collapse cases, we found that it is often accompanied by the discriminator function having sharp gradients around some real data points (See Figure $1 ^ { 2 }$ ). This intuitively makes sense from the definition of mode collapse discussed earlier. Such sharp gradients encourage the generator to map multiple $z$ vectors to a single output $x$ and lead the game towards a degenerate equilibrium. Now, a simple strategy to mitigate this failure case would be to regularize the discriminator using the following penalty -
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+
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$$
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+
\lambda \cdot \mathbb { E } _ { x \sim P _ { r e a l } , \delta \sim N _ { d } ( 0 , c I ) } \big [ \| \nabla _ { \mathbf { x } } D _ { \theta } ( x + \delta ) \| ^ { 2 } \big ]
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+
$$
|
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+
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This strategy indeed improves the stability of GAN training. We show the results of a toy experiment with one hidden layer neural networks in Figure 2 and Figure 3 to demonstrate this. This partly explains the success of WGAN and gradient penalties in the recent literature (Gulrajani et al., 2017; Qi, 2017), and why they improve the training stability of GANs, despite being motivated by reasoning based on unrealistic assumptions. However, we noticed that this scheme in its current form can be brittle and if over-penalized, the discriminator can end up assigning both a real point $x$ and noise $x + \delta$ , the same probability of being real. Thus, a better choice of penalty is -
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$$
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\lambda \cdot \mathbb { E } _ { x \sim P _ { r e a l } , \delta \sim N _ { d } ( 0 , c I ) } \big [ \operatorname* { m a x } \big ( 0 , \| \nabla _ { \mathbf { x } } D _ { \theta } ( x + \delta ) \| ^ { 2 } - k \big ) \big ]
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+
$$
|
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+
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+
Finally, due to practical optimization considerations (this has also been observed in Gulrajani et al.
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(2017)), we instead use the penalty shown below in all our experiments.
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$$
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\lambda \cdot \mathbb { E } _ { x \sim P _ { r e a l } , \delta \sim N _ { d } ( 0 , c I ) } \big [ \| \nabla _ { \mathbf { x } } D _ { \theta } ( x + \delta ) \| - k \big ] ^ { 2 }
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+
$$
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+
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+
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Figure 1: One hidden layer networks as $G$ and $D$ (MNIST). On the left, we plot inception score against time for vanilla GAN training and on the right, we plot the squared norm of discriminator’s gradients around real data points for the same experiment. Notice how this quantity changes before, during and after mode collapse events.
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+
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Figure 2: One hidden layer networks as $G$ and $D$ (MNIST). On the left, losses for both the players are shown for vanilla GAN training and on the right, we added a regularization term to penalize the gradients of $D ( x )$ around real data points. Notice the improved stability.
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+
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This still works as long as small perturbations of real data, $x + \delta$ are likely to lie off the data-manifold, which is true in the case of image domain and some other settings. Because, in these cases, we do want our discriminator to assign different probabilities of being real to training data and noisy samples. We caution the practitioners to keep this important point in mind while making their choice of penalty. All of the above schemes have the same effect of constraining the norm of discriminator’s gradients around real points to be small and can therefore, mitigate the mode collapse situation. We refer to GAN training using these penalty schemes or heuristics as the DRAGAN algorithm.
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+
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+
Additional details:
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+
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+
• We use the vanilla GAN objective in our experiments, but our penalty improves stability using other objective functions as well. This is demonstrated in section 3.3.
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+
• The penalty scheme used in our experiments is the one shown in equation 1.
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+
• We use small pixel-level noise but it is possible to find better ways of imposing this penalty. However, this exploration is beyond the scope of our paper.
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• The optimal configuration of the hyperparameters for DRAGAN depends on the architecture, dataset and data domain. We set them to be $\lambda \sim 1 0$ , $k = 1$ and $c \sim 1 0$ in most of our experiments.
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+
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+
# 2.5 COUPLED VS LOCAL PENALTIES
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+
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+
Several recent works have also proposed regularization schemes which constrain the discriminator’s gradients in the ambient data space, so as to improve the stability of GAN training. Despite being from different motivations, WGAN-GP and LS-GAN are closely related approaches to ours. First, we show that these two approaches are very similar, which is not widely known in the literature. Qi (2017) introduced LS-GAN with the idea of maintaining a margin between losses assigned to real and fake samples. Further, they also impose Lipschitz constraint on $D$ and the two conditions together result in a situation where the following holds for any real and fake sample pair (roughly) -
|
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+
|
| 151 |
+
|
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+
Figure 3: One hidden layer networks as $G$ and $D$ (MNIST). On the left, inception score plot is shown for vanilla GAN training and on the right, we added a regularization term to penalize the gradients of $D ( x )$ around real data points. Notice how mode collapse is mitigated.
|
| 153 |
+
|
| 154 |
+
$$
|
| 155 |
+
D _ { \theta } ( x ) - D _ { \theta } ( G _ { \phi } ( z ) ) \approx | | x , G _ { \phi } ( z ) | |
|
| 156 |
+
$$
|
| 157 |
+
|
| 158 |
+
The authors argue that the resulting discriminator function would have non-vanishing gradients almost everywhere between real and fake samples (section 6 of Qi (2017)). Next, Gulrajani et al. (2017) proposed an extension to address various shortcomings of the original WGAN and they impose the following condition on $D$ -
|
| 159 |
+
|
| 160 |
+
$$
|
| 161 |
+
| | \nabla _ { x } D _ { \theta } ( \hat { x } ) | | \approx 1
|
| 162 |
+
$$
|
| 163 |
+
|
| 164 |
+
where $\hat { x } = ( \epsilon ) x + ( 1 - \epsilon ) G _ { \phi } ( z )$ is some point on the line between a real and a fake sample, both chosen independently at random. This leads to $D$ having norm-1 gradients almost everywhere between real and fake samples. Notice that this behavior is very similar to that of LS-GAN’s discriminator function. Thus, WGAN-GP is a slight variation of the original LS-GAN algorithm and we refer to these methods as “coupled penalties”.
|
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+
|
| 166 |
+
On a side note, we also want to point out that WGAN-GP’s penalty doesn’t actually follow from KR-duality as claimed in their paper. By Lemma 1 of Gulrajani et al. (2017), the optimal discriminator $D ^ { * }$ will have norm-1 gradients (almost everywhere) only between those $x$ and $G _ { \phi } ( z )$ pairs which are sampled from the optimal coupling or joint distribution $\pi ^ { * }$ . Therefore, there is no basis for WGAN-GP’s penalty (equation 3) where arbitrary pairs of real and fake samples are used. This fact adds more credence to our theory regarding why gradient penalties might be mitigating mode collapse.
|
| 167 |
+
|
| 168 |
+
The most important distinction between coupled penalties and our methods is that we only impose gradient constraints in local regions around real samples. We refer to these penalty schemes as “local penalties”. Coupled penalties impose gradient constraints between real and generated samples and we point out some potential issues that arise from this:
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| 169 |
+
|
| 170 |
+
• With adversarial training finding applications beyond fitting implicit generative models, penalties which depend on generated samples can be prohibitive.
|
| 171 |
+
• The resulting class of functions when coupled penalties are used will be highly restricted compared to our method and this affects modeling performance. We refer the reader to Figure 4 and appendix section 5.2.2 to see this effect.
|
| 172 |
+
• Our algorithm works with AGD, while WGAN-GP needs multiple inner iterations to optimize D. This is because the generated samples can be anywhere in the data space and they change from one iteration to the next. In contrast, we consistently regularize $D _ { \theta } ( x )$ only along the real data manifold.
|
| 173 |
+
|
| 174 |
+
To conclude, appropriate constraining of the discriminator’s gradients can mitigate mode collapse but we should be careful so that it doesn’t have any negative effects. We pointed out some issues with coupled penalties and how local penalties can help. We refer the reader to section 3 for further experimental results.
|
| 175 |
+
|
| 176 |
+
|
| 177 |
+
Figure 4: Swissroll experiment (different phases of training) - Vanilla GAN (top), WGAN-GP (middle), and DRAGAN (bottom). Real samples are marked orange and generated samples are green. Level sets of $D _ { \theta } ( x )$ are shown in the background where yellow is high and purple is low.
|
| 178 |
+
|
| 179 |
+
# 3 EXPERIMENTAL RESULTS
|
| 180 |
+
|
| 181 |
+
In section 3.1, we compare the modeling performance of our algorithm against vanilla GAN and WGAN variants in the standard DCGAN/CIFAR-10 setup. Section 3.2 demonstrates DRAGAN’s improved stability across a variety of architectures. In section 3.3, we show that our method also works with other objective functions. Appendix contains samples for inspection, some of the missing plots and additional results. Throughout, we use inception score (Salimans et al., 2016) which is a well-studied and reliable metric in the literature, and sample quality to measure the performance.
|
| 182 |
+
|
| 183 |
+
# 3.1 INCEPTION SCORES FOR CIFAR-10 USING DCGAN ARCHITECTURE
|
| 184 |
+
|
| 185 |
+
DCGAN is a family of architectures designed to perform well with the vanilla training procedure. They are ubiquitous in the GAN literature owing to the instability of vanilla GAN in general settings. We use this architecture to model CIFAR-10 and compare against vanilla GAN, WGAN and WGANGP. As WGANs need 5 discriminator iterations for every generator iteration, comparing the modeling performance can be tricky. To address this, we report two scores for vanilla GAN and DRAGAN - one using the same number of generator iterations as WGANs and one using the same number of discriminator iterations. The results are shown in Figure 5 and samples are included in the appendix (Figure 8). Notice that DRAGAN beats WGAN variants in both the configurations, while vanilla GAN is only slightly better. A key point to note here is that our algorithm is fast compared to WGANs, so in practice, the performance will be closer to the DRAGANd case. In the next section, we will show that if we move away from this specific architecture family, vanilla GAN training can become highly unstable and that DRAGAN penalty mitigates this issue.
|
| 186 |
+
|
| 187 |
+
# 3.2 MEASURING STABILITY AND PERFORMANCE ACROSS ARCHITECTURES
|
| 188 |
+
|
| 189 |
+
Ideally, we would want our training procedure to perform well in a stable fashion across a variety of architectures (other than DCGANs). Similar to Arjovsky et al. (2017) and Gulrajani et al. (2017), we remove the stabilizing components of DCGAN architecture and demonstrate improved stability $\&$ modeling performance compared to vanilla GAN training (see appendix section 5.2.3). However, this is a small set of architectures and it is not clear if there is an improvement in general.
|
| 190 |
+
|
| 191 |
+
To address this, we introduce a metric termed the BogoNet score to compare the stability & performance of different GAN training procedures. The basic idea is to choose random architectures for players $G$ and $D$ independently, and evaluate the performance of different algorithms in the resulting games. A good algorithm should achieve stable performance without failing to learn or resulting in mode collapse, despite the potentially imbalanced architectures. In our experiment, each player is assigned a network from a diverse pool of architectures belonging to three different families (MLP, ResNet, DCGAN).
|
| 192 |
+
|
| 193 |
+
|
| 194 |
+
Figure 5: Comparison of modeling performance on CIFAR10
|
| 195 |
+
|
| 196 |
+
<table><tr><td>Algorithm</td><td>Score</td></tr><tr><td>WGAN</td><td>3.25</td></tr><tr><td>WGAN-GP</td><td>5.99</td></tr><tr><td>DRAGAN9</td><td>6.11</td></tr><tr><td>DRAGANd</td><td>6.90</td></tr><tr><td>Vanilla GANg</td><td>6.3</td></tr><tr><td>Vanilla GANd</td><td>6.99</td></tr></table>
|
| 197 |
+
|
| 198 |
+
(b) Inception scores
|
| 199 |
+
|
| 200 |
+
Table 1: Summary of inception score statistics across 100 architectures
|
| 201 |
+
|
| 202 |
+
<table><tr><td rowspan=1 colspan=1>Algorithm</td><td rowspan=1 colspan=2>Final score</td><td rowspan=1 colspan=2>Area under curve</td><td rowspan=1 colspan=1>Qual. score</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Mean</td><td rowspan=1 colspan=1>Std</td><td rowspan=1 colspan=1>Mean</td><td rowspan=1 colspan=1>Std</td><td rowspan=1 colspan=1>Total</td></tr><tr><td rowspan=1 colspan=1>Vanilla GAN</td><td rowspan=1 colspan=1>2.91</td><td rowspan=1 colspan=1>1.44</td><td rowspan=1 colspan=1>277.72</td><td rowspan=1 colspan=1>126.09</td><td rowspan=1 colspan=1>92.5</td></tr><tr><td rowspan=1 colspan=1>DRAGAN</td><td rowspan=1 colspan=1>3.70</td><td rowspan=1 colspan=1>1.71</td><td rowspan=1 colspan=1>312.15</td><td rowspan=1 colspan=1>135.35</td><td rowspan=1 colspan=1>157.5</td></tr><tr><td rowspan=1 colspan=1>WGAN-GP</td><td rowspan=1 colspan=1>3.49</td><td rowspan=1 colspan=1>1.30</td><td rowspan=1 colspan=1>300.09</td><td rowspan=1 colspan=1>100.96</td><td rowspan=1 colspan=1>1</td></tr></table>
|
| 203 |
+
|
| 204 |
+
To demonstrate that our algorithm performs better compared to vanilla GAN training and WGAN-GP, we created 100 such instances of hard games. Each instance is trained using these algorithms on CIFAR-10 (under similar conditions for a fixed number of generator iterations, which gives a slight advantage to WGAN-GP) and we plot how inception score changes over time. For each algorithm, we calculated the average of final inception scores and area under the curve (AUC) over all 100 instances. The results are shown in Table 1. Notice that we beat the other algorithms in both metrics, which indicates some improvement in stability and modeling performance.
|
| 205 |
+
|
| 206 |
+
Further, we perform some qualitative analysis to verify that BogoNet score indeed captures the improvements in stability. We create another set of 50 hard architectures and compare DRAGAN against vanilla GAN training. Each instance is allotted 5 points and we split this bounty between the two algorithms depending on their performance. If both perform well or perform poorly, they get 2.5 points each, so that we nullify the effect of such non-differentiating architectures. However, if one algorithm achieves stable performance compared to the other (in terms of failure to learn or mode collapses), we assign it higher portions of the bounty. Results were judged by two of the authors in a blind manner: The curves were shown side-by-side with the choice of algorithm for each side being randomized and unlabeled. The vanilla GAN received an average score of 92.5 while our algorithm achieved an average score of 157.5 and this correlates with BogoNet score from earlier. See appendix section 5.3 for some additional details regarding this experiment.
|
| 207 |
+
|
| 208 |
+
# 3.3 STABILITY USING DIFFERENT OBJECTIVE FUNCTIONS
|
| 209 |
+
|
| 210 |
+
Our algorithm improves stability across a variety of objective functions and we demonstrate this using the following experiment. Nowozin et al. (2016) show that we can interpret GAN training as minimizing various $f$ -divergences when an appropriate game objective function is used. We show experiments using the objective functions developed for Forward KL, Reverse KL, Pearson $\chi ^ { 2 }$ , Squared Hellinger, and Total Variation divergence minimization. We use a hard architecture from the previous subsection to demonstrate the improvements in stability. Our algorithm is stable in all cases except for the total variation case, while the vanilla algorithm failed in all the cases (see Figure 6 for two examples and Figure 15 in appendix for all five). Thus, practitioners can now choose their game objective from a larger set of functions and use DRAGAN (unlike WGANs which requires a specific objective function).
|
| 211 |
+
|
| 212 |
+
|
| 213 |
+
Figure 6: Inception score plots for two divergence measures, demonstrating superior stability for our algorithm.
|
| 214 |
+
|
| 215 |
+
# 4 CONCLUSIONS
|
| 216 |
+
|
| 217 |
+
In this paper, we propose to study GAN training process as regret minimization, which is in contrast to the popular view that there is consistent minimization of a divergence between real and generated distributions. We analyze the convergence of GAN training from this new point of view and hypothesize that mode collapse occurs due to the existence of undesirable local equilibria. A simple observation is made about how the mode collapse situation often exhibits sharp gradients of the discriminator function around some real data points. This characterization partly explains the workings of previously proposed WGAN and gradient penalties, and motivates our novel penalty scheme. We show evidence of improved stability using DRAGAN and the resulting improvements in modeling performance across a variety of settings. We leave it to future works to explore our ideas in more depth and come up with improved training algorithms.
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+
# REFERENCES
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Martin Arjovsky and Léon Bottou. Towards principled methods for training generative adversarial networks. arXiv preprint arXiv:1701.04862, 2017.
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| 222 |
+
Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein gan. arXiv preprint arXiv:1701.07875, 2017.
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Nicolo Cesa-Bianchi and Gábor Lugosi. Prediction, learning, and games. Cambridge university press, 2006.
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Nicolo Cesa-Bianchi, Alex Conconi, and Claudio Gentile. On the generalization ability of on-line learning algorithms. IEEE Transactions on Information Theory, 50(9):2050–2057, 2004.
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Tong Che, Yanran Li, Athul Paul Jacob, Yoshua Bengio, and Wenjie Li. Mode regularized generative adversarial networks. arXiv preprint arXiv:1612.02136, 2016.
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Yoav Freund and Robert E Schapire. Adaptive game playing using multiplicative weights. Games and Economic Behavior, 29(1-2):79–103, 1999.
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Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 27, pp. 2672–2680. Curran Associates, Inc., 2014. URL http://papers.nips. cc/paper/5423-generative-adversarial-nets.pdf.
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+
Ian J. Goodfellow. NIPS 2016 tutorial: Generative adversarial networks. CoRR, abs/1701.00160, 2017. URL http://arxiv.org/abs/1701.00160.
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Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron Courville. Improved training of wasserstein gans. arXiv preprint arXiv:1704.00028, 2017.
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Elad Hazan, Karan Singh, and Cyril Zhang. Efficient regret minimization in non-convex games. arXiv preprint arXiv:1708.00075, 2017.
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Elad Hazan et al. Introduction to online convex optimization. Foundations and Trends® in Optimization, 2(3-4):157–325, 2016.
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Panayotis Mertikopoulos, Christos Papadimitriou, and Georgios Piliouras. Cycles in adversarial regularized learning. arXiv preprint arXiv:1709.02738, 2017.
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Luke Metz, Ben Poole, David Pfau, and Jascha Sohl-Dickstein. Unrolled generative adversarial networks. CoRR, abs/1611.02163, 2016. URL http://arxiv.org/abs/1611.02163.
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J. von Neumann. Zur theorie der gesellschaftsspiele. Mathematische Annalen, 100:295–320, 1928. URL http://eudml.org/doc/159291.
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Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V Vazirani. Algorithmic game theory, volume 1. Cambridge University Press Cambridge, 2007.
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Sebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-gan: Training generative neural samplers using variational divergence minimization. In Advances in Neural Information Processing Systems, pp. 271–279, 2016.
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Guo-Jun Qi. Loss-sensitive generative adversarial networks on lipschitz densities. CoRR, abs/1701.06264, 2017. URL http://arxiv.org/abs/1701.06264.
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Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks. arXiv:1511.06434 [cs], November 2015. URL http://arxiv.org/abs/1511.06434. arXiv: 1511.06434.
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Tim Salimans, Ian J. Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. CoRR, abs/1606.03498, 2016. URL http://arxiv. org/abs/1606.03498.
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+
Maurice Sion. On general minimax theorems. Pacific J. Math, 8(1):171–176, 1958.
|
| 241 |
+
Junbo Zhao, Michael Mathieu, and Yann LeCun. Energy-based generative adversarial network. arXiv preprint arXiv:1609.03126, 2016.
|
| 242 |
+
Martin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the 20th International Conference on Machine Learning (ICML-03), pp. 928–936, 2003.
|
| 243 |
+
|
| 244 |
+
# 5 APPENDIX
|
| 245 |
+
|
| 246 |
+
# 5.1 SAMPLES AND LATENT SPACE WALKS
|
| 247 |
+
|
| 248 |
+
In this section, we provide samples from an additional experiment run on CelebA dataset (Figure 7). The samples from the experiment in section 3.1 are shown in Figure 8. Further, Radford et al. (2015) suggest that walking on the manifold learned by the generator can expose signs of memorization. We use DCGAN architecture to model MNIST and CelebA datasets using DRAGAN penalty, and the latent space walks of the learned models are shown in Figure 9 and Figure 10. The results demonstrate that the generator is indeed learning smooth transitions between different images, when our algorithm is used.
|
| 249 |
+
|
| 250 |
+
|
| 251 |
+
Figure 7: Modeling CelebA with DRAGAN using DCGAN architecture.
|
| 252 |
+
|
| 253 |
+
|
| 254 |
+
Figure 8: Modeling CIFAR-10 using DCGAN architecture.
|
| 255 |
+
|
| 256 |
+
|
| 257 |
+
Figure 9: Latent space walk of the model learned on MNIST using DRAGAN
|
| 258 |
+
|
| 259 |
+
|
| 260 |
+
Figure 10: Latent space walk of the model learned on CelebA using DRAGAN
|
| 261 |
+
|
| 262 |
+
# 5.2 ADDITIONAL EXPERIMENTS
|
| 263 |
+
|
| 264 |
+
# 5.2.1 ONE HIDDEN LAYER NETWORK TO MODEL MNIST
|
| 265 |
+
|
| 266 |
+
We design a simple experiment where $G$ and $D$ are both fully connected networks with just one hidden layer. Vanilla GAN performs poorly even in this simple case and we observe severe mode collapses. In contrast, our algorithm is stable throughout and obtains decent quality samples despite the constrained setup.
|
| 267 |
+
|
| 268 |
+
|
| 269 |
+
Figure 11: One hidden layer network to model MNIST - Inception score plots
|
| 270 |
+
|
| 271 |
+
|
| 272 |
+
Figure 12: One hidden layer network to model MNIST - Samples
|
| 273 |
+
|
| 274 |
+
# 5.2.2 8-GAUSSIANS EXPERIMENT
|
| 275 |
+
|
| 276 |
+
We analyze the performance of WGAN-GP and DRAGAN on the 8-Gaussians dataset. As it can be seen in Figure 13, both of them approximately converge to the real distribution but notice that in the case of WGAN-GP, $D _ { \theta } ( x )$ seems overly constrained in the data space. In contrast, DRAGAN’s discriminator is more flexible.
|
| 277 |
+
|
| 278 |
+
|
| 279 |
+
|
| 280 |
+
Figure 13: Comparing the performance of WGAN-GP and DRAGAN on the 8-Gaussians dataset. Orange is real samples, green is generated samples. The level sets of $D _ { \theta } ( x )$ are shown in the background, with yellow as high and purple as low.
|
| 281 |
+
|
| 282 |
+
# 5.2.3 STABILITY ACROSS DCGAN ARCHITECTURE VARIATIONS
|
| 283 |
+
|
| 284 |
+
DCGAN architecture has been designed following specific guidelines to make it stable (Radford et al., 2015). We restate the suggested rules here.
|
| 285 |
+
|
| 286 |
+
1. Use all-convolutional networks which learn their own spatial downsampling (discriminator)
|
| 287 |
+
or upsampling (generator)
|
| 288 |
+
2. Remove fully connected hidden layers for deeper architectures
|
| 289 |
+
3. Use batch normalization in both the generator and the discriminator
|
| 290 |
+
4. Use ReLU activation in the generator for all layers except the output layer, which uses tanh
|
| 291 |
+
5. Use LeakyReLU activation in the discriminator for all layers
|
| 292 |
+
|
| 293 |
+
We show below that such constraints can be relaxed when using our algorithm and still maintain training stability. Below, we present a series of experiments in which we remove different stabilizing components from the DCGAN architecture and analyze the performance of our algorithm. Specifically, we choose the following four architectures which are difficult to train (in each case, we start with base DCGAN architecture and apply the changes) -
|
| 294 |
+
|
| 295 |
+
• No BN and a constant number of filters in the generator • 4-layer 512-dim ReLU MLP generator • tanh nonlinearities everywhere • tanh nonlinearity in the generator and 4-layer 512-dim L
|
| 296 |
+
|
| 297 |
+
Notice that, in each case, our algorithm is stable while the vanilla GAN training fails. A similar approach is used to demonstrate the stability of training procedures in Arjovsky et al. (2017) and Gulrajani et al. (2017).
|
| 298 |
+
|
| 299 |
+
|
| 300 |
+
(a) tanh activation
|
| 301 |
+
|
| 302 |
+
|
| 303 |
+
(b) FC generator
|
| 304 |
+
|
| 305 |
+
|
| 306 |
+
Figure 14: Comparing performance of DRAGAN and Vanilla GAN training in the hard variations of DCGAN architecture.
|
| 307 |
+
|
| 308 |
+
# 5.2.4 STABILITY ACROSS OBJECTIVE FUNCTIONS
|
| 309 |
+
|
| 310 |
+
Due to space limitations, we only showed plots for two cases in section 3.3. Below we show the results for all five cases.
|
| 311 |
+
|
| 312 |
+
|
| 313 |
+
|
| 314 |
+
|
| 315 |
+
Figure 15: Comparing performance of DRAGAN and Vanilla GAN training using different objective functions.
|
| 316 |
+
|
| 317 |
+
# 5.3 BOGONET DETAILS
|
| 318 |
+
|
| 319 |
+
We used three families of architectures with probabilities - DCGAN (0.6), ResNet (0.2), MLP (0.2). Next, we further parameterized each family to create additional variation. For instance, the DCGAN family can result in networks with or without batch normalization, have LeakyReLU or Tanh nonlinearities. The number and width of filters, latent space dimensionality are some other possible variations in our experiment. Similarly, the number of layers and hidden units in each layer for MLP are chosen randomly. For ResNets, we chose their depth randomly. This creates a set of hard games which test the stability of a given training algorithm.
|
| 320 |
+
|
| 321 |
+
We showed qualitative analysis of the inception score plots in section 3.2 to verify that BogoNet score indeed captures the improvements in stability. Below, we show some examples of how the bounty splits were done. The plots in Figure 14 were scored as (averages are shown in DRAGAN, Vanilla GAN order):
|
| 322 |
+
|
| 323 |
+
A - (5, 0), B - (3.5, 1.5), C – (2.25, 2.75), D – (2, 3)
|
parse/train/ryepFJbA-/ryepFJbA-_content_list.json
ADDED
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| 1 |
+
[
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| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
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"text": "ON CONVERGENCE AND STABILITY OF GANS ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
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|
| 7 |
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| 8 |
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| 9 |
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],
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| 12 |
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"page_idx": 0
|
| 13 |
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},
|
| 14 |
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{
|
| 15 |
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"type": "text",
|
| 16 |
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"text": "Anonymous authors Paper under double-blind review ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
183,
|
| 19 |
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|
| 20 |
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|
| 21 |
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|
| 22 |
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],
|
| 23 |
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"page_idx": 0
|
| 24 |
+
},
|
| 25 |
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{
|
| 26 |
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"type": "text",
|
| 27 |
+
"text": "ABSTRACT ",
|
| 28 |
+
"text_level": 1,
|
| 29 |
+
"bbox": [
|
| 30 |
+
454,
|
| 31 |
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|
| 32 |
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|
| 33 |
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224
|
| 34 |
+
],
|
| 35 |
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"page_idx": 0
|
| 36 |
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},
|
| 37 |
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{
|
| 38 |
+
"type": "text",
|
| 39 |
+
"text": "We propose studying GAN training dynamics as regret minimization, which is in contrast to the popular view that there is consistent minimization of a divergence between real and generated distributions. We analyze the convergence of GAN training from this new point of view to understand why mode collapse happens. We hypothesize the existence of undesirable local equilibria in this non-convex game to be responsible for mode collapse. We observe that these local equilibria often exhibit sharp gradients of the discriminator function around some real data points. We demonstrate that these degenerate local equilibria can be avoided with a gradient penalty scheme called DRAGAN. We show that DRAGAN enables faster training, achieves improved stability with fewer mode collapses, and leads to generator networks with better modeling performance across a variety of architectures and objective functions. ",
|
| 40 |
+
"bbox": [
|
| 41 |
+
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| 42 |
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|
| 43 |
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|
| 44 |
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410
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| 45 |
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],
|
| 46 |
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"page_idx": 0
|
| 47 |
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},
|
| 48 |
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{
|
| 49 |
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"type": "text",
|
| 50 |
+
"text": "1 INTRODUCTION ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
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176,
|
| 54 |
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| 55 |
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|
| 56 |
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| 57 |
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],
|
| 58 |
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"page_idx": 0
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| 59 |
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|
| 60 |
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{
|
| 61 |
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"type": "text",
|
| 62 |
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"text": "Generative modeling involves taking a set of samples drawn from an unknown data generating distribution $P _ { r e a l }$ and finding an estimate $P _ { m o d e l }$ that closely resembles it. Generative adversarial networks (GAN) (Goodfellow et al., 2014) is a powerful framework used for fitting implicit generative models. The basic setup consists of two networks, the generator and the discriminator, playing against each other in a repeated zero-sum game setting. The goal here is to reach an equilibrium where $P _ { r e a l }$ , $P _ { m o d e l }$ are close, and the alternating gradient updates procedure (AGD) is used to achieve this. However, this process is highly unstable and often results in mode collapse (Goodfellow, 2017). This calls for an deeper investigation into training dynamics of GANs. ",
|
| 63 |
+
"bbox": [
|
| 64 |
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174,
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| 65 |
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|
| 66 |
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| 67 |
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| 68 |
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],
|
| 69 |
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"page_idx": 0
|
| 70 |
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},
|
| 71 |
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{
|
| 72 |
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"type": "text",
|
| 73 |
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"text": "In this paper, we propose studying GAN training dynamics as a repeated game in which both the players are using no-regret algorithms (Cesa-Bianchi & Lugosi, 2006) and discuss how AGD 1 falls under this paradigm. In contrast, much of the theory (Goodfellow et al., 2014; Arjovsky & Bottou, 2017) and recent developments (Nowozin et al., 2016; Arjovsky et al., 2017; Gulrajani et al., 2017) are based on the unrealistic assumption that the discriminator is playing optimally (in the function space) at each step and as a result, there is consistent minimization of a divergence between real and generated distributions. This corresponds to at least one player using the best-response algorithm (in the function space), and the resulting game dynamics can be completely different in both these cases (Nisan et al., 2007). Thus, there is a clear disconnect between theoretical arguments used as motivation in recent literature and what actually happens in practice. ",
|
| 74 |
+
"bbox": [
|
| 75 |
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|
| 76 |
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|
| 77 |
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| 78 |
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|
| 79 |
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],
|
| 80 |
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|
| 81 |
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},
|
| 82 |
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{
|
| 83 |
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"type": "text",
|
| 84 |
+
"text": "We would like to point out that the latter view can still be useful for reasoning about the asymptotic equilibrium situation but we argue that regret minimization is the more appropriate way to think about GAN training dynamics. So, we analyze the convergence of GAN training from this new point of view to understand why mode collapse happens. We start with a short analysis of the artificial convex-concave case of the GAN game in section 2.2. This setting has a unique solution and guaranteed convergence (of averaged iterates) using no-regret algorithms can be shown with standard arguments from game theory literature. Here, we make explicit, the critical (previously not widely known) connection between AGD used in GAN training and regret minimization. This immediately yields a novel proof for the asymptotic convergence of GAN training, in the non-parametric limit. Prior to our work, such a result (Goodfellow et al., 2014) required a strong assumption that the discriminator is optimal at each step. ",
|
| 85 |
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"bbox": [
|
| 86 |
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| 87 |
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| 89 |
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| 90 |
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],
|
| 91 |
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"page_idx": 0
|
| 92 |
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},
|
| 93 |
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{
|
| 94 |
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"type": "text",
|
| 95 |
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"text": "However, these convergence results do not hold when the game objective function is non-convex, which is the practical case when deep neural networks are used. In non-convex games, global regret minimization and equilibrium computation are computationally hard in general. Recent gametheoretic literature indicates that AGD can end up cycling (Mertikopoulos et al., 2017) or converging to a (potentially bad) local equilibrium, under some conditions (Hazan et al., 2017). We hypothesize these to be the reasons for cycling and mode collapse observed during GAN training, respectively (section 2.3). In this work, we do not explore the cycling issue but focus our attention on the mode collapse problem. In contrast to our hypothesis, the prevalent view of mode collapse and instability (Arjovsky & Bottou, 2017) is that it results from attempting to minimize a strong divergence during training. However, as we argued earlier, GAN training with AGD does not consistently minimize a divergence and therefore, such a theory is not suitable to discuss convergence or to address the stability issue. ",
|
| 96 |
+
"bbox": [
|
| 97 |
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|
| 98 |
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|
| 99 |
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|
| 100 |
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|
| 101 |
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],
|
| 102 |
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"page_idx": 1
|
| 103 |
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},
|
| 104 |
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{
|
| 105 |
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"type": "text",
|
| 106 |
+
"text": "Next, if mode collapse is indeed the result of an undesirable local equilibrium, a natural question then is how we can avoid it? We make a simple observation that, in the GAN game, mode collapse situations are often accompanied by sharp gradients of the discriminator function around some real data points (section 2.4). Therefore, a simple strategy to mitigate mode collapse is to regularize the discriminator so as to constrain its gradients in the ambient data space. We demonstrate that this improves the stability using a toy experiment with one hidden layer neural networks. This gives rise to a new explanation for why WGAN and gradient penalties might be improving the stability of GAN training – they are mitigating the mode collapse problem by keeping the gradients of the discriminator function small in data space. From this motivation, we propose a training algorithm involving a novel gradient penalty scheme called DRAGAN (Deep Regret Analytic Generative Adversarial Networks) which enables faster training, achieves improved stability and modeling performance (over WGAN-GP (Gulrajani et al., 2017) which is the state-of-the-art stable training procedure) across a variety of architectures and objective functions. ",
|
| 107 |
+
"bbox": [
|
| 108 |
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|
| 109 |
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|
| 110 |
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|
| 111 |
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| 112 |
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],
|
| 113 |
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"page_idx": 1
|
| 114 |
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},
|
| 115 |
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{
|
| 116 |
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"type": "text",
|
| 117 |
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"text": "Below, we provide a short literature review. Several recent works focus on stabilizing the training of GANs. While some solutions (Radford et al., 2015; Salimans et al., 2016) require the usage of specific architectures (or) modeling objectives, some (Che et al., 2016; Zhao et al., 2016) significantly deviate from the original GAN framework. Other promising works in this direction (Metz et al., 2016; Arjovsky et al., 2017; Qi, 2017; Gulrajani et al., 2017) impose a significant computational overhead. Thus, a fast and versatile method for consistent stable training of GANs is still missing in the literature. Our work is aimed at addressing this. ",
|
| 118 |
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|
| 119 |
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| 120 |
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| 121 |
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| 122 |
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| 123 |
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],
|
| 124 |
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"page_idx": 1
|
| 125 |
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},
|
| 126 |
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{
|
| 127 |
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"type": "text",
|
| 128 |
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"text": "To summarize, our contributions are as follows: ",
|
| 129 |
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|
| 130 |
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| 131 |
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| 132 |
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| 133 |
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| 134 |
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],
|
| 135 |
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|
| 136 |
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},
|
| 137 |
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{
|
| 138 |
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"type": "text",
|
| 139 |
+
"text": "• We propose a new way of reasoning about the GAN training dynamics - by viewing AGD as regret minimization. \n• We provide a novel proof for the asymptotic convergence of GAN training in the nonparametric limit and it does not require the discriminator to be optimal at each step. \n• We discuss how AGD can converge to a potentially bad local equilibrium in non-convex games and hypothesize this to be responsible for mode collapse during GAN training. \n• We characterize mode collapse situations with sharp gradients of the discriminator function around some real data points. \n• A novel gradient penalty scheme called DRAGAN is introduced based on this observation and we demonstrate that it mitigates the mode collapse issue. ",
|
| 140 |
+
"bbox": [
|
| 141 |
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|
| 142 |
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| 143 |
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| 144 |
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|
| 145 |
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],
|
| 146 |
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"page_idx": 1
|
| 147 |
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},
|
| 148 |
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{
|
| 149 |
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"type": "text",
|
| 150 |
+
"text": "2 THEORETICAL ANALYSIS OF GAN TRAINING DYNAMICS ",
|
| 151 |
+
"text_level": 1,
|
| 152 |
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"bbox": [
|
| 153 |
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| 154 |
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| 155 |
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| 156 |
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|
| 157 |
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],
|
| 158 |
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"page_idx": 1
|
| 159 |
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},
|
| 160 |
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{
|
| 161 |
+
"type": "text",
|
| 162 |
+
"text": "We start with a brief description of the GAN framework (section 2.1). We discuss guaranteed convergence in the artificial convex-concave case using no-regret algorithms, and make a critical connection between GAN training process (AGD) and regret minimization (section 2.2). This immediately yields a novel proof for the asymptotic convergence of GAN training in the nonparametric limit. Then, we consider the practical non-convex case and discuss how AGD can converge to a potentially bad local equilibrium here (section 2.3). We characterize mode collapse situations with sharp gradients of the discriminator function around real samples and this provides an effective strategy to avoid them. This naturally leads to the introduction of our gradient penalty scheme DRAGAN (section 2.4). We end with a discussion and comparison with other gradient penalties in the literature (section 2.5). ",
|
| 163 |
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"bbox": [
|
| 164 |
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| 165 |
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| 168 |
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|
| 169 |
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"page_idx": 1
|
| 170 |
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|
| 171 |
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|
| 172 |
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"type": "text",
|
| 173 |
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"text": "",
|
| 174 |
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"bbox": [
|
| 175 |
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| 180 |
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|
| 181 |
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},
|
| 182 |
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{
|
| 183 |
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"type": "text",
|
| 184 |
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"text": "2.1 BACKGROUND ",
|
| 185 |
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"text_level": 1,
|
| 186 |
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| 192 |
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| 193 |
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|
| 194 |
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{
|
| 195 |
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"type": "text",
|
| 196 |
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"text": "The GAN framework can be viewed as a repeated zero-sum game, consisting of two players - the generator, which produces synthetic data given some noise source and the discriminator, which is trained to distinguish generator’s samples from the real data. The generator model G is parameterized by $\\phi$ , takes a noise vector $\\mathbf { z }$ as input, and produces a synthetic sample $G _ { \\phi } ( \\mathbf { z } )$ . The discriminator model $\\mathrm { D }$ is parameterized by $\\theta$ , takes a sample $\\mathbf { x }$ as input and computes $D _ { \\theta } ( \\mathbf { x } )$ , which can be interpreted as the probability that $\\mathbf { x }$ is real. ",
|
| 197 |
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"page_idx": 2
|
| 204 |
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},
|
| 205 |
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{
|
| 206 |
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"type": "text",
|
| 207 |
+
"text": "The models $\\mathbf { G }$ , D can be selected from any arbitrary class of functions – in practice, GANs typical rely on deep networks for both. Their cost functions are defined as ",
|
| 208 |
+
"bbox": [
|
| 209 |
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| 210 |
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| 211 |
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| 213 |
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],
|
| 214 |
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"page_idx": 2
|
| 215 |
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},
|
| 216 |
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{
|
| 217 |
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"type": "equation",
|
| 218 |
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"img_path": "images/cea4737c7cbdc5c3b03df186a1a41ba8ec5d5c09f4f9072b415b437dc26447f7.jpg",
|
| 219 |
+
"text": "$$\n\\begin{array} { r l } & { J ^ { ( D ) } ( \\phi , \\theta ) : = - \\mathbb { E } _ { x \\sim p _ { r e a l } } \\log D _ { \\theta } ( x ) - \\mathbb { E } _ { \\mathbf { z } } \\log ( 1 - D _ { \\theta } ( G _ { \\phi } ( z ) ) ) , \\mathrm { ~ a r ~ } } \\\\ & { J ^ { ( G ) } ( \\phi , \\theta ) : = - J ^ { ( D ) } ( \\phi , \\theta ) } \\end{array}\n$$",
|
| 220 |
+
"text_format": "latex",
|
| 221 |
+
"bbox": [
|
| 222 |
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| 223 |
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| 224 |
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|
| 225 |
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|
| 226 |
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],
|
| 227 |
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"page_idx": 2
|
| 228 |
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},
|
| 229 |
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{
|
| 230 |
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"type": "text",
|
| 231 |
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"text": "And the complete game can be specified as - ",
|
| 232 |
+
"bbox": [
|
| 233 |
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| 234 |
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| 235 |
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| 236 |
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| 237 |
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],
|
| 238 |
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"page_idx": 2
|
| 239 |
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},
|
| 240 |
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{
|
| 241 |
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"type": "equation",
|
| 242 |
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"img_path": "images/4bea38eca5a6a131257aafbbcb1a724cb6d8b4ab1f31233bbc203efed5a41eac.jpg",
|
| 243 |
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"text": "$$\n\\underset { \\phi } { \\operatorname* { m i n } } \\underset { \\theta } { \\operatorname* { m a x } } \\left\\{ J ( \\phi , \\theta ) = \\mathbb { E } _ { x \\sim p _ { r e a l } } \\log D _ { \\theta } ( x ) + \\mathbb { E } _ { \\mathbf { z } } \\log ( 1 - D _ { \\theta } ( G _ { \\phi } ( z ) ) ) \\right\\}\n$$",
|
| 244 |
+
"text_format": "latex",
|
| 245 |
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"bbox": [
|
| 246 |
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|
| 247 |
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| 248 |
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| 249 |
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|
| 250 |
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],
|
| 251 |
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"page_idx": 2
|
| 252 |
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},
|
| 253 |
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{
|
| 254 |
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"type": "text",
|
| 255 |
+
"text": "The generator distribution $P _ { m o d e l }$ asymptotically converges to the real distribution $P _ { r e a l }$ if updates are made in the function space and the discriminator is optimal at each step (Goodfellow et al., 2014). ",
|
| 256 |
+
"bbox": [
|
| 257 |
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| 258 |
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|
| 259 |
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| 260 |
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| 261 |
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],
|
| 262 |
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"page_idx": 2
|
| 263 |
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},
|
| 264 |
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{
|
| 265 |
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"type": "text",
|
| 266 |
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"text": "2.2 CONVEX-CONCAVE CASE AND NO-REGRET ALGORITHMS ",
|
| 267 |
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"bbox": [
|
| 268 |
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| 269 |
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| 271 |
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| 272 |
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|
| 273 |
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"page_idx": 2
|
| 274 |
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},
|
| 275 |
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{
|
| 276 |
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"type": "text",
|
| 277 |
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"text": "According to Sion’s theorem (Sion, 1958), if $\\Phi \\subset \\mathbb { R } ^ { m }$ , $\\Theta \\subset \\mathbb { R } ^ { n }$ such that they are compact and convex sets, and the function $J : \\Phi \\times \\Theta \\mathbb { R }$ is convex in its first argument and concave in its second, then we have - ",
|
| 278 |
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| 284 |
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|
| 285 |
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},
|
| 286 |
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{
|
| 287 |
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"type": "equation",
|
| 288 |
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"img_path": "images/d98535379518beea655b8a76763eaeae867f4c00e57b306e3d220f109c5e3abb.jpg",
|
| 289 |
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"text": "$$\n\\operatorname* { m i n } _ { \\phi \\in \\Phi } \\operatorname* { m a x } _ { \\theta \\in \\Theta } J ( \\phi , \\theta ) = \\operatorname* { m a x } _ { \\theta \\in \\Theta } \\operatorname* { m i n } _ { \\phi \\in \\Phi } J ( \\phi , \\theta )\n$$",
|
| 290 |
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"text_format": "latex",
|
| 291 |
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"bbox": [
|
| 292 |
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| 293 |
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| 295 |
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| 296 |
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| 297 |
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|
| 298 |
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},
|
| 299 |
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{
|
| 300 |
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"type": "text",
|
| 301 |
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"text": "That is, an equilibrium is guaranteed to exist in this setting where players’ payoffs correspond to the unique value of the game (Neumann, 1928). ",
|
| 302 |
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"text": "A natural question then is how we can find such an equilibrium. A simple procedure that players can use is best-response algorithms (BRD). In each round, best-responding players play their optimal strategy given their opponent’s current strategy. Despite its simplicity, BRD are often computationally intractable and they don’t lead to convergence even in simple games. In contrast, a technique that is both efficient and provably works is regret minimization. If both players update their parameters using no-regret algorithms, then it is easy to show that their averaged iterates will converge to an equilibrium pair (Nisan et al., 2007). Let us first define no-regret algorithms. ",
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"text": "Definition 2.1 (No-regret algorithm). Given a sequence of convex loss functions $L _ { 1 } , L _ { 2 } , \\dots :$ $K \\mathbb { R }$ , an algorithm that selects a sequence of $k _ { t }$ ’s, each of which may only depend on previously observed $L _ { 1 } , \\dots , L _ { t - 1 }$ , is said to have no regret if $\\begin{array} { r } { \\frac { R ( T ) } { T } = o ( 1 ) } \\end{array}$ , where we define ",
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"text": "$$\n\\begin{array} { r } { R ( T ) : = \\sum _ { t = 1 } ^ { T } L _ { t } ( k _ { t } ) - \\operatorname* { m i n } _ { k \\in K } \\sum _ { t = 1 } ^ { T } L _ { t } ( k ) } \\end{array}\n$$",
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"text": "We can apply no-regret learning to our problem of equilibrium finding in the GAN game $J ( \\cdot , \\cdot )$ as follows. The generator imagines the function $J ( \\cdot , \\theta _ { t } )$ as its loss function on round $t$ , and similarly the discriminator imagines computes the average itera $- J ( \\phi _ { t } , \\cdot )$ unctiand $t$ $T$ oun. If f play, each playeris the equilibrium $\\begin{array} { r } { \\stackrel { \\cdot } { \\phi } _ { T } : = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\phi _ { \\underline { { t } } } } \\end{array}$ $\\begin{array} { r } { \\bar { \\theta } _ { T } : = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\theta _ { t } } \\end{array}$ $V ^ { * }$ value of the game, and the players suffer regret $R _ { 1 } ( T )$ and $\\bar { R _ { 2 } ( T ) }$ respectively, then one can show using standard arguments (Freund & Schapire, 1999) that - ",
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"text": "$$\n\\begin{array} { r } { V ^ { * } - \\frac { R _ { 2 } ( T ) } { T } \\le \\operatorname* { m a x } _ { \\theta \\in \\Theta } J \\big ( \\bar { \\phi } _ { T } , \\theta \\big ) - \\frac { R _ { 2 } ( T ) } { T } \\le \\operatorname* { m i n } _ { \\phi \\in \\Phi } J \\big ( \\phi , \\bar { \\theta } _ { T } \\big ) + \\frac { R _ { 1 } ( T ) } { T } \\le V ^ { * } + \\frac { R _ { 1 } ( T ) } { T } . } \\end{array}\n$$",
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"text": "In other words, $\\bar { \\theta } _ { T }$ and $\\bar { \\phi } _ { T }$ are \"almost optimal\" solutions to the game, where the \"almost\" approximation factor is given by the average regret terms R1(T )+R2(T ) . Under the no-regret condition, the former will vanish, and hence we can guarantee convergence in the limit. Next, we define a popular family of no-regret algorithms. ",
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"text": "Definition 2.2 (Follow The Regularized Leader). FTRL (Hazan et al., 2016) selects $k _ { t }$ on round $t$ by solving for arg $\\begin{array} { r } { \\operatorname* { m i n } _ { k \\in \\boldsymbol { K } } \\{ \\bar { \\sum } _ { s = 1 } ^ { t - 1 } L _ { s } ( k ) + \\frac { 1 } { \\eta } \\Omega ( k ) \\} } \\end{array}$ , where $\\Omega ( \\cdot )$ is some convex regularization function and $\\eta$ is a learning rate. ",
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"text": "Remark: Roughly speaking, if you select the regularization as $\\begin{array} { r } { \\Omega ( \\cdot ) = \\frac { 1 } { 2 } \\| \\cdot \\| ^ { 2 } } \\end{array}$ , then FTRL becomes the well-known online gradient descent or OGD (Zinkevich, 2003). Ignoring the case of constraint violations, OGD can be written in a simple iterative form: $k _ { t } = k _ { t - 1 } - \\eta \\nabla L _ { t - 1 } ( k _ { t - 1 } )$ . ",
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"text": "The typical GAN training procedure using alternating gradient updates (or simultaneous gradient updates) is almost this - both the players applying online gradient descent. Notice that the $\\operatorname* { m i n } / \\operatorname* { m a x }$ objective function in GANs involves a stochastic component, with two randomized inputs given on each round, $x$ and $z$ which are sampled from the data distribution and a standard multivariate normal, respectively. Let us write $J _ { x , z } ( \\phi , \\theta \\bar ) : = \\log { D _ { \\theta } ( x ) } + \\log ( 1 - D _ { \\theta } ( G _ { \\phi } ( z ) ) )$ . Taking expectations with respect to $\\mathbf { x }$ and $\\mathbf { z }$ , we define the full (non-stochastic) game as $J ( \\phi , \\theta ) = \\mathbb { E } _ { \\mathbf { x } , \\mathbf { z } } \\left[ J _ { x , z } ( \\phi , \\theta ) \\right]$ . But the above online training procedure is still valid with stochastic inputs. That is, the equilibrium computation would proceed similarly, where on each round we sample $x _ { t }$ and $z _ { t }$ , and follow the updates ",
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"text": "$$\n\\phi _ { t + 1 } \\phi _ { t } - \\eta \\nabla _ { \\phi } J _ { x _ { t } , z _ { t } } ( \\phi _ { t } , \\theta _ { t } ) . \\quad \\mathrm { a n d } \\quad \\theta _ { t + 1 } \\theta _ { t } + \\eta ^ { ' } \\nabla _ { \\theta } J _ { x _ { t } , z _ { t } } ( \\phi _ { t } , \\theta _ { t } )\n$$",
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"text": "On a side note, a benefit of this stochastic perspective is that we can get a generalization bound on the mean parameters $\\bar { \\phi } _ { T }$ after $T$ rounds of optimization. The celebrated \"online-to-batch conversion\" (Cesa-Bianchi et al., 2004) implies that $\\mathbb { E } _ { \\mathbf { x } , \\mathbf { z } } \\big [ J _ { x , z } ( \\bar { \\phi } _ { T } , \\theta ) \\big ]$ , for any $\\theta$ , is no more than the optimal value $\\mathbb { E } _ { \\mathbf { x } , \\mathbf { z } } [ J _ { x , z } ( \\phi ^ { * } , \\theta ) ]$ plus an \"estimation error\" bounded by $\\mathbb { E } \\left[ { \\frac { R _ { 1 } ( T ) + R _ { 2 } ( T ) } { T } } \\right]$ , where the expectation is taken with respect to the sequence of samples observed along the way, and any randomness in the algorithm. Analogously, this applies to $\\bar { \\theta } _ { T } ^ { \\star }$ as well. A limitation of this result, however, is that it requires a fresh sample $x _ { t }$ to be used on every round. ",
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"text": "To summarize, we discussed in this subsection about how the artificial convex-concave case is easy to solve through regret minimization. While this is a standard result in game theory and online learning literature, it is not widely known in the GAN literature. For instance, Salimans et al. (2016) and Goodfellow (2017) discuss a toy game which is convex-concave and show cycling behavior. But, the simple solution in that case is to just average the iterates. Further, we made explicit, the critical connection between regret minimization and alternating gradient updates procedure used for GAN training. Now, Goodfellow et al. (2014) argue that, if $G$ and $D$ have enough capacity (in the non-parametric limit) and updates are made in the function space, then the GAN game can be considered convex-concave. Thus, our analysis based on regret minimization immediately yields a novel proof for the asymptotic convergence of GANs, without requiring that the discriminator be optimal at each step. ",
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"text": "Moreover, the connection between regret minimization and GAN training process gives a novel way to reason about its dynamics. In contrast, the popular view of GAN training as consistently minimizing a divergence arises if the discriminator uses BRD (in the function space) and thus, it has little to do with the actual training process of GANs. As a result, this calls into question the motivation behind many recent developments like WGAN and gradient penalties among others, which improve the training stability of GANs. In the next subsection, we discuss the practical non-convex case and why training instability arises. This provides the necessary ideas to investigate mode collapse from our new perspective. ",
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"text": "2.3 NON-CONVEX CASE AND LOCAL EQUILIBRIA ",
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"text": "In practice, we choose $G$ , $D$ to be deep neural networks and the function $J ( \\phi , \\theta )$ need not be convexconcave anymore. The nice properties we had in the convex-concave case like the existence of a unique solution and guaranteed convergence through regret minimization no longer hold. In fact, regret minimization and equilibrium computation are computationally hard in general non-convex settings. However, analogous to the case of non-convex optimization (also intractable) where we focus on finding local minima, we can look for tractable solution concepts in non-convex games. ",
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"text": "Recent work by Hazan et al. (2017) introduces the notion of local regret and shows that if both the players use a smoothed variant of OGD to minimize this quantity, then the non-convex game converges to some form of local equilibrium, under mild assumptions. The usual training procedure of GANs (AGD) corresponds to using a window size of 1 in their formulation. Thus, GAN training will eventually converge (approximately) to a local equilibrium which is described below or the updates will cycle. We leave it to future works to explore the equally important cycling issue and focus here on the former case. ",
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"text": "Definition 2.3 (Local Equilibrium). A pair $( \\phi ^ { * } , \\theta ^ { * } )$ is called an $\\epsilon$ -approximate local equilibrium if it holds that ",
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"text": "$$\n\\begin{array} { l } { { \\forall \\phi ^ { ' } , \\vert \\vert \\phi ^ { ' } - \\phi ^ { * } \\vert \\vert \\le \\eta : J ( \\phi ^ { * } , \\theta ^ { * } ) \\le J ( \\phi ^ { ' } , \\theta ^ { * } ) + \\epsilon } } \\\\ { { \\forall \\theta ^ { ' } , \\vert \\vert \\theta ^ { ' } - \\theta ^ { * } \\vert \\vert \\le \\eta : J ( \\phi ^ { * } , \\theta ^ { * } ) \\ge J ( \\phi ^ { * } , \\theta ^ { ' } ) - \\epsilon } } \\end{array}\n$$",
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"text": "That is, in a local equilibrium, both the players do not have much of an incentive to switch to any other strategy within a small neighborhood of their current strategies. Now, we turn our attention to the mode collapse issue which poses a significant challenge to the GAN training process. The training is said to have resulted in mode collapse if the generator ends up mapping multiple z vectors to the same output $\\mathbf { x }$ , which is assigned a high probability of being real by the discriminator (Goodfellow, 2017). We hypothesize this to be the result of the game converging to bad local equilibria. ",
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"text": "The prevalent view of mode collapse and instability in GAN training (Arjovsky & Bottou, 2017) is that it is caused due to the supports of real and model distributions being disjoint or lying on low-dimensional manifolds. The argument is that this would result in strong distance measures like KL-divergence or JS-divergence getting maxed out, and the generator cannot get useful gradients to learn. In fact, this is the motivation for the introduction of WGAN (Arjovsky et al., 2017). But, as we argued earlier, GAN training does not consistently minimize a divergence as that would require using intractable best-response algorithms. Hence, such a theory is not suitable to discuss convergence or to address the instability of GAN training. Our new view of GAN training process as regret minimization is closer to what is used in practice and provides an alternate explanation for mode collapse - the existence of undesirable local equilibria. The natural question now is how we can avoid them? ",
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"text": "2.4 MODE COLLAPSE AND GRADIENT PENALTIES ",
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"text": "The problem of dealing with multiple equilibria in games and how to avoid undesirable ones is an important question in algorithmic game theory (Nisan et al., 2007). In this work, we constrain ourselves to the GAN game and aim to characterize the undesirable local equilibria (mode collapse) in an effort to avoid them. In this direction, after empirically studying multiple mode collapse cases, we found that it is often accompanied by the discriminator function having sharp gradients around some real data points (See Figure $1 ^ { 2 }$ ). This intuitively makes sense from the definition of mode collapse discussed earlier. Such sharp gradients encourage the generator to map multiple $z$ vectors to a single output $x$ and lead the game towards a degenerate equilibrium. Now, a simple strategy to mitigate this failure case would be to regularize the discriminator using the following penalty - ",
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"text": "$$\n\\lambda \\cdot \\mathbb { E } _ { x \\sim P _ { r e a l } , \\delta \\sim N _ { d } ( 0 , c I ) } \\big [ \\| \\nabla _ { \\mathbf { x } } D _ { \\theta } ( x + \\delta ) \\| ^ { 2 } \\big ]\n$$",
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"text": "This strategy indeed improves the stability of GAN training. We show the results of a toy experiment with one hidden layer neural networks in Figure 2 and Figure 3 to demonstrate this. This partly explains the success of WGAN and gradient penalties in the recent literature (Gulrajani et al., 2017; Qi, 2017), and why they improve the training stability of GANs, despite being motivated by reasoning based on unrealistic assumptions. However, we noticed that this scheme in its current form can be brittle and if over-penalized, the discriminator can end up assigning both a real point $x$ and noise $x + \\delta$ , the same probability of being real. Thus, a better choice of penalty is - ",
|
| 589 |
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"type": "equation",
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"img_path": "images/01755ede9cddf95c4cbca79977adab2fcb8e69b86894c92afdb8d8a71e38531a.jpg",
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"text": "$$\n\\lambda \\cdot \\mathbb { E } _ { x \\sim P _ { r e a l } , \\delta \\sim N _ { d } ( 0 , c I ) } \\big [ \\operatorname* { m a x } \\big ( 0 , \\| \\nabla _ { \\mathbf { x } } D _ { \\theta } ( x + \\delta ) \\| ^ { 2 } - k \\big ) \\big ]\n$$",
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"type": "text",
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| 612 |
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"text": "Finally, due to practical optimization considerations (this has also been observed in Gulrajani et al. \n(2017)), we instead use the penalty shown below in all our experiments. ",
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"img_path": "images/5011db7721b8181f031a8d97854413ed19cd0345f6fa658805985acc1570aa62.jpg",
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"text": "$$\n\\lambda \\cdot \\mathbb { E } _ { x \\sim P _ { r e a l } , \\delta \\sim N _ { d } ( 0 , c I ) } \\big [ \\| \\nabla _ { \\mathbf { x } } D _ { \\theta } ( x + \\delta ) \\| - k \\big ] ^ { 2 }\n$$",
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"img_path": "images/60d0ba3aafa834ef689a9a664b53b2eeea17dedc1dbc3bedf725f1116ca44757.jpg",
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"image_caption": [
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| 638 |
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"Figure 1: One hidden layer networks as $G$ and $D$ (MNIST). On the left, we plot inception score against time for vanilla GAN training and on the right, we plot the squared norm of discriminator’s gradients around real data points for the same experiment. Notice how this quantity changes before, during and after mode collapse events. "
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"img_path": "images/d65b27c7cb6a60e92836c2b33369e5c1b231d2b25768acc523e3c0a7f3b42736.jpg",
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"image_caption": [
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| 653 |
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"Figure 2: One hidden layer networks as $G$ and $D$ (MNIST). On the left, losses for both the players are shown for vanilla GAN training and on the right, we added a regularization term to penalize the gradients of $D ( x )$ around real data points. Notice the improved stability. "
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],
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"type": "text",
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"text": "This still works as long as small perturbations of real data, $x + \\delta$ are likely to lie off the data-manifold, which is true in the case of image domain and some other settings. Because, in these cases, we do want our discriminator to assign different probabilities of being real to training data and noisy samples. We caution the practitioners to keep this important point in mind while making their choice of penalty. All of the above schemes have the same effect of constraining the norm of discriminator’s gradients around real points to be small and can therefore, mitigate the mode collapse situation. We refer to GAN training using these penalty schemes or heuristics as the DRAGAN algorithm. ",
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| 667 |
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},
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"type": "text",
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"text": "Additional details: ",
|
| 678 |
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"type": "text",
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"text": "• We use the vanilla GAN objective in our experiments, but our penalty improves stability using other objective functions as well. This is demonstrated in section 3.3. \n• The penalty scheme used in our experiments is the one shown in equation 1. \n• We use small pixel-level noise but it is possible to find better ways of imposing this penalty. However, this exploration is beyond the scope of our paper. \n• The optimal configuration of the hyperparameters for DRAGAN depends on the architecture, dataset and data domain. We set them to be $\\lambda \\sim 1 0$ , $k = 1$ and $c \\sim 1 0$ in most of our experiments. ",
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"type": "text",
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"text": "2.5 COUPLED VS LOCAL PENALTIES ",
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"text_level": 1,
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"text": "Several recent works have also proposed regularization schemes which constrain the discriminator’s gradients in the ambient data space, so as to improve the stability of GAN training. Despite being from different motivations, WGAN-GP and LS-GAN are closely related approaches to ours. First, we show that these two approaches are very similar, which is not widely known in the literature. Qi (2017) introduced LS-GAN with the idea of maintaining a margin between losses assigned to real and fake samples. Further, they also impose Lipschitz constraint on $D$ and the two conditions together result in a situation where the following holds for any real and fake sample pair (roughly) - ",
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"type": "image",
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"img_path": "images/2d12f09eef44f72791080e64a67614c1e4c57e796445fdcaea72ff7b773c9ac3.jpg",
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"image_caption": [
|
| 724 |
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"Figure 3: One hidden layer networks as $G$ and $D$ (MNIST). On the left, inception score plot is shown for vanilla GAN training and on the right, we added a regularization term to penalize the gradients of $D ( x )$ around real data points. Notice how mode collapse is mitigated. "
|
| 725 |
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],
|
| 726 |
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"image_footnote": [],
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"type": "text",
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"text": "",
|
| 738 |
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"bbox": [
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"type": "equation",
|
| 748 |
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"img_path": "images/3c7b0e13b16cde3ef7924dcb6b1f55eae73facdf21996455421d731b35c45962.jpg",
|
| 749 |
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"text": "$$\nD _ { \\theta } ( x ) - D _ { \\theta } ( G _ { \\phi } ( z ) ) \\approx | | x , G _ { \\phi } ( z ) | |\n$$",
|
| 750 |
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"text_format": "latex",
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| 751 |
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"bbox": [
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| 759 |
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| 760 |
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"type": "text",
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| 761 |
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"text": "The authors argue that the resulting discriminator function would have non-vanishing gradients almost everywhere between real and fake samples (section 6 of Qi (2017)). Next, Gulrajani et al. (2017) proposed an extension to address various shortcomings of the original WGAN and they impose the following condition on $D$ - ",
|
| 762 |
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"bbox": [
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"type": "equation",
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"img_path": "images/344979f5412e152eeb1b997d7ef93f733d76c15c8ce36073bbd53c6003f780fe.jpg",
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| 773 |
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"text": "$$\n| | \\nabla _ { x } D _ { \\theta } ( \\hat { x } ) | | \\approx 1\n$$",
|
| 774 |
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"text_format": "latex",
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| 775 |
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"bbox": [
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| 784 |
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"type": "text",
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| 785 |
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"text": "where $\\hat { x } = ( \\epsilon ) x + ( 1 - \\epsilon ) G _ { \\phi } ( z )$ is some point on the line between a real and a fake sample, both chosen independently at random. This leads to $D$ having norm-1 gradients almost everywhere between real and fake samples. Notice that this behavior is very similar to that of LS-GAN’s discriminator function. Thus, WGAN-GP is a slight variation of the original LS-GAN algorithm and we refer to these methods as “coupled penalties”. ",
|
| 786 |
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"bbox": [
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"type": "text",
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| 796 |
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"text": "On a side note, we also want to point out that WGAN-GP’s penalty doesn’t actually follow from KR-duality as claimed in their paper. By Lemma 1 of Gulrajani et al. (2017), the optimal discriminator $D ^ { * }$ will have norm-1 gradients (almost everywhere) only between those $x$ and $G _ { \\phi } ( z )$ pairs which are sampled from the optimal coupling or joint distribution $\\pi ^ { * }$ . Therefore, there is no basis for WGAN-GP’s penalty (equation 3) where arbitrary pairs of real and fake samples are used. This fact adds more credence to our theory regarding why gradient penalties might be mitigating mode collapse. ",
|
| 797 |
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"bbox": [
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| 804 |
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| 805 |
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|
| 806 |
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"type": "text",
|
| 807 |
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"text": "The most important distinction between coupled penalties and our methods is that we only impose gradient constraints in local regions around real samples. We refer to these penalty schemes as “local penalties”. Coupled penalties impose gradient constraints between real and generated samples and we point out some potential issues that arise from this: ",
|
| 808 |
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"bbox": [
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| 815 |
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| 816 |
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|
| 817 |
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"type": "text",
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| 818 |
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"text": "• With adversarial training finding applications beyond fitting implicit generative models, penalties which depend on generated samples can be prohibitive. \n• The resulting class of functions when coupled penalties are used will be highly restricted compared to our method and this affects modeling performance. We refer the reader to Figure 4 and appendix section 5.2.2 to see this effect. \n• Our algorithm works with AGD, while WGAN-GP needs multiple inner iterations to optimize D. This is because the generated samples can be anywhere in the data space and they change from one iteration to the next. In contrast, we consistently regularize $D _ { \\theta } ( x )$ only along the real data manifold. ",
|
| 819 |
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| 829 |
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"text": "To conclude, appropriate constraining of the discriminator’s gradients can mitigate mode collapse but we should be careful so that it doesn’t have any negative effects. We pointed out some issues with coupled penalties and how local penalties can help. We refer the reader to section 3 for further experimental results. ",
|
| 830 |
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},
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{
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| 839 |
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"type": "image",
|
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"img_path": "images/7599d4f33b3ef214e704679a9c30d2442686de23f3f908d8f6d8f1b4dd024757.jpg",
|
| 841 |
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"image_caption": [
|
| 842 |
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"Figure 4: Swissroll experiment (different phases of training) - Vanilla GAN (top), WGAN-GP (middle), and DRAGAN (bottom). Real samples are marked orange and generated samples are green. Level sets of $D _ { \\theta } ( x )$ are shown in the background where yellow is high and purple is low. "
|
| 843 |
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],
|
| 844 |
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| 845 |
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},
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| 853 |
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{
|
| 854 |
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"type": "text",
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| 855 |
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"text": "3 EXPERIMENTAL RESULTS ",
|
| 856 |
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"text_level": 1,
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| 857 |
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},
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|
| 866 |
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"type": "text",
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| 867 |
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"text": "In section 3.1, we compare the modeling performance of our algorithm against vanilla GAN and WGAN variants in the standard DCGAN/CIFAR-10 setup. Section 3.2 demonstrates DRAGAN’s improved stability across a variety of architectures. In section 3.3, we show that our method also works with other objective functions. Appendix contains samples for inspection, some of the missing plots and additional results. Throughout, we use inception score (Salimans et al., 2016) which is a well-studied and reliable metric in the literature, and sample quality to measure the performance. ",
|
| 868 |
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},
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| 877 |
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"type": "text",
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| 878 |
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"text": "3.1 INCEPTION SCORES FOR CIFAR-10 USING DCGAN ARCHITECTURE",
|
| 879 |
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"text_level": 1,
|
| 880 |
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},
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"type": "text",
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| 890 |
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"text": "DCGAN is a family of architectures designed to perform well with the vanilla training procedure. They are ubiquitous in the GAN literature owing to the instability of vanilla GAN in general settings. We use this architecture to model CIFAR-10 and compare against vanilla GAN, WGAN and WGANGP. As WGANs need 5 discriminator iterations for every generator iteration, comparing the modeling performance can be tricky. To address this, we report two scores for vanilla GAN and DRAGAN - one using the same number of generator iterations as WGANs and one using the same number of discriminator iterations. The results are shown in Figure 5 and samples are included in the appendix (Figure 8). Notice that DRAGAN beats WGAN variants in both the configurations, while vanilla GAN is only slightly better. A key point to note here is that our algorithm is fast compared to WGANs, so in practice, the performance will be closer to the DRAGANd case. In the next section, we will show that if we move away from this specific architecture family, vanilla GAN training can become highly unstable and that DRAGAN penalty mitigates this issue. ",
|
| 891 |
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| 898 |
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},
|
| 899 |
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{
|
| 900 |
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"type": "text",
|
| 901 |
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"text": "3.2 MEASURING STABILITY AND PERFORMANCE ACROSS ARCHITECTURES ",
|
| 902 |
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"text_level": 1,
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| 903 |
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},
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| 911 |
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{
|
| 912 |
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"type": "text",
|
| 913 |
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"text": "Ideally, we would want our training procedure to perform well in a stable fashion across a variety of architectures (other than DCGANs). Similar to Arjovsky et al. (2017) and Gulrajani et al. (2017), we remove the stabilizing components of DCGAN architecture and demonstrate improved stability $\\&$ modeling performance compared to vanilla GAN training (see appendix section 5.2.3). However, this is a small set of architectures and it is not clear if there is an improvement in general. ",
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| 914 |
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"bbox": [
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"type": "text",
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"text": "To address this, we introduce a metric termed the BogoNet score to compare the stability & performance of different GAN training procedures. The basic idea is to choose random architectures for players $G$ and $D$ independently, and evaluate the performance of different algorithms in the resulting games. A good algorithm should achieve stable performance without failing to learn or resulting in mode collapse, despite the potentially imbalanced architectures. In our experiment, each player is assigned a network from a diverse pool of architectures belonging to three different families (MLP, ResNet, DCGAN). ",
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"img_path": "images/b082f4f9dcf00a9e09e461643346f431a03dfbd41016cb3e560a08d21c95a83d.jpg",
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"image_caption": [
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"Figure 5: Comparison of modeling performance on CIFAR10 "
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"type": "table",
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"img_path": "images/9ef6ebdb1c335d40442a82e34c966185712254ea7ac51aab28308e34ce6f8f9f.jpg",
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"table_caption": [],
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"table_footnote": [
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| 953 |
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"(b) Inception scores "
|
| 954 |
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],
|
| 955 |
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"table_body": "<table><tr><td>Algorithm</td><td>Score</td></tr><tr><td>WGAN</td><td>3.25</td></tr><tr><td>WGAN-GP</td><td>5.99</td></tr><tr><td>DRAGAN9</td><td>6.11</td></tr><tr><td>DRAGANd</td><td>6.90</td></tr><tr><td>Vanilla GANg</td><td>6.3</td></tr><tr><td>Vanilla GANd</td><td>6.99</td></tr></table>",
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"type": "table",
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"img_path": "images/cef281d5e664e3e27c8bedd06831526b83f0c27f044c425149a38f76e89a2137.jpg",
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"table_caption": [
|
| 968 |
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"Table 1: Summary of inception score statistics across 100 architectures "
|
| 969 |
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],
|
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"table_footnote": [],
|
| 971 |
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"table_body": "<table><tr><td rowspan=1 colspan=1>Algorithm</td><td rowspan=1 colspan=2>Final score</td><td rowspan=1 colspan=2>Area under curve</td><td rowspan=1 colspan=1>Qual. score</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Mean</td><td rowspan=1 colspan=1>Std</td><td rowspan=1 colspan=1>Mean</td><td rowspan=1 colspan=1>Std</td><td rowspan=1 colspan=1>Total</td></tr><tr><td rowspan=1 colspan=1>Vanilla GAN</td><td rowspan=1 colspan=1>2.91</td><td rowspan=1 colspan=1>1.44</td><td rowspan=1 colspan=1>277.72</td><td rowspan=1 colspan=1>126.09</td><td rowspan=1 colspan=1>92.5</td></tr><tr><td rowspan=1 colspan=1>DRAGAN</td><td rowspan=1 colspan=1>3.70</td><td rowspan=1 colspan=1>1.71</td><td rowspan=1 colspan=1>312.15</td><td rowspan=1 colspan=1>135.35</td><td rowspan=1 colspan=1>157.5</td></tr><tr><td rowspan=1 colspan=1>WGAN-GP</td><td rowspan=1 colspan=1>3.49</td><td rowspan=1 colspan=1>1.30</td><td rowspan=1 colspan=1>300.09</td><td rowspan=1 colspan=1>100.96</td><td rowspan=1 colspan=1>1</td></tr></table>",
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"type": "text",
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"text": "To demonstrate that our algorithm performs better compared to vanilla GAN training and WGAN-GP, we created 100 such instances of hard games. Each instance is trained using these algorithms on CIFAR-10 (under similar conditions for a fixed number of generator iterations, which gives a slight advantage to WGAN-GP) and we plot how inception score changes over time. For each algorithm, we calculated the average of final inception scores and area under the curve (AUC) over all 100 instances. The results are shown in Table 1. Notice that we beat the other algorithms in both metrics, which indicates some improvement in stability and modeling performance. ",
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"type": "text",
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"text": "Further, we perform some qualitative analysis to verify that BogoNet score indeed captures the improvements in stability. We create another set of 50 hard architectures and compare DRAGAN against vanilla GAN training. Each instance is allotted 5 points and we split this bounty between the two algorithms depending on their performance. If both perform well or perform poorly, they get 2.5 points each, so that we nullify the effect of such non-differentiating architectures. However, if one algorithm achieves stable performance compared to the other (in terms of failure to learn or mode collapses), we assign it higher portions of the bounty. Results were judged by two of the authors in a blind manner: The curves were shown side-by-side with the choice of algorithm for each side being randomized and unlabeled. The vanilla GAN received an average score of 92.5 while our algorithm achieved an average score of 157.5 and this correlates with BogoNet score from earlier. See appendix section 5.3 for some additional details regarding this experiment. ",
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"type": "text",
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"text": "3.3 STABILITY USING DIFFERENT OBJECTIVE FUNCTIONS ",
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| 1005 |
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"text_level": 1,
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"text": "Our algorithm improves stability across a variety of objective functions and we demonstrate this using the following experiment. Nowozin et al. (2016) show that we can interpret GAN training as minimizing various $f$ -divergences when an appropriate game objective function is used. We show experiments using the objective functions developed for Forward KL, Reverse KL, Pearson $\\chi ^ { 2 }$ , Squared Hellinger, and Total Variation divergence minimization. We use a hard architecture from the previous subsection to demonstrate the improvements in stability. Our algorithm is stable in all cases except for the total variation case, while the vanilla algorithm failed in all the cases (see Figure 6 for two examples and Figure 15 in appendix for all five). Thus, practitioners can now choose their game objective from a larger set of functions and use DRAGAN (unlike WGANs which requires a specific objective function). ",
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},
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"img_path": "images/08edb8a49c57f2c9b5265ce764b2a3bfe91b069519917cce9d29be5b8b42f86f.jpg",
|
| 1028 |
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"image_caption": [
|
| 1029 |
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"Figure 6: Inception score plots for two divergence measures, demonstrating superior stability for our algorithm. "
|
| 1030 |
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],
|
| 1031 |
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|
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| 1041 |
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"type": "text",
|
| 1042 |
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"text": "4 CONCLUSIONS ",
|
| 1043 |
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"text_level": 1,
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| 1044 |
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| 1052 |
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| 1053 |
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"type": "text",
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| 1054 |
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"text": "In this paper, we propose to study GAN training process as regret minimization, which is in contrast to the popular view that there is consistent minimization of a divergence between real and generated distributions. We analyze the convergence of GAN training from this new point of view and hypothesize that mode collapse occurs due to the existence of undesirable local equilibria. A simple observation is made about how the mode collapse situation often exhibits sharp gradients of the discriminator function around some real data points. This characterization partly explains the workings of previously proposed WGAN and gradient penalties, and motivates our novel penalty scheme. We show evidence of improved stability using DRAGAN and the resulting improvements in modeling performance across a variety of settings. We leave it to future works to explore our ideas in more depth and come up with improved training algorithms. ",
|
| 1055 |
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"bbox": [
|
| 1056 |
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| 1057 |
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},
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{
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"type": "text",
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| 1065 |
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"text": "REFERENCES ",
|
| 1066 |
+
"text_level": 1,
|
| 1067 |
+
"bbox": [
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+
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},
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{
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+
"type": "text",
|
| 1077 |
+
"text": "Martin Arjovsky and Léon Bottou. Towards principled methods for training generative adversarial networks. arXiv preprint arXiv:1701.04862, 2017. \nMartin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein gan. arXiv preprint arXiv:1701.07875, 2017. \nNicolo Cesa-Bianchi and Gábor Lugosi. Prediction, learning, and games. Cambridge university press, 2006. \nNicolo Cesa-Bianchi, Alex Conconi, and Claudio Gentile. On the generalization ability of on-line learning algorithms. IEEE Transactions on Information Theory, 50(9):2050–2057, 2004. \nTong Che, Yanran Li, Athul Paul Jacob, Yoshua Bengio, and Wenjie Li. Mode regularized generative adversarial networks. arXiv preprint arXiv:1612.02136, 2016. \nYoav Freund and Robert E Schapire. Adaptive game playing using multiplicative weights. Games and Economic Behavior, 29(1-2):79–103, 1999. \nIan Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 27, pp. 2672–2680. Curran Associates, Inc., 2014. URL http://papers.nips. cc/paper/5423-generative-adversarial-nets.pdf. \nIan J. Goodfellow. NIPS 2016 tutorial: Generative adversarial networks. CoRR, abs/1701.00160, 2017. URL http://arxiv.org/abs/1701.00160. \nIshaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron Courville. Improved training of wasserstein gans. arXiv preprint arXiv:1704.00028, 2017. \nElad Hazan, Karan Singh, and Cyril Zhang. Efficient regret minimization in non-convex games. arXiv preprint arXiv:1708.00075, 2017. \nElad Hazan et al. Introduction to online convex optimization. Foundations and Trends® in Optimization, 2(3-4):157–325, 2016. \nPanayotis Mertikopoulos, Christos Papadimitriou, and Georgios Piliouras. Cycles in adversarial regularized learning. arXiv preprint arXiv:1709.02738, 2017. \nLuke Metz, Ben Poole, David Pfau, and Jascha Sohl-Dickstein. Unrolled generative adversarial networks. CoRR, abs/1611.02163, 2016. URL http://arxiv.org/abs/1611.02163. \nJ. von Neumann. Zur theorie der gesellschaftsspiele. Mathematische Annalen, 100:295–320, 1928. URL http://eudml.org/doc/159291. \nNoam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V Vazirani. Algorithmic game theory, volume 1. Cambridge University Press Cambridge, 2007. \nSebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-gan: Training generative neural samplers using variational divergence minimization. In Advances in Neural Information Processing Systems, pp. 271–279, 2016. \nGuo-Jun Qi. Loss-sensitive generative adversarial networks on lipschitz densities. CoRR, abs/1701.06264, 2017. URL http://arxiv.org/abs/1701.06264. \nAlec Radford, Luke Metz, and Soumith Chintala. Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks. arXiv:1511.06434 [cs], November 2015. URL http://arxiv.org/abs/1511.06434. arXiv: 1511.06434. \nTim Salimans, Ian J. Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. CoRR, abs/1606.03498, 2016. URL http://arxiv. org/abs/1606.03498. \nMaurice Sion. On general minimax theorems. Pacific J. Math, 8(1):171–176, 1958. \nJunbo Zhao, Michael Mathieu, and Yann LeCun. Energy-based generative adversarial network. arXiv preprint arXiv:1609.03126, 2016. \nMartin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the 20th International Conference on Machine Learning (ICML-03), pp. 928–936, 2003. ",
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"text": "",
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},
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{
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"type": "text",
|
| 1099 |
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"text": "5 APPENDIX ",
|
| 1100 |
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"text_level": 1,
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"bbox": [
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},
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{
|
| 1110 |
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"type": "text",
|
| 1111 |
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"text": "5.1 SAMPLES AND LATENT SPACE WALKS ",
|
| 1112 |
+
"text_level": 1,
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| 1113 |
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"bbox": [
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},
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| 1121 |
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{
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| 1122 |
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"type": "text",
|
| 1123 |
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"text": "In this section, we provide samples from an additional experiment run on CelebA dataset (Figure 7). The samples from the experiment in section 3.1 are shown in Figure 8. Further, Radford et al. (2015) suggest that walking on the manifold learned by the generator can expose signs of memorization. We use DCGAN architecture to model MNIST and CelebA datasets using DRAGAN penalty, and the latent space walks of the learned models are shown in Figure 9 and Figure 10. The results demonstrate that the generator is indeed learning smooth transitions between different images, when our algorithm is used. ",
|
| 1124 |
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},
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| 1132 |
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{
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| 1133 |
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"type": "image",
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| 1134 |
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"img_path": "images/253e30feb5d444bfa0f9d2a46c2370020a9aa842e68a8c3101be4d1f05d075f1.jpg",
|
| 1135 |
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"image_caption": [
|
| 1136 |
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"Figure 7: Modeling CelebA with DRAGAN using DCGAN architecture. "
|
| 1137 |
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],
|
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},
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{
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"type": "image",
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"img_path": "images/00a3a72b049e3e37dc6001b4c685352f4e4d6b5af0259c6be6aa8569f5ba080a.jpg",
|
| 1150 |
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"image_caption": [
|
| 1151 |
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"Figure 8: Modeling CIFAR-10 using DCGAN architecture. "
|
| 1152 |
+
],
|
| 1153 |
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"image_footnote": [],
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| 1161 |
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},
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| 1162 |
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{
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| 1163 |
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"type": "image",
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| 1164 |
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"img_path": "images/706df446adbab90e162115d9e647d7d1ef935b7f035a159850b57539a35c150d.jpg",
|
| 1165 |
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"image_caption": [
|
| 1166 |
+
"Figure 9: Latent space walk of the model learned on MNIST using DRAGAN "
|
| 1167 |
+
],
|
| 1168 |
+
"image_footnote": [],
|
| 1169 |
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| 1171 |
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| 1173 |
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"page_idx": 13
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| 1176 |
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},
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| 1177 |
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{
|
| 1178 |
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"type": "image",
|
| 1179 |
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"img_path": "images/315d94d7f7741e9c64b718ad1502e3ef8549793fb818bbe79ade1e9ee7ab1389.jpg",
|
| 1180 |
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"image_caption": [
|
| 1181 |
+
"Figure 10: Latent space walk of the model learned on CelebA using DRAGAN "
|
| 1182 |
+
],
|
| 1183 |
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"image_footnote": [],
|
| 1184 |
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"bbox": [
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| 1185 |
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| 1186 |
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"page_idx": 13
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| 1191 |
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},
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| 1192 |
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{
|
| 1193 |
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"type": "text",
|
| 1194 |
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"text": "5.2 ADDITIONAL EXPERIMENTS ",
|
| 1195 |
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"text_level": 1,
|
| 1196 |
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},
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{
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| 1205 |
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"type": "text",
|
| 1206 |
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"text": "5.2.1 ONE HIDDEN LAYER NETWORK TO MODEL MNIST",
|
| 1207 |
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"text_level": 1,
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| 1208 |
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"bbox": [
|
| 1209 |
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174,
|
| 1210 |
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842,
|
| 1211 |
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580,
|
| 1212 |
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857
|
| 1213 |
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],
|
| 1214 |
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"page_idx": 13
|
| 1215 |
+
},
|
| 1216 |
+
{
|
| 1217 |
+
"type": "text",
|
| 1218 |
+
"text": "We design a simple experiment where $G$ and $D$ are both fully connected networks with just one hidden layer. Vanilla GAN performs poorly even in this simple case and we observe severe mode collapses. In contrast, our algorithm is stable throughout and obtains decent quality samples despite the constrained setup. ",
|
| 1219 |
+
"bbox": [
|
| 1220 |
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174,
|
| 1221 |
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867,
|
| 1222 |
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|
| 1223 |
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|
| 1224 |
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|
| 1225 |
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"page_idx": 13
|
| 1226 |
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},
|
| 1227 |
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{
|
| 1228 |
+
"type": "image",
|
| 1229 |
+
"img_path": "images/21a2f5cc05da6cef15ec214b6e79bdb86cf01dca6d822a107a4e2db884743d92.jpg",
|
| 1230 |
+
"image_caption": [
|
| 1231 |
+
"Figure 11: One hidden layer network to model MNIST - Inception score plots "
|
| 1232 |
+
],
|
| 1233 |
+
"image_footnote": [],
|
| 1234 |
+
"bbox": [
|
| 1235 |
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183,
|
| 1236 |
+
109,
|
| 1237 |
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802,
|
| 1238 |
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289
|
| 1239 |
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],
|
| 1240 |
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"page_idx": 14
|
| 1241 |
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},
|
| 1242 |
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{
|
| 1243 |
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"type": "image",
|
| 1244 |
+
"img_path": "images/0746c7e25322ed76843ed863c3f4a1e3db5bf220d06a38a957cd03102051a8a5.jpg",
|
| 1245 |
+
"image_caption": [
|
| 1246 |
+
"Figure 12: One hidden layer network to model MNIST - Samples "
|
| 1247 |
+
],
|
| 1248 |
+
"image_footnote": [],
|
| 1249 |
+
"bbox": [
|
| 1250 |
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263,
|
| 1251 |
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352,
|
| 1252 |
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738,
|
| 1253 |
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498
|
| 1254 |
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],
|
| 1255 |
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"page_idx": 14
|
| 1256 |
+
},
|
| 1257 |
+
{
|
| 1258 |
+
"type": "text",
|
| 1259 |
+
"text": "5.2.2 8-GAUSSIANS EXPERIMENT ",
|
| 1260 |
+
"text_level": 1,
|
| 1261 |
+
"bbox": [
|
| 1262 |
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176,
|
| 1263 |
+
549,
|
| 1264 |
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421,
|
| 1265 |
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564
|
| 1266 |
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],
|
| 1267 |
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"page_idx": 14
|
| 1268 |
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},
|
| 1269 |
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{
|
| 1270 |
+
"type": "text",
|
| 1271 |
+
"text": "We analyze the performance of WGAN-GP and DRAGAN on the 8-Gaussians dataset. As it can be seen in Figure 13, both of them approximately converge to the real distribution but notice that in the case of WGAN-GP, $D _ { \\theta } ( x )$ seems overly constrained in the data space. In contrast, DRAGAN’s discriminator is more flexible. ",
|
| 1272 |
+
"bbox": [
|
| 1273 |
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173,
|
| 1274 |
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573,
|
| 1275 |
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826,
|
| 1276 |
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630
|
| 1277 |
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],
|
| 1278 |
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"page_idx": 14
|
| 1279 |
+
},
|
| 1280 |
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{
|
| 1281 |
+
"type": "image",
|
| 1282 |
+
"img_path": "images/9b616a01a4caead05a9e60150cc795f9edabc4a1e5a148a0bfcf30d36f239ef0.jpg",
|
| 1283 |
+
"image_caption": [],
|
| 1284 |
+
"image_footnote": [],
|
| 1285 |
+
"bbox": [
|
| 1286 |
+
200,
|
| 1287 |
+
655,
|
| 1288 |
+
794,
|
| 1289 |
+
911
|
| 1290 |
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],
|
| 1291 |
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"page_idx": 14
|
| 1292 |
+
},
|
| 1293 |
+
{
|
| 1294 |
+
"type": "text",
|
| 1295 |
+
"text": "Figure 13: Comparing the performance of WGAN-GP and DRAGAN on the 8-Gaussians dataset. Orange is real samples, green is generated samples. The level sets of $D _ { \\theta } ( x )$ are shown in the background, with yellow as high and purple as low. ",
|
| 1296 |
+
"bbox": [
|
| 1297 |
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173,
|
| 1298 |
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99,
|
| 1299 |
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826,
|
| 1300 |
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143
|
| 1301 |
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],
|
| 1302 |
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"page_idx": 15
|
| 1303 |
+
},
|
| 1304 |
+
{
|
| 1305 |
+
"type": "text",
|
| 1306 |
+
"text": "5.2.3 STABILITY ACROSS DCGAN ARCHITECTURE VARIATIONS ",
|
| 1307 |
+
"text_level": 1,
|
| 1308 |
+
"bbox": [
|
| 1309 |
+
174,
|
| 1310 |
+
194,
|
| 1311 |
+
633,
|
| 1312 |
+
209
|
| 1313 |
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],
|
| 1314 |
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"page_idx": 15
|
| 1315 |
+
},
|
| 1316 |
+
{
|
| 1317 |
+
"type": "text",
|
| 1318 |
+
"text": "DCGAN architecture has been designed following specific guidelines to make it stable (Radford et al., 2015). We restate the suggested rules here. ",
|
| 1319 |
+
"bbox": [
|
| 1320 |
+
174,
|
| 1321 |
+
224,
|
| 1322 |
+
825,
|
| 1323 |
+
253
|
| 1324 |
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],
|
| 1325 |
+
"page_idx": 15
|
| 1326 |
+
},
|
| 1327 |
+
{
|
| 1328 |
+
"type": "text",
|
| 1329 |
+
"text": "1. Use all-convolutional networks which learn their own spatial downsampling (discriminator) \nor upsampling (generator) \n2. Remove fully connected hidden layers for deeper architectures \n3. Use batch normalization in both the generator and the discriminator \n4. Use ReLU activation in the generator for all layers except the output layer, which uses tanh \n5. Use LeakyReLU activation in the discriminator for all layers ",
|
| 1330 |
+
"bbox": [
|
| 1331 |
+
205,
|
| 1332 |
+
268,
|
| 1333 |
+
825,
|
| 1334 |
+
424
|
| 1335 |
+
],
|
| 1336 |
+
"page_idx": 15
|
| 1337 |
+
},
|
| 1338 |
+
{
|
| 1339 |
+
"type": "text",
|
| 1340 |
+
"text": "We show below that such constraints can be relaxed when using our algorithm and still maintain training stability. Below, we present a series of experiments in which we remove different stabilizing components from the DCGAN architecture and analyze the performance of our algorithm. Specifically, we choose the following four architectures which are difficult to train (in each case, we start with base DCGAN architecture and apply the changes) - ",
|
| 1341 |
+
"bbox": [
|
| 1342 |
+
174,
|
| 1343 |
+
440,
|
| 1344 |
+
826,
|
| 1345 |
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511
|
| 1346 |
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],
|
| 1347 |
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"page_idx": 15
|
| 1348 |
+
},
|
| 1349 |
+
{
|
| 1350 |
+
"type": "text",
|
| 1351 |
+
"text": "• No BN and a constant number of filters in the generator • 4-layer 512-dim ReLU MLP generator • tanh nonlinearities everywhere • tanh nonlinearity in the generator and 4-layer 512-dim L ",
|
| 1352 |
+
"bbox": [
|
| 1353 |
+
217,
|
| 1354 |
+
527,
|
| 1355 |
+
599,
|
| 1356 |
+
636
|
| 1357 |
+
],
|
| 1358 |
+
"page_idx": 15
|
| 1359 |
+
},
|
| 1360 |
+
{
|
| 1361 |
+
"type": "text",
|
| 1362 |
+
"text": "Notice that, in each case, our algorithm is stable while the vanilla GAN training fails. A similar approach is used to demonstrate the stability of training procedures in Arjovsky et al. (2017) and Gulrajani et al. (2017). ",
|
| 1363 |
+
"bbox": [
|
| 1364 |
+
174,
|
| 1365 |
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652,
|
| 1366 |
+
826,
|
| 1367 |
+
695
|
| 1368 |
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],
|
| 1369 |
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"page_idx": 15
|
| 1370 |
+
},
|
| 1371 |
+
{
|
| 1372 |
+
"type": "image",
|
| 1373 |
+
"img_path": "images/f08b1ae4765593c628424fef09f6125c39f1f98e2d3d07e01f8a12b5e2fe6dda.jpg",
|
| 1374 |
+
"image_caption": [
|
| 1375 |
+
"(a) tanh activation "
|
| 1376 |
+
],
|
| 1377 |
+
"image_footnote": [],
|
| 1378 |
+
"bbox": [
|
| 1379 |
+
210,
|
| 1380 |
+
729,
|
| 1381 |
+
460,
|
| 1382 |
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893
|
| 1383 |
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],
|
| 1384 |
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"page_idx": 15
|
| 1385 |
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},
|
| 1386 |
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{
|
| 1387 |
+
"type": "image",
|
| 1388 |
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"img_path": "images/7dfd4d64a8431aaf60c273ea45682704b7332c6ff928d3ac858d451ab7d7f023.jpg",
|
| 1389 |
+
"image_caption": [
|
| 1390 |
+
"(b) FC generator "
|
| 1391 |
+
],
|
| 1392 |
+
"image_footnote": [],
|
| 1393 |
+
"bbox": [
|
| 1394 |
+
524,
|
| 1395 |
+
737,
|
| 1396 |
+
781,
|
| 1397 |
+
895
|
| 1398 |
+
],
|
| 1399 |
+
"page_idx": 15
|
| 1400 |
+
},
|
| 1401 |
+
{
|
| 1402 |
+
"type": "image",
|
| 1403 |
+
"img_path": "images/8d15a3fe42dd37e5d63a7ca85bd42e77e855485f63134adcb4f70974bf0eb1cf.jpg",
|
| 1404 |
+
"image_caption": [
|
| 1405 |
+
"Figure 14: Comparing performance of DRAGAN and Vanilla GAN training in the hard variations of DCGAN architecture. "
|
| 1406 |
+
],
|
| 1407 |
+
"image_footnote": [],
|
| 1408 |
+
"bbox": [
|
| 1409 |
+
209,
|
| 1410 |
+
107,
|
| 1411 |
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784,
|
| 1412 |
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299
|
| 1413 |
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],
|
| 1414 |
+
"page_idx": 16
|
| 1415 |
+
},
|
| 1416 |
+
{
|
| 1417 |
+
"type": "text",
|
| 1418 |
+
"text": "5.2.4 STABILITY ACROSS OBJECTIVE FUNCTIONS ",
|
| 1419 |
+
"text_level": 1,
|
| 1420 |
+
"bbox": [
|
| 1421 |
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178,
|
| 1422 |
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380,
|
| 1423 |
+
526,
|
| 1424 |
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393
|
| 1425 |
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],
|
| 1426 |
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"page_idx": 16
|
| 1427 |
+
},
|
| 1428 |
+
{
|
| 1429 |
+
"type": "text",
|
| 1430 |
+
"text": "Due to space limitations, we only showed plots for two cases in section 3.3. Below we show the results for all five cases. ",
|
| 1431 |
+
"bbox": [
|
| 1432 |
+
174,
|
| 1433 |
+
409,
|
| 1434 |
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825,
|
| 1435 |
+
438
|
| 1436 |
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],
|
| 1437 |
+
"page_idx": 16
|
| 1438 |
+
},
|
| 1439 |
+
{
|
| 1440 |
+
"type": "image",
|
| 1441 |
+
"img_path": "images/318d07b48ece6f833bf757cca8eef723de3f0f3b498fed4e51b99ddc70725895.jpg",
|
| 1442 |
+
"image_caption": [],
|
| 1443 |
+
"image_footnote": [],
|
| 1444 |
+
"bbox": [
|
| 1445 |
+
202,
|
| 1446 |
+
477,
|
| 1447 |
+
785,
|
| 1448 |
+
944
|
| 1449 |
+
],
|
| 1450 |
+
"page_idx": 16
|
| 1451 |
+
},
|
| 1452 |
+
{
|
| 1453 |
+
"type": "image",
|
| 1454 |
+
"img_path": "images/0689a07198cc09bec2e05fb575a141417894d6e5b92e18d11aa07da3f8df33e0.jpg",
|
| 1455 |
+
"image_caption": [
|
| 1456 |
+
"Figure 15: Comparing performance of DRAGAN and Vanilla GAN training using different objective functions. "
|
| 1457 |
+
],
|
| 1458 |
+
"image_footnote": [],
|
| 1459 |
+
"bbox": [
|
| 1460 |
+
367,
|
| 1461 |
+
114,
|
| 1462 |
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624,
|
| 1463 |
+
299
|
| 1464 |
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],
|
| 1465 |
+
"page_idx": 17
|
| 1466 |
+
},
|
| 1467 |
+
{
|
| 1468 |
+
"type": "text",
|
| 1469 |
+
"text": "5.3 BOGONET DETAILS",
|
| 1470 |
+
"text_level": 1,
|
| 1471 |
+
"bbox": [
|
| 1472 |
+
176,
|
| 1473 |
+
364,
|
| 1474 |
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352,
|
| 1475 |
+
378
|
| 1476 |
+
],
|
| 1477 |
+
"page_idx": 17
|
| 1478 |
+
},
|
| 1479 |
+
{
|
| 1480 |
+
"type": "text",
|
| 1481 |
+
"text": "We used three families of architectures with probabilities - DCGAN (0.6), ResNet (0.2), MLP (0.2). Next, we further parameterized each family to create additional variation. For instance, the DCGAN family can result in networks with or without batch normalization, have LeakyReLU or Tanh nonlinearities. The number and width of filters, latent space dimensionality are some other possible variations in our experiment. Similarly, the number of layers and hidden units in each layer for MLP are chosen randomly. For ResNets, we chose their depth randomly. This creates a set of hard games which test the stability of a given training algorithm. ",
|
| 1482 |
+
"bbox": [
|
| 1483 |
+
173,
|
| 1484 |
+
390,
|
| 1485 |
+
825,
|
| 1486 |
+
489
|
| 1487 |
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],
|
| 1488 |
+
"page_idx": 17
|
| 1489 |
+
},
|
| 1490 |
+
{
|
| 1491 |
+
"type": "text",
|
| 1492 |
+
"text": "We showed qualitative analysis of the inception score plots in section 3.2 to verify that BogoNet score indeed captures the improvements in stability. Below, we show some examples of how the bounty splits were done. The plots in Figure 14 were scored as (averages are shown in DRAGAN, Vanilla GAN order): ",
|
| 1493 |
+
"bbox": [
|
| 1494 |
+
173,
|
| 1495 |
+
496,
|
| 1496 |
+
825,
|
| 1497 |
+
551
|
| 1498 |
+
],
|
| 1499 |
+
"page_idx": 17
|
| 1500 |
+
},
|
| 1501 |
+
{
|
| 1502 |
+
"type": "text",
|
| 1503 |
+
"text": "A - (5, 0), B - (3.5, 1.5), C – (2.25, 2.75), D – (2, 3) ",
|
| 1504 |
+
"bbox": [
|
| 1505 |
+
173,
|
| 1506 |
+
558,
|
| 1507 |
+
513,
|
| 1508 |
+
573
|
| 1509 |
+
],
|
| 1510 |
+
"page_idx": 17
|
| 1511 |
+
}
|
| 1512 |
+
]
|
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|
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parse/train/ryepFJbA-/ryepFJbA-_model.json
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|
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ADDED
|
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| 1 |
+
# Deep Reinforcement Learning at the Edge of the Statistical Precipice
|
| 2 |
+
|
| 3 |
+
Max Schwarzer MILA, Université de Montréal
|
| 4 |
+
|
| 5 |
+
Rishabh Agarwal∗ Google Research, Brain Team MILA, Université de Montréal
|
| 6 |
+
|
| 7 |
+
Pablo Samuel Castro Google Research, Brain Team
|
| 8 |
+
|
| 9 |
+
Aaron Courville MILA, Université de Montréal
|
| 10 |
+
|
| 11 |
+
Marc G. Bellemare Google Research, Brain Team
|
| 12 |
+
|
| 13 |
+
# Abstract
|
| 14 |
+
|
| 15 |
+
Deep reinforcement learning (RL) algorithms are predominantly evaluated by comparing their relative performance on a large suite of tasks. Most published results on deep RL benchmarks compare point estimates of aggregate performance such as mean and median scores across tasks, ignoring the statistical uncertainty implied by the use of a finite number of training runs. Beginning with the Arcade Learning Environment (ALE), the shift towards computationally-demanding benchmarks has led to the practice of evaluating only a small number of runs per task, exacerbating the statistical uncertainty in point estimates. In this paper, we argue that reliable evaluation in the few-run deep RL regime cannot ignore the uncertainty in results without running the risk of slowing down progress in the field. We illustrate this point using a case study on the Atari $1 0 0 \mathrm { k }$ benchmark, where we find substantial discrepancies between conclusions drawn from point estimates alone versus a more thorough statistical analysis. With the aim of increasing the field’s confidence in reported results with a handful of runs, we advocate for reporting interval estimates of aggregate performance and propose performance profiles to account for the variability in results, as well as present more robust and efficient aggregate metrics, such as interquartile mean scores, to achieve small uncertainty in results. Using such statistical tools, we scrutinize performance evaluations of existing algorithms on other widely used RL benchmarks including the ALE, Procgen, and the DeepMind Control Suite, again revealing discrepancies in prior comparisons. Our findings call for a change in how we evaluate performance in deep RL, for which we present a more rigorous evaluation methodology, accompanied with an open-source library rliable2, to prevent unreliable results from stagnating the field.
|
| 16 |
+
|
| 17 |
+
# 1 Introduction
|
| 18 |
+
|
| 19 |
+
Research in artificial intelligence, and particularly deep reinforcement learning (RL), relies on evaluating aggregate performance on a diverse suite of tasks to assess progress. Quantitative evaluation on a suite of tasks, such as Atari games [5], reveals strengths and limitations of methods while simultaneously guiding researchers towards methods with promising results. Performance of RL algorithms is usually summarized with a point estimate of task performance measure, such as mean and median performance across tasks, aggregated over independent training runs.
|
| 20 |
+
|
| 21 |
+
estimates. While evaluating more runs per task has been prescribed to reduce uncertainty and obtain reliable estimates [20, 41, 49], 3-10 runs are prevalent in deep RL as it is often computationally prohibitive to evaluate more runs. For example, 5 runs each on $5 0 +$ Atari 2600 games in ALE using standard protocol requires more than 1000 GPU training days [15]. As we move towards more challenging and complex RL benchmarks (e.g., StarCraft [110]), evaluating more than a handful of runs will become increasingly demanding due to increased amount of compute and data needed to tackle such tasks. Additional confounding factors, such as exploration in the low-data regime, exacerbates the performance variability in deep RL – as seen on the Atari $1 0 0 \mathrm { k }$ benchmark [50] – often requiring many more runs to achieve negligible statistical uncertainty in reported estimates.
|
| 22 |
+
|
| 23 |
+
Ignoring the statistical uncertainty in deep RL results gives a false impression of fast scientific progress in the field. It inevitably evades the question: “Would similar findings be obtained with new independent runs under different random conditions?” This could steer researchers towards superficially beneficial methods [11, 12, 25], often at the expense of better methods being neglected or even rejected early [67, 74] as such methods fail to outperform inferior methods simply due to less favorable random conditions. Furthermore, only reporting point estimates obscures nuances in comparisons [85] and can erroneously lead the field to conclude which methods are state-ofthe-art [63, 84], ensuing wasted effort when applied in practice [108]. Moreover, not report
|
| 24 |
+
|
| 25 |
+
|
| 26 |
+
Figure 1: Number of runs in RL over the years. Beginning with DQN [75] on the ALE, 5 or less runs are common in the field. Here, we show representative RL papers with empirical results, in the order of their publication year: TD-learning [99], Sparse coding [100], Options [102], Tetris (CEM) [103], Batch-Q [31], ALE [5], DQN [75], AlphaGo [96], A3C [76], DDPG [62], ES [88], PPO [92], SAC [36], Rainbow [42], AlphaStar [110], GoExplore [28], OpenAI Five [8], Balloon navigation [7] and MuZero [91].
|
| 27 |
+
|
| 28 |
+
ing the uncertainty in deep RL results makes them difficult to reproduce except under the exact same random conditions, which could lead to a reproducibility crisis similar to the one that plagues other fields [4, 44, 78]. Finally, unreliable results could erode trust in deep RL research itself [45].
|
| 29 |
+
|
| 30 |
+
In this work, we show that recent deep RL papers compare unreliable point estimates, which are dominated by statistical uncertainty, as well as exploit non-standard evaluation protocols, using a case study on Atari 100k (Section 3). Then, we illustrate how to reliably evaluate performance with only a handful of runs using a more rigorous evaluation methodology that accounts for uncertainty in results (Section 4). To exemplify the necessity of such methodology, we scrutinize performance evaluations of existing algorithms on widely used benchmarks, including the ALE [5] (Atari $1 0 0 \mathrm { k }$ , Atari 200M), Procgen [18] and DeepMind Control Suite [104], again revealing discrepancies in prior comparisons (Section 5). Our findings call for a change in how we evaluate performance in deep RL, for which we present a better methodology to prevent unreliable results from stagnating the field.
|
| 31 |
+
|
| 32 |
+
How do we reliably evaluate performance on deep RL benchmarks with only a handful of runs? As a practical solution that is easily applicable with 3-10 runs per task, we identify three statistical tools (Table 1) for improving the quality of experimental reporting. Since any performance estimate based on a finite number of runs is a random variable, we argue that it should be treated as such. Specifically, we argue for reporting aggregate performance measures using interval estimates via stratified bootstrap confidence intervals, as opposed to point estimates. Among prevalent aggregate measures, mean can be easily dominated by performance on a few outlier tasks, while median has high variability and zero performance on nearly half of the tasks does not change it. To address these deficiencies, we present more efficient and robust alternatives, such as interquartile mean, which are not unduly affected by outliers and have small uncertainty even with a handful of runs. Furthermore, to reveal the variability in performance across tasks, we propose reporting performance distributions across all runs. Compared to prior work [5, 83], these distributions result in performance profiles [26] that are statistically unbiased, more robust to outliers, and require fewer runs for smaller uncertainty.
|
| 33 |
+
|
| 34 |
+
# 2 Formalism
|
| 35 |
+
|
| 36 |
+
We consider the setting in which a reinforcement learning algorithm is evaluated on $M$ tasks. For each of these tasks, we perform $N$ independent runs3 which each provide a scalar, normalized score $x _ { m , n }$ , $m = 1 , \ldots , M$ and $n = 1 , \ldots , N$ . These normalized scores are obtained by linearly rescaling per-task scores4 based on two reference points; for example, performance on the Atari games is typically normalized with respect to a random agent and an average human, who are assigned a normalized score of 0 and 1 respectively [75]. We denote the set of normalized scores by $x _ { 1 : M , 1 : N }$ .
|
| 37 |
+
|
| 38 |
+
Table 1: Our recommendations for reliable evaluation, easily applicable with a handful of runs. Refer to Section 4 for details about recommendations and Section 5 for their application to widely-used RL benchmarks.
|
| 39 |
+
|
| 40 |
+
<table><tr><td>Desideratum</td><td>Current Evaluation Protocol</td><td>Our Recommendation</td></tr><tr><td>Uncertainty in aggregate performance</td><td>Point estimates · Ignore statistical uncertainty ·Hinder resultsreproducibility</td><td>Interval estimates via stratified bootstrap confidence intervals</td></tr><tr><td>Variability in performance across tasks and runs</td><td>Tables with mean scores per task · Overwhelming beyond a few tasks ·Standard deviations often omitted · Incomplete picture for multimodal and heavy-tailed distributions</td><td>Performance profiles (score distributions) · Show tail distribution of scores on com- bined runs across tasks · Allow qualitative comparisons ·Easily read any score percentile</td></tr><tr><td>Aggregate metrics for sum- marizing performance across tasks</td><td>Mean ·Often dominated by performance on outlier tasks Median · Requires large number of runs to claim improvements · Poor indicator of overall perfor-</td><td>Interquartile Mean (IQM) across all runs ·Performance on middle 5O% of com- bined runs · Robust to outlier scores but more statis- tically efficient than median To show other aspects of performance gains, report average probability of improvement</td></tr></table>
|
| 41 |
+
|
| 42 |
+
In most experiments, there is inherent randomness in the scores obtained from different runs. This randomness can arise from stochasticity in the task, exploratory choices made during learning, randomized initial parameters, but also software and hardware considerations such as non-determinism in GPUs and in machine learning frameworks [116]. Thus, we model the algorithm’s normalized score on the $m ^ { t h }$ task as a real-valued random variable $X _ { m }$ . Then, the score $x _ { m , n }$ is a realization of the random variable $X _ { m , n }$ , which is identically distributed as $X _ { m }$ . For $\tau \in \mathbb { R }$ , we define the tail distribution function of $X _ { m }$ as $F _ { m } ( \tau ) = \mathrm { P } ( \dot { X } _ { m } > \tau )$ . For any collection of scores $y _ { 1 : K }$ , the empirical tail distribution function is given by $\begin{array} { r } { \hat { F } ( \tau ; y _ { 1 : K } ) = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \mathbb { 1 } [ y _ { k } > \tau ] } \end{array}$ . In particular, we write $\hat { F } _ { m } ( \tau ) = \hat { F } ( \tau ; x _ { m , 1 : N } )$ .
|
| 43 |
+
|
| 44 |
+
The aggregate performance of an algorithm maps the set of normalized scores $x _ { 1 : M , 1 : N }$ to a scalar
|
| 45 |
+
value. Two prwe denote by $\begin{array} { r } { \bar { x } _ { m } = \frac { 1 } { N } \bar { \sum _ { n = 1 } ^ { N } } \bar { x _ { m , n } } } \end{array}$ fo t metrics are the age score on task $m$ an and across $N$ dian normalized scores. Ifruns, then these aggregate $\left( \hat { x } _ { 1 : M } \right)$ $\left( \hat { x } _ { 1 : M } \right)$
|
| 46 |
+
median over the task means since they are computed from a finite set of $N$ runs. Since $\hat { x } _ { m }$ is a
|
| 47 |
+
realization of the random variable $\begin{array} { r } { \bar { X } _ { m } = \frac { 1 } { N } \sum _ { n = 1 } ^ { \bar { N } } X _ { m , n } } \end{array}$ , the sample mean and median scores are
|
| 48 |
+
point estimates of the random variables Mean $\left( \hat { X } _ { 1 : M } \right)$ and Median $\left( \hat { X } _ { 1 : M } \right)$ respectively. We call true
|
| 49 |
+
mean and true median the metrics that would be obtained if we had unlimited experimental capacity $N \to \infty$ ), given by Mea $\mathsf { 1 } \big ( \mathbb { E } [ X _ { 1 : M } ] \big )$ and Median $\left( \mathbb { E } [ X _ { 1 : M } ] \right)$ respectively.
|
| 50 |
+
|
| 51 |
+
Confidence intervals (CIs) for a finite-sample score can be interpreted as an estimate of plausible values for the true score. A $\alpha \times 1 0 0 \%$ CI computes an interval such that if we rerun the experiment and construct the CI using a different set of runs, the fraction of calculated CIs (which would differ for each set of runs) that contain the true score would tend towards $\alpha \times 1 0 0 \%$ , where $\alpha \in [ 0 , 1 ]$ is the nominal coverage rate. $9 5 \%$ CIs are typically used in practice. If the true score lies outside the $9 5 \%$ CI, then a sampling event has occurred which had a probability of $5 \%$ of happening by chance.
|
| 52 |
+
|
| 53 |
+
|
| 54 |
+
Figure 2: Left. Distribution of median normalized scores computed using 100,000 different sets of $N$ runs subsampled uniformly with replacement from 100 runs. For a given algorithm, the sampling distribution shows the variation in the median scores when re-estimated using a different set of runs. The reported point estimates of median in publications, as shown by dashed lines, do not provide any information about the variability in median scores and severely overestimate or underestimate the expected median. We use the same number of runs as reported by publications: $N = 5$ runs for DER, OTR and DrQ, $N = 1 0$ runs for SPR and $N = 2 0$ runs for CURL. Right. $9 5 \%$ CIs for median and IQM scores (Section 4.3) for varying $N$ . There is a substantial uncertainty in median scores even with 50 runs. IQM has much smaller CIs than median. Note that when CIs overlap, properly accounting for uncertainty entails computing CIs for score differences (Figure A.15).
|
| 55 |
+
|
| 56 |
+
Remark. Following Amrhein et al. [2], Romer [87], Wasserstein et al. [112], we recommend using confidence intervals for measuring the uncertainty in results and showing effect sizes (e.g., performance improvements over baseline) that are compatible with the given data. Furthermore, we emphasize using statistical thinking but avoid statistical significance tests (e.g., $p$ -value $< 0 . 0 5 )$ because of their dichotomous nature (significant vs. not significant) and common misinterpretations [33, 35, 73] such as 1) lack of statistically significant results does not demonstrate the absence of effect (Figure 2, right), and 2) given enough data, any trivial effect can be statistically significant but may not be practically significant.
|
| 57 |
+
|
| 58 |
+
# 3 Case Study: The Atari 100k benchmark
|
| 59 |
+
|
| 60 |
+
We begin with a case study to illustrate the pitfalls arising from the naïve use of point estimates in the few-run regime. Our case study concerns the Atari 100k benchmark [50], an offshoot of the ALE for evaluating data-efficiency in deep RL. In this benchmark, algorithms are evaluated on only $1 0 0 \mathrm { k }$ steps (2-3 hours of game-play) for each of its 26 games, versus 200M frames in the ALE benchmark. Prior reported results on this benchmark have been computed mostly from 3 [39, 55, 59, 72, 89, 95] or 5 runs [50, 51, 53, 54, 64, 66, 86, 107, 115], and more rarely, 10 [65, 93] or 20 runs [56].
|
| 61 |
+
|
| 62 |
+
Our case study compares the performance of five recent deep RL algorithms, namely: (1) DER [107] and (2) OTR [51], (3) DrQ5 [53], (4) CURL [56], and (5) SPR [93]. We chose these methods as representative of influential algorithms within this benchmark. Since good performance on one game can result in unduly high sample means without providing much information about performance on other games, it is common to measure performance on Atari $1 0 0 \mathrm { k }$ using sample medians. Refer to Appendix A.2 for more details about the experimental setup.
|
| 63 |
+
|
| 64 |
+
We investigate statistical variations in the few-run regime by evaluating 100 independent runs for each algorithm, where the score for a run is the average returns obtained in 100 evaluation episodes taking place after training. Each run corresponds to training one algorithm on each of the 26 games in Atari $1 0 0 \mathrm { k }$ . This provides us with $2 6 \times 1 0 0$ scores per algorithm, which we then subsample with replacement to 3–100 runs. The subsampled scores are then used to produce a collection of point estimates whose statistical variability can be measured. We begin by using this experimental protocol to highlight statistical concerns regarding median normalized scores.
|
| 65 |
+
|
| 66 |
+
High variability in reported results. Our first observation is that the sample medians reported in the literature exhibit substantial variability when viewed as random quantities that depend on a small number of sample runs (Figure 2, left). This shows that there is a fairly large potential for drawing erroneous conclusions based on point estimates alone. As a concrete example, our analysis suggests that DER may in fact be better than OTR, unlike what the reported point estimates suggest. We conclude that in the few-run regime, point estimates are unlikely to provide definitive answers to the question: “Would we draw the same conclusions were we to re-evaluate our algorithm with a different set of runs?”
|
| 67 |
+
|
| 68 |
+
Substantial bias in sample medians. The sample median is a biased estimator of the true median: $\mathbb { E } [ \mathbf { M e d i a n } ( \bar { X } _ { 1 : M } ) ] \neq$ Median $\left( \mathbb { E } [ X _ { 1 : M } ] \right)$ in general. In the few-run regime, we find that this bias can dominate the comparison between algorithms, as evidenced in Fig
|
| 69 |
+
|
| 70 |
+
|
| 71 |
+
Figure 3: Expected sample median of task means. The expected score for $N$ runs is computed by repeatedly subsampling $N$ runs with replacement out of 100 runs for 100,000 times.
|
| 72 |
+
|
| 73 |
+
ure 3. For example, the score difference between sample medians with 5 and 100 runs for SPR $( + 0 . 0 3$ points) is about $36 \%$ of its mean improvement over $\mathrm { D r Q } ( \varepsilon )$ $( + 0 . 0 8$ points). Adding to the issue, the magnitude and sign of this bias strongly depends on the algorithm being evaluated.
|
| 74 |
+
|
| 75 |
+
Statistical concerns cannot be satisfactorily addressed with few runs. While claiming improvements with 3 or fewer runs may naturally raise eyebrows, folk wisdom in experimental RL suggests that 20 or 30 runs are enough. By calculating $9 5 \%$ confidence interval6 on sample medians for a varying number of runs (Figure 2, right), we find that this number is closer to 50–100 runs in Atari $1 0 0 \mathrm { k }$ – far too many to be computationally feasible for most research projects.
|
| 76 |
+
|
| 77 |
+
Consider a setting in which an algorithm is known to be better – what is the reliability of median and IQM (Section 4.3) for accurately assessing performance differences as the number of runs varies? Specifically, we consider two identical $N$ -run experiments involving SPR, except that we artificially inflate one of the experiments’ scores by a fixed fraction or lift of $+ \ell \%$ (Figure 4). In particular, $\ell = 0$ corresponds to running the same experiment twice but with different runs. We find that statistically defensible improvements with median scores is only achieved for 25 runs $\ell = 2 5$ ) and 100 runs $\ell = 1 0$ ). With $\ell = 0$ , even 100 runs are insufficient, with deviations of $2 0 \%$ possible.
|
| 78 |
+
|
| 79 |
+
Changes in evaluation protocols invalidates comparisons to prior work. A typical and relatively safe approach for measuring the performance of an RL algorithm is to average the scores received in their final training episodes [69]. However, the field has seen a number of alternative protocols used, including reporting the maximum evaluation score achieved during training [1, 3, 75] or across multiple runs [32, 47, 82]. A similar protocol is also used by CURL and SUNRISE [59] (Appendix A.4).
|
| 80 |
+
|
| 81 |
+
Results produced under alternative protocols involving maximum are generally incomparable with end-performance reported results. On Atari 100k, we find that the two protocols produce substantially different results (Figure 5), of a magnitude greater than the actual difference in score. In particular, evaluating DER with CURL’s protocol results in scores far above those reported for CURL. In other words, this gap in evaluation procedures resulted in CURL being assessed as achieving a greater true median than DER, where our experiment gives strong support to DER being superior. Similarly, we find that a lot of SUNRISE’s improvement over DER can be explained by the change in evaluation protocol (Figure 5). Refer to Appendix A.4 for discussion on pitfalls of such alternative protocols.
|
| 82 |
+
|
| 83 |
+
# 4 Recommendations and Tools for Reliable Evaluation
|
| 84 |
+
|
| 85 |
+
Our case study shows that the increase in the number of runs required to address the statistical uncertainty issues is typically infeasible for computationally demanding deep RL benchmarks. In this section, we identify three tools for improving the quality of experimental reporting in the few-run regime, all aligned with the principle of accounting for statistical uncertainty in results.
|
| 86 |
+
|
| 87 |
+
# 4.1 Stratified Bootstrap Confidence Intervals
|
| 88 |
+
|
| 89 |
+
We first reaffirm the importance of reporting interval estimates to indicate the range within which an algorithm’s aggregate performance is believed to lie. Concretely, we propose using bootstrap CIs [29] with stratified sampling for aggregate performance, a method that can be applied to small sample sizes and is better justified than reporting sample standard deviations in this context. While prior work has recommended using bootstrap CIs for reporting uncertainty in single task mean scores with $N$ runs [16, 20, 41], this is less useful when $N$ is small (Figure A.18), as bootstrapping assumes that re-sampling from the data approximates sampling from the true distribution. We can do better by aggregating samples across tasks, for a total of $M N$ random samples.
|
| 90 |
+
|
| 91 |
+
|
| 92 |
+
Figure 4: Detecting score lifts. Left. $9 5 \%$ CIs for observed lift with median scores, and Right. $9 5 \%$ CIs for observed lift with IQM (Section 4.3) when comparing SPR with an algorithm that performs $\ell \%$ better. IQM requires fewer runs than median for small uncertainty.
|
| 93 |
+
|
| 94 |
+
|
| 95 |
+
Figure 5: Normalized DER scores with non-standard evaluation protocols. Gains from SUNRISE and CURL over DER can mostly be explained by such protocols.
|
| 96 |
+
|
| 97 |
+
|
| 98 |
+
Figure 6: Validating $95 \%$ Stratified Bootstrap CIs for a varying number of runs for median and IQM scores for DER. The true coverage $\%$ is computed by sampling 10,000 sets of K runs without replacement from 200 runs and checking the fraction of $9 5 \%$ CIs that contains the true estimate approximation based on 200 runs. Note that we evaluate additional 100 runs for DER for an accurate point estimate. Percentile CIs has the best coverage while achieving a small width compared to other methods. Also, CI widths for IQM are much smaller than that of median. We also note that with 3 runs, bootstrap CIs underestimate the true $9 5 \%$ CIs and might require a larger nominal coverage rate to achieve true $9 5 \%$ coverage.
|
| 99 |
+
|
| 100 |
+
To compute the stratified bootstrap CIs, we re-sample runs with replacement independently for each task to construct an empirical bootstrap sample with $N$ runs each for $M$ tasks from which we calculate a statistic and repeat this process many times to approximate the sampling distribution of the statistic. We measure the reliability of this technique in Atari $1 0 0 \mathrm { k }$ for variable $N$ , by comparing the nominal coverage of $9 5 \%$ to the “true” coverage from the estimated CIs (Figure 6) for different bootstrap methods (see [30] and Appendix A.5). We find that percentile CIs provide good interval estimates for as few as $N = 1 0$ runs for both median and IQM scores (Section 4.3).
|
| 101 |
+
|
| 102 |
+
# 4.2 Performance Profiles
|
| 103 |
+
|
| 104 |
+
Most deep RL benchmarks yield scores that vary widely between tasks and may be heavy-tailed, multimodal, or possess outliers (e.g., Figure A.14). In this regime, both point estimates, such as mean and median scores, and interval estimates of these quantities paint an incomplete picture of an algorithm’s performance [24, Section 3]. Instead, we recommend the use of performance profiles [26], commonly used in benchmarking optimization software. While performance profiles from Dolan and Moré [26] correspond to empirical cumulative distribution functions without any uncertainty estimates, profiles proposed herein visualize the empirical tail distribution function (Section 2) of a random score (higher curve is better), with pointwise confidence bands based on stratified bootstrap.
|
| 105 |
+
|
| 106 |
+
By representing the entire set of normalized scores $x _ { 1 : M , 1 : N }$ visually, performance profiles reveal performance variability across tasks much better than interval estimates of aggregate metrics. Although tables containing per-task mean scores and standard deviations can reveal this variability, such tables tend to be overwhelming for more than a few tasks.7 In addition, performance profiles are robust to outlier runs and insensitive to small changes in performance across all tasks [26].
|
| 107 |
+
|
| 108 |
+
In this paper, we propose the use of a performance profile we call run-score distributions or simply score distributions (Figure 7, right), particularly well-suited to the few-run regime. A score distribution shows the fraction of runs above a certain normalized score and is given by
|
| 109 |
+
|
| 110 |
+
|
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+
Figure 7: Performance profiles on Atari 100k based on score distributions (left), which we recommend, and average score distributions (right). Shaded regions show pointwise $9 5 \%$ confidence bands based on percentile bootstrap with stratified sampling. The profiles on the left are more robust to outliers and have smaller confidence bands. We use 10 runs to show the robustness of profiles with a few runs. For SimPLe [50], we use the 5 runs from their reported results. The $\tau$ value where the profiles intersect $y = 0 . 5$ shows the median while for a non-negative random variable, area under the performance profile corresponds to the mean.
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$$
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\hat { F } _ { X } ( \tau ) = \hat { F } ( \tau ; x _ { 1 : M , 1 : N } ) = \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \hat { F } _ { m } ( \tau ) = \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \mathbb { 1 } [ x _ { m , n } > \tau ] .
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$$
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One advantage of the score distribution is that it is an unbiased estimator of the underlying distribution $\begin{array} { r } { F ( \tau ) = \frac { 1 } { N } \sum _ { m = 1 } ^ { M } F _ { m } ( \tau ) } \end{array}$ . Another advantage is that an outlier run with extremely high score can change the output of score distribution for any $\tau$ by at most a value of $\frac { 1 } { M N }$ .
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It is useful to contrast score distributions to average-score distributions, originally proposed in the context of the ALE [5] as a generalization of the median score. Average-score distributions correspond to the performance profile of a random variable $\bar { X }$ , $\hat { F } _ { \bar { X } } ( \tau ) = \hat { F } ( \tau ; \bar { x } _ { 1 : M } )$ , which shows the fraction of tasks on which an algorithm performs better than a certain score. However, such distributions are a biased estimate of the thing they seek to represent. Run-score distributions are more robust than average-score distributions, as they are a step function in $1 / M N$ versus $1 / M$ intervals, and typically has less variance: $\begin{array} { r } { \sigma _ { X } ^ { 2 } = \frac { 1 } { M ^ { 2 } N } \sum _ { m = 1 } ^ { M } F _ { m } ( \tau ) ( 1 - F _ { m } ( \tau ) ) \quad } \end{array}$ versus $\begin{array} { r } { \sigma _ { \bar { X } } ^ { 2 } = \frac { 1 } { M ^ { 2 } } \sum _ { m = 1 } ^ { M } F _ { \bar { X } _ { m } } ( \tau ) ( 1 - F _ { \bar { X } _ { m } } ( \tau ) ) } \end{array}$ . Figure 7 illustrates these differences.
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# 4.3 Robust and Efficient Aggregate Metrics
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Performance profiles allow us to compare different methods at a glance. If one curve is strictly above another, the better method is said to stochastically dominate8 the other [27, 61]. In RL benchmarks with a large number of tasks, however, stochastic dominance is rarely observed: performance profiles often intersect at multiple points. Finer quantitative comparisons must therefore entail aggregate metrics.
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We can extract a number of aggregate metrics from score distributions, including median (mixing runs and tasks) and mean normalized scores (matching our usual definition). As we already argued that these metrics are deficient, we now consider interesting alternatives also derived from score distributions.
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Figure 8: Aggregate metrics. For a non-negative random variable $X$ , IQM corresponds to the red shaded region while optimality gap corresponds to the orange shaded region in the performance profile of $X$ .
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culates the mean score of the remaining $50 \%$ runs $\scriptstyle ( = \lfloor N M / 2 \rfloor$ for $N$ runs each on $M$ tasks). IQM interpolates between mean and median across runs, which are $0 \%$ and almost $5 0 \%$ trimmed means respectively. Compared to sample median, IQM is a better indicator of overall performance as it is calculated using $50 \%$ of the combined runs while median only depends on the performance ordering across tasks and not on the magnitude except at most 2 tasks. For example, zero scores on nearly half of the tasks does not affect the median while IQM exhibits a severe degradation. Compared to mean, IQM is robust to outliers, yet has considerably less bias than median (Figure A.17). While median is more robust to outliers than IQM, this robustness comes at the expense of statistical efficiency, which is crucial in the few-run regime: IQM results in much smaller CIs (Figure 2 (right) and 6) and is able to detect a given improvement with far fewer runs (Figures 4 and A.15).
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Figure 9: Aggregate metrics on Atari 200M with $9 5 \%$ CIs based on 55 games with sticky actions [69]. Higher mean, median and IQM scores and lower optimality gap are better. The CIs are estimated using the percentile bootstrap with stratified sampling. IQM typically results in smaller CIs than median scores. Large values of mean scores relative to median and IQM indicate being dominated by a few high performing tasks, for example, DreamerV2 and M-IQN obtain normalized scores above 50 on the game JAMESBOND. Optimality gap is less susceptible to outliers compared to mean scores. We compare DQN (Nature) [75], DQN with Adam optimizer, C51 [6], REM [1], Rainbow [42], IQN [22], Munchausen-IQN (M-IQN) [109], and DreamerV2 [38]. All results are based on 5 runs per game except for M-IQN and DreamerV2 which report results with 3 and 11 runs.
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As a robust alternative to mean, we recommend using the optimality gap: the amount by which the algorithm fails to meet a minimum score of $\gamma = 1 . 0$ (orange region in Figure 8). This assumes that a score of 1.0 is a desirable target beyond which improvements are not very important, for example when the aim is to obtain human-level performance [e.g., 3, 23]. Naturally, the threshold $\gamma$ may be chosen differently, which we discuss further in Appendix A.7.
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If one is interested in knowing how robust an improvement from an algorithm $X$ over an algorithm $Y$ is, another possible metric to consider is the average probability of improvement – this metric shows how likely it is for $X$ to outperform $Y$ on a randomly selected task. Specifically, $P ( X >$ $\begin{array} { r } { Y ) = \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \overset { \textstyle } { P } ( X _ { m } > Y _ { m } ) , } \end{array}$ , where $P ( X _ { m } > Y _ { m } )$ (Equation A.2) is the probability that $X$ is better than $Y$ on task $m$ . Note that, unlike IQM and optimality gap, this metric does not account for the size of improvement. While finding the best aggregate metric is still an open question and is often dependent on underlying normalized score distribution, our proposed alternatives avoid the failure modes of prevalent metrics while being robust and requiring fewer runs to reduce uncertainty.
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# 5 Re-evaluating Evaluation on Deep RL Benchmarks
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Arcade Learning Environment. Training RL agents for 200M frames on the ALE [5, 69] is the most widely recognized benchmark in deep RL. We revisit some popular methods which demonstrated progress on this benchmark and reveal discrepancies in their findings as a consequence of ignoring the uncertainty in their results (Figure 9). For example, DreamerV2 [38] exhibits a large amount of uncertainty in aggregate scores. While M-IQN [109] claimed better performance than Dopamine Rainbow9 [42] in terms of median normalized scores, their interval estimates strikingly overlap. Similarly, while C51 [5] is considered substantially better than DQN [75], the interval estimates as well as performance profiles for DQN (Adam) and C51 overlap significantly.
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Figure 9 reveals an interesting limitation of aggregate metrics: depending on the choice of metric, the ordering between algorithms changes (e.g., Median vs. IQM). The inconsistency in ranking across aggregate metrics arises from the fact that such metrics only capture a specific aspect of overall performance across tasks and runs. Additionally, the change of algorithm ranking between optimality gap and IQM/median scores reveal that while recent algorithms typically show performance gains relative to humans on average, their performance seems to be worse on games below human performance. Since performance profiles capture the full picture, they would often illustrate why such inconsistencies exist. For example, optimality gap and IQM can be both read as areas in the profile (Figure 8). The performance profile in Figure 10 (left) illustrates the nuances present when comparing different algorithms. For example, IQN seems to be better than Rainbow for $\tau \geq 2$ , but worse for $\tau < 2$ . Similarly, the profiles of DreamerV2 and M-IQN for $\tau < 8$ intersect at multiple points. To compare sample efficiency of the agents, we also present their IQM scores as a function of number of frames in Figure 10 (right).
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Figure 10: Atari 200M evaluation. Left. Score distributions using human-normalized scores obtained after training for 200M frames. Right. Sample-efficiency of agents as a function of number of frames measured via IQM human-normalized scores. Shaded regions show pointwise $9 5 \%$ percentile stratified bootstrap CIs.
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Figure 11: DeepMind Control Suite evaluation results, averaged across 6 tasks, on the 100k and $5 0 0 \mathrm { k }$ benchmark. We compare $\mathrm { S A C + A E }$ [114], SLAC [58], Dreamer [37], CURL [98], RAD [57], DrQ [53], PISAC [60], SUNRISE [59], and CURL-D2RL [97]. The ordering of the algorithms in the left figure is based on their claimed relative performance – all algorithms except Dreamer claimed improvement over at least one algorithm placed below them. (a) Interval estimates show $9 5 \%$ stratified bootstrap CIs for methods with individual runs provided by their respective authors and $9 5 \%$ studentized CIs for CURL, CURL-D2RL, and SUNRISE. Normalized scores are computed by dividing by the maximum score $( = 1 0 0 0 )$ . (b) Score distributions. (c) The $i ^ { t h }$ column in the rank distribution plots show the probability that a given method is assigned rank $_ { i }$ averaged across all tasks. The ranks are estimated using 200,000 stratified bootstrap re-samples.
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DeepMind Control Suite. Recent continuous control papers benchmark performance on 6 tasks in DM Control [104] at $1 0 0 \mathrm { k }$ and $5 0 0 \mathrm { k }$ steps. Typically, such papers claim improvement based on higher mean scores per task regardless of the variability in those scores. However, we find that when accounting for uncertainty in results, most algorithms do not consistently rank above algorithms they claimed to improve upon (Figure 11c and 11b). Furthermore, there are huge overlaps in $9 5 \%$ CIs of mean normalized scores for most algorithms (Figure 11a). These findings suggest that a lot of the reported improvements are spurious, resulting from randomness in the experimental protocol.
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Procgen benchmark. Procgen [18] is a popular benchmark, consisting of 16 diverse tasks, for evaluating generalization in RL. Recent papers report mean PPO-normalized scores on this benchmark to emphasize the gains relative to PPO [92] as most methods are built on top of it. However, Figure 12 (left) shows that PPO-normalized scores typically have a heavy-tailed distribution making the mean scores highly dependent on performance on a small fraction of tasks. Instead, we recommend using normalization based on the estimated minimum and maximum scores on ProcGen [18] and reporting aggregate metrics based on such scores (Figure A.32). While publications sometimes make binary claims about whether they improve over prior methods, such improvements are inherently probabilistic. To reveal this discrepancy, we investigate the following question: “What is the probability that an algorithm which claimed improvement over a prior algorithm performs better than it?” (Figure 12, right). While this probability does not distinguish between two algorithms which uniformly improve on all tasks by $1 \%$ and $100 \%$ , it does highlight how likely an improvement is. For example, there is only a $4 0 - 5 0 \%$ chance that UCB-DrAC [81] improves upon PLR [48]. We note that a number of improvements reported in the existing literature are only $5 0 - 7 0 \%$ likely.
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Figure 12: Procgen evaluation results based on easy mode comparisons [80] with 16 tasks. Left. Score distributions which compare PPO [92], MixReg [111], UCB-DrAC [81], PLR [48], PPG [19] and IDAAC [80]. Shaded regions indicate $9 5 \%$ percentile stratified bootstrap CIs. Right. Each row shows the probability of improvement, with $9 5 \%$ bootstrap CIs, that the algorithm $X$ on the left outperforms algorithm $Y$ on the right, given that $X$ was claimed to be better than $Y$ . For all algorithms, results are based on 10 runs per task.
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# 6 Discussion
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We saw, both in our case study on the Atari $1 0 0 \mathrm { k }$ benchmark and with our analysis of other widely-used RL benchmarks, that statistical issues can have a sizeable influence on reported results, in particular when point estimates are used or evaluation protocols are not kept constant within comparisons. Despite earlier calls for more experimental rigor in deep RL [16, 20, 21, 41, 49, 83] (discussed in Appendix A.3), our analysis shows that the field has not yet found sure footing in this regards.
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In part, this is because the issue of reproducibility is a complex one; where our work is concerned with our confidence about and interpretation of reported results (what Goodman et al. [34] calls results reproducibility), others [79] have highlighted that there might be missing information about the experiments themselves (methods reproducibility). We remark that the problem is not solved by fixing random seeds, as has sometimes been proposed [52, 77], since it does not really address the question of whether an algorithm would perform well under similar conditions but with different seeds. Furthermore, fixed seeds might benefit certain algorithms more than others. Nor can the problem be solved by the use of dichotomous statistical significance tests, as discussed in Section 2.
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One way to minimize the risks associated with statistical effects is to report results in a more complete fashion, paying close attention to bias and uncertainty within these estimates. To this end, our recommendations are summarized in Table 1. To further support RL researchers in this endeavour, we released an easy-to-use Python library, rliable along with a Colab notebook for implementing our recommendations, as well as all the individual runs used in our experiments10. Again, we emphasize the importance of published papers providing results for all runs to allow for future statistical analyses.
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A barrier to adoption of evaluation protocols proposed in this work, and more generally, rigorous evaluation, is whether there are clear incentives for researchers to do so, as more rigor generally entails more nuanced and tempered claims. Arguably, doing good and reproducible science is one such incentive. We hope that our findings about erroneous conclusions in published papers would encourage researchers to avoid fooling themselves, even if that requires tempered claims. That said, a more pragmatic incentive would be if conferences and reviewers required more rigorous evaluation for publication, e.g., NeurIPS 2021 checklist asks whether error bars are reported. Moving towards reliable evaluation is an ongoing process and we believe that this paper would greatly benefit it.
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Given the substantial influence of statistical considerations in experiments involving 40-year old Atari 2600 video games and low-DOF robotic simulations, we argue that it is unlikely that an increase in available computation will resolve the problem for the future generation of RL benchmarks. Instead, just as a well-prepared rock-climber can skirt the edge of the steepest precipices, it seems likely that ongoing progress in reinforcement learning will require greater experimental discipline.
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# Societal Impacts
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This paper calls for statistical sophistication in deep RL research by accounting for statistical uncertainty in reported results. However, statistical sophistication can introduce new forms of statistical abuses and monitoring the literature for such abuses should be an ongoing priority for the research community. Moving towards reliable evaluation and reproducible research is an ongoing process and this paper only partly addresses it by providing tools for more reliable evaluation. That said, while accounting for uncertainty in results is not a panacea, it provides a strong foundation for trustworthy results on which the community can build upon, with increased confidence. In terms of broader societal impact of this work, we do not see any foreseeable strongly negative impacts. However, this paper could positively impact society by constituting a step forwards in rigorous few-run evaluation regime, which reduces computational burden on researchers and is “greener” than evaluating a large number of runs.
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# Acknowledgments
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We thank Xavier Bouthillier, Dumitru Erhan, Marlos C. Machado, David Ha, Fabio Viola, Fernando Diaz, Stephanie Chan, Jacob Buckman, Danijar Hafner and anonymous NeurIPS’ reviewers for providing valuable feedback for an earlier draft of this work. We also acknowledge Matteo Hessel, David Silver, Tom Schaul, Csaba Szepesvári, Hado van Hasselt, Rosanne Liu, Simon Kornblith, Aviral Kumar, George Tucker, Kevin Murphy, Ankit Anand, Aravind Srinivas, Matthew Botvinick, Clare Lyle, Kimin Lee, Misha Laskin, Ankesh Anand, Joelle Pineau and Braham Synder for helpful discussions. We also thank all the authors who provided individual runs for their corresponding publications. We are also grateful for general support from Google Research teams in Montréal and elsewhere.
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# Checklist
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1. For all authors...
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
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(b) Did you describe the limitations of your work? [Yes] See Section 6.
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(c) Did you discuss any potential negative societal impacts of your work? [Yes]
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(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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2. If you are including theoretical results...
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(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
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3. If you ran experiments...
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(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See Appendix A.1.
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(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix A.2
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| 307 |
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(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]
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| 308 |
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(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix A.2
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4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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(a) If your work uses existing assets, did you cite the creators? [Yes]
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(b) Did you mention the license of the assets? [Yes] Apache License, Version 2.0
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(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]
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(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]
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(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
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5. If you used crowdsourcing or conducted research with human subjects...
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(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
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(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
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(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
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|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "Deep Reinforcement Learning at the Edge of the Statistical Precipice ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
207,
|
| 8 |
+
122,
|
| 9 |
+
792,
|
| 10 |
+
172
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Max Schwarzer MILA, Université de Montréal ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
398,
|
| 19 |
+
227,
|
| 20 |
+
598,
|
| 21 |
+
253
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],
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| 23 |
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| 24 |
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},
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| 25 |
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{
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"type": "text",
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"text": "Rishabh Agarwal∗ Google Research, Brain Team MILA, Université de Montréal ",
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"text": "Pablo Samuel Castro Google Research, Brain Team ",
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"text": "Aaron Courville MILA, Université de Montréal ",
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"text": "Marc G. Bellemare Google Research, Brain Team ",
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"type": "text",
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"text": "Abstract ",
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"text_level": 1,
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"type": "text",
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"text": "Deep reinforcement learning (RL) algorithms are predominantly evaluated by comparing their relative performance on a large suite of tasks. Most published results on deep RL benchmarks compare point estimates of aggregate performance such as mean and median scores across tasks, ignoring the statistical uncertainty implied by the use of a finite number of training runs. Beginning with the Arcade Learning Environment (ALE), the shift towards computationally-demanding benchmarks has led to the practice of evaluating only a small number of runs per task, exacerbating the statistical uncertainty in point estimates. In this paper, we argue that reliable evaluation in the few-run deep RL regime cannot ignore the uncertainty in results without running the risk of slowing down progress in the field. We illustrate this point using a case study on the Atari $1 0 0 \\mathrm { k }$ benchmark, where we find substantial discrepancies between conclusions drawn from point estimates alone versus a more thorough statistical analysis. With the aim of increasing the field’s confidence in reported results with a handful of runs, we advocate for reporting interval estimates of aggregate performance and propose performance profiles to account for the variability in results, as well as present more robust and efficient aggregate metrics, such as interquartile mean scores, to achieve small uncertainty in results. Using such statistical tools, we scrutinize performance evaluations of existing algorithms on other widely used RL benchmarks including the ALE, Procgen, and the DeepMind Control Suite, again revealing discrepancies in prior comparisons. Our findings call for a change in how we evaluate performance in deep RL, for which we present a more rigorous evaluation methodology, accompanied with an open-source library rliable2, to prevent unreliable results from stagnating the field. ",
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"type": "text",
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"text": "1 Introduction ",
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"text": "Research in artificial intelligence, and particularly deep reinforcement learning (RL), relies on evaluating aggregate performance on a diverse suite of tasks to assess progress. Quantitative evaluation on a suite of tasks, such as Atari games [5], reveals strengths and limitations of methods while simultaneously guiding researchers towards methods with promising results. Performance of RL algorithms is usually summarized with a point estimate of task performance measure, such as mean and median performance across tasks, aggregated over independent training runs. ",
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"type": "text",
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"text": "estimates. While evaluating more runs per task has been prescribed to reduce uncertainty and obtain reliable estimates [20, 41, 49], 3-10 runs are prevalent in deep RL as it is often computationally prohibitive to evaluate more runs. For example, 5 runs each on $5 0 +$ Atari 2600 games in ALE using standard protocol requires more than 1000 GPU training days [15]. As we move towards more challenging and complex RL benchmarks (e.g., StarCraft [110]), evaluating more than a handful of runs will become increasingly demanding due to increased amount of compute and data needed to tackle such tasks. Additional confounding factors, such as exploration in the low-data regime, exacerbates the performance variability in deep RL – as seen on the Atari $1 0 0 \\mathrm { k }$ benchmark [50] – often requiring many more runs to achieve negligible statistical uncertainty in reported estimates. ",
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"text": "Ignoring the statistical uncertainty in deep RL results gives a false impression of fast scientific progress in the field. It inevitably evades the question: “Would similar findings be obtained with new independent runs under different random conditions?” This could steer researchers towards superficially beneficial methods [11, 12, 25], often at the expense of better methods being neglected or even rejected early [67, 74] as such methods fail to outperform inferior methods simply due to less favorable random conditions. Furthermore, only reporting point estimates obscures nuances in comparisons [85] and can erroneously lead the field to conclude which methods are state-ofthe-art [63, 84], ensuing wasted effort when applied in practice [108]. Moreover, not report",
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"type": "image",
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"img_path": "images/baef0f62fa415bcccbff96b8bd1f5704c329c1997876481d2ea9785708be657d.jpg",
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"image_caption": [
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"Figure 1: Number of runs in RL over the years. Beginning with DQN [75] on the ALE, 5 or less runs are common in the field. Here, we show representative RL papers with empirical results, in the order of their publication year: TD-learning [99], Sparse coding [100], Options [102], Tetris (CEM) [103], Batch-Q [31], ALE [5], DQN [75], AlphaGo [96], A3C [76], DDPG [62], ES [88], PPO [92], SAC [36], Rainbow [42], AlphaStar [110], GoExplore [28], OpenAI Five [8], Balloon navigation [7] and MuZero [91]. "
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"text": "ing the uncertainty in deep RL results makes them difficult to reproduce except under the exact same random conditions, which could lead to a reproducibility crisis similar to the one that plagues other fields [4, 44, 78]. Finally, unreliable results could erode trust in deep RL research itself [45]. ",
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"text": "In this work, we show that recent deep RL papers compare unreliable point estimates, which are dominated by statistical uncertainty, as well as exploit non-standard evaluation protocols, using a case study on Atari 100k (Section 3). Then, we illustrate how to reliably evaluate performance with only a handful of runs using a more rigorous evaluation methodology that accounts for uncertainty in results (Section 4). To exemplify the necessity of such methodology, we scrutinize performance evaluations of existing algorithms on widely used benchmarks, including the ALE [5] (Atari $1 0 0 \\mathrm { k }$ , Atari 200M), Procgen [18] and DeepMind Control Suite [104], again revealing discrepancies in prior comparisons (Section 5). Our findings call for a change in how we evaluate performance in deep RL, for which we present a better methodology to prevent unreliable results from stagnating the field. ",
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"text": "How do we reliably evaluate performance on deep RL benchmarks with only a handful of runs? As a practical solution that is easily applicable with 3-10 runs per task, we identify three statistical tools (Table 1) for improving the quality of experimental reporting. Since any performance estimate based on a finite number of runs is a random variable, we argue that it should be treated as such. Specifically, we argue for reporting aggregate performance measures using interval estimates via stratified bootstrap confidence intervals, as opposed to point estimates. Among prevalent aggregate measures, mean can be easily dominated by performance on a few outlier tasks, while median has high variability and zero performance on nearly half of the tasks does not change it. To address these deficiencies, we present more efficient and robust alternatives, such as interquartile mean, which are not unduly affected by outliers and have small uncertainty even with a handful of runs. Furthermore, to reveal the variability in performance across tasks, we propose reporting performance distributions across all runs. Compared to prior work [5, 83], these distributions result in performance profiles [26] that are statistically unbiased, more robust to outliers, and require fewer runs for smaller uncertainty. ",
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"type": "text",
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"text": "2 Formalism ",
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"text": "We consider the setting in which a reinforcement learning algorithm is evaluated on $M$ tasks. For each of these tasks, we perform $N$ independent runs3 which each provide a scalar, normalized score $x _ { m , n }$ , $m = 1 , \\ldots , M$ and $n = 1 , \\ldots , N$ . These normalized scores are obtained by linearly rescaling per-task scores4 based on two reference points; for example, performance on the Atari games is typically normalized with respect to a random agent and an average human, who are assigned a normalized score of 0 and 1 respectively [75]. We denote the set of normalized scores by $x _ { 1 : M , 1 : N }$ . ",
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"type": "table",
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"img_path": "images/0df8afd636b0cf3bb5c44b8acbe857672c39ad88e6ab0d463fa74089e0075d16.jpg",
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"table_caption": [
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| 212 |
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"Table 1: Our recommendations for reliable evaluation, easily applicable with a handful of runs. Refer to Section 4 for details about recommendations and Section 5 for their application to widely-used RL benchmarks. "
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],
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"table_footnote": [],
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"table_body": "<table><tr><td>Desideratum</td><td>Current Evaluation Protocol</td><td>Our Recommendation</td></tr><tr><td>Uncertainty in aggregate performance</td><td>Point estimates · Ignore statistical uncertainty ·Hinder resultsreproducibility</td><td>Interval estimates via stratified bootstrap confidence intervals</td></tr><tr><td>Variability in performance across tasks and runs</td><td>Tables with mean scores per task · Overwhelming beyond a few tasks ·Standard deviations often omitted · Incomplete picture for multimodal and heavy-tailed distributions</td><td>Performance profiles (score distributions) · Show tail distribution of scores on com- bined runs across tasks · Allow qualitative comparisons ·Easily read any score percentile</td></tr><tr><td>Aggregate metrics for sum- marizing performance across tasks</td><td>Mean ·Often dominated by performance on outlier tasks Median · Requires large number of runs to claim improvements · Poor indicator of overall perfor-</td><td>Interquartile Mean (IQM) across all runs ·Performance on middle 5O% of com- bined runs · Robust to outlier scores but more statis- tically efficient than median To show other aspects of performance gains, report average probability of improvement</td></tr></table>",
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"text": "",
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"type": "text",
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"text": "In most experiments, there is inherent randomness in the scores obtained from different runs. This randomness can arise from stochasticity in the task, exploratory choices made during learning, randomized initial parameters, but also software and hardware considerations such as non-determinism in GPUs and in machine learning frameworks [116]. Thus, we model the algorithm’s normalized score on the $m ^ { t h }$ task as a real-valued random variable $X _ { m }$ . Then, the score $x _ { m , n }$ is a realization of the random variable $X _ { m , n }$ , which is identically distributed as $X _ { m }$ . For $\\tau \\in \\mathbb { R }$ , we define the tail distribution function of $X _ { m }$ as $F _ { m } ( \\tau ) = \\mathrm { P } ( \\dot { X } _ { m } > \\tau )$ . For any collection of scores $y _ { 1 : K }$ , the empirical tail distribution function is given by $\\begin{array} { r } { \\hat { F } ( \\tau ; y _ { 1 : K } ) = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\mathbb { 1 } [ y _ { k } > \\tau ] } \\end{array}$ . In particular, we write $\\hat { F } _ { m } ( \\tau ) = \\hat { F } ( \\tau ; x _ { m , 1 : N } )$ . ",
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"type": "text",
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"text": "The aggregate performance of an algorithm maps the set of normalized scores $x _ { 1 : M , 1 : N }$ to a scalar \nvalue. Two prwe denote by $\\begin{array} { r } { \\bar { x } _ { m } = \\frac { 1 } { N } \\bar { \\sum _ { n = 1 } ^ { N } } \\bar { x _ { m , n } } } \\end{array}$ fo t metrics are the age score on task $m$ an and across $N$ dian normalized scores. Ifruns, then these aggregate $\\left( \\hat { x } _ { 1 : M } \\right)$ $\\left( \\hat { x } _ { 1 : M } \\right)$ \nmedian over the task means since they are computed from a finite set of $N$ runs. Since $\\hat { x } _ { m }$ is a \nrealization of the random variable $\\begin{array} { r } { \\bar { X } _ { m } = \\frac { 1 } { N } \\sum _ { n = 1 } ^ { \\bar { N } } X _ { m , n } } \\end{array}$ , the sample mean and median scores are \npoint estimates of the random variables Mean $\\left( \\hat { X } _ { 1 : M } \\right)$ and Median $\\left( \\hat { X } _ { 1 : M } \\right)$ respectively. We call true \nmean and true median the metrics that would be obtained if we had unlimited experimental capacity $N \\to \\infty$ ), given by Mea $\\mathsf { 1 } \\big ( \\mathbb { E } [ X _ { 1 : M } ] \\big )$ and Median $\\left( \\mathbb { E } [ X _ { 1 : M } ] \\right)$ respectively. ",
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"type": "text",
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"text": "Confidence intervals (CIs) for a finite-sample score can be interpreted as an estimate of plausible values for the true score. A $\\alpha \\times 1 0 0 \\%$ CI computes an interval such that if we rerun the experiment and construct the CI using a different set of runs, the fraction of calculated CIs (which would differ for each set of runs) that contain the true score would tend towards $\\alpha \\times 1 0 0 \\%$ , where $\\alpha \\in [ 0 , 1 ]$ is the nominal coverage rate. $9 5 \\%$ CIs are typically used in practice. If the true score lies outside the $9 5 \\%$ CI, then a sampling event has occurred which had a probability of $5 \\%$ of happening by chance. ",
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"type": "image",
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"img_path": "images/dbb6f647a07253dbec04886e2c95a85d12dfd055069a80853090b2b8c4850e33.jpg",
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"image_caption": [
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| 272 |
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"Figure 2: Left. Distribution of median normalized scores computed using 100,000 different sets of $N$ runs subsampled uniformly with replacement from 100 runs. For a given algorithm, the sampling distribution shows the variation in the median scores when re-estimated using a different set of runs. The reported point estimates of median in publications, as shown by dashed lines, do not provide any information about the variability in median scores and severely overestimate or underestimate the expected median. We use the same number of runs as reported by publications: $N = 5$ runs for DER, OTR and DrQ, $N = 1 0$ runs for SPR and $N = 2 0$ runs for CURL. Right. $9 5 \\%$ CIs for median and IQM scores (Section 4.3) for varying $N$ . There is a substantial uncertainty in median scores even with 50 runs. IQM has much smaller CIs than median. Note that when CIs overlap, properly accounting for uncertainty entails computing CIs for score differences (Figure A.15). "
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"text": "Remark. Following Amrhein et al. [2], Romer [87], Wasserstein et al. [112], we recommend using confidence intervals for measuring the uncertainty in results and showing effect sizes (e.g., performance improvements over baseline) that are compatible with the given data. Furthermore, we emphasize using statistical thinking but avoid statistical significance tests (e.g., $p$ -value $< 0 . 0 5 )$ because of their dichotomous nature (significant vs. not significant) and common misinterpretations [33, 35, 73] such as 1) lack of statistically significant results does not demonstrate the absence of effect (Figure 2, right), and 2) given enough data, any trivial effect can be statistically significant but may not be practically significant. ",
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"type": "text",
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"text": "3 Case Study: The Atari 100k benchmark ",
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"type": "text",
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"text": "We begin with a case study to illustrate the pitfalls arising from the naïve use of point estimates in the few-run regime. Our case study concerns the Atari 100k benchmark [50], an offshoot of the ALE for evaluating data-efficiency in deep RL. In this benchmark, algorithms are evaluated on only $1 0 0 \\mathrm { k }$ steps (2-3 hours of game-play) for each of its 26 games, versus 200M frames in the ALE benchmark. Prior reported results on this benchmark have been computed mostly from 3 [39, 55, 59, 72, 89, 95] or 5 runs [50, 51, 53, 54, 64, 66, 86, 107, 115], and more rarely, 10 [65, 93] or 20 runs [56]. ",
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"text": "Our case study compares the performance of five recent deep RL algorithms, namely: (1) DER [107] and (2) OTR [51], (3) DrQ5 [53], (4) CURL [56], and (5) SPR [93]. We chose these methods as representative of influential algorithms within this benchmark. Since good performance on one game can result in unduly high sample means without providing much information about performance on other games, it is common to measure performance on Atari $1 0 0 \\mathrm { k }$ using sample medians. Refer to Appendix A.2 for more details about the experimental setup. ",
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"text": "We investigate statistical variations in the few-run regime by evaluating 100 independent runs for each algorithm, where the score for a run is the average returns obtained in 100 evaluation episodes taking place after training. Each run corresponds to training one algorithm on each of the 26 games in Atari $1 0 0 \\mathrm { k }$ . This provides us with $2 6 \\times 1 0 0$ scores per algorithm, which we then subsample with replacement to 3–100 runs. The subsampled scores are then used to produce a collection of point estimates whose statistical variability can be measured. We begin by using this experimental protocol to highlight statistical concerns regarding median normalized scores. ",
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"text": "High variability in reported results. Our first observation is that the sample medians reported in the literature exhibit substantial variability when viewed as random quantities that depend on a small number of sample runs (Figure 2, left). This shows that there is a fairly large potential for drawing erroneous conclusions based on point estimates alone. As a concrete example, our analysis suggests that DER may in fact be better than OTR, unlike what the reported point estimates suggest. We conclude that in the few-run regime, point estimates are unlikely to provide definitive answers to the question: “Would we draw the same conclusions were we to re-evaluate our algorithm with a different set of runs?” ",
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"text": "",
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"text": "Substantial bias in sample medians. The sample median is a biased estimator of the true median: $\\mathbb { E } [ \\mathbf { M e d i a n } ( \\bar { X } _ { 1 : M } ) ] \\neq$ Median $\\left( \\mathbb { E } [ X _ { 1 : M } ] \\right)$ in general. In the few-run regime, we find that this bias can dominate the comparison between algorithms, as evidenced in Fig",
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"img_path": "images/54c2f614649e9bb0207d69a1082eea7875a56549fabdd02ad13b6dd73af7eb1d.jpg",
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"image_caption": [
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"Figure 3: Expected sample median of task means. The expected score for $N$ runs is computed by repeatedly subsampling $N$ runs with replacement out of 100 runs for 100,000 times. "
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"text": "ure 3. For example, the score difference between sample medians with 5 and 100 runs for SPR $( + 0 . 0 3$ points) is about $36 \\%$ of its mean improvement over $\\mathrm { D r Q } ( \\varepsilon )$ $( + 0 . 0 8$ points). Adding to the issue, the magnitude and sign of this bias strongly depends on the algorithm being evaluated. ",
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"text": "Statistical concerns cannot be satisfactorily addressed with few runs. While claiming improvements with 3 or fewer runs may naturally raise eyebrows, folk wisdom in experimental RL suggests that 20 or 30 runs are enough. By calculating $9 5 \\%$ confidence interval6 on sample medians for a varying number of runs (Figure 2, right), we find that this number is closer to 50–100 runs in Atari $1 0 0 \\mathrm { k }$ – far too many to be computationally feasible for most research projects. ",
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"text": "Consider a setting in which an algorithm is known to be better – what is the reliability of median and IQM (Section 4.3) for accurately assessing performance differences as the number of runs varies? Specifically, we consider two identical $N$ -run experiments involving SPR, except that we artificially inflate one of the experiments’ scores by a fixed fraction or lift of $+ \\ell \\%$ (Figure 4). In particular, $\\ell = 0$ corresponds to running the same experiment twice but with different runs. We find that statistically defensible improvements with median scores is only achieved for 25 runs $\\ell = 2 5$ ) and 100 runs $\\ell = 1 0$ ). With $\\ell = 0$ , even 100 runs are insufficient, with deviations of $2 0 \\%$ possible. ",
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"text": "Changes in evaluation protocols invalidates comparisons to prior work. A typical and relatively safe approach for measuring the performance of an RL algorithm is to average the scores received in their final training episodes [69]. However, the field has seen a number of alternative protocols used, including reporting the maximum evaluation score achieved during training [1, 3, 75] or across multiple runs [32, 47, 82]. A similar protocol is also used by CURL and SUNRISE [59] (Appendix A.4). ",
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"text": "Results produced under alternative protocols involving maximum are generally incomparable with end-performance reported results. On Atari 100k, we find that the two protocols produce substantially different results (Figure 5), of a magnitude greater than the actual difference in score. In particular, evaluating DER with CURL’s protocol results in scores far above those reported for CURL. In other words, this gap in evaluation procedures resulted in CURL being assessed as achieving a greater true median than DER, where our experiment gives strong support to DER being superior. Similarly, we find that a lot of SUNRISE’s improvement over DER can be explained by the change in evaluation protocol (Figure 5). Refer to Appendix A.4 for discussion on pitfalls of such alternative protocols. ",
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"type": "text",
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"text": "4 Recommendations and Tools for Reliable Evaluation ",
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"text": "Our case study shows that the increase in the number of runs required to address the statistical uncertainty issues is typically infeasible for computationally demanding deep RL benchmarks. In this section, we identify three tools for improving the quality of experimental reporting in the few-run regime, all aligned with the principle of accounting for statistical uncertainty in results. ",
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"text": "4.1 Stratified Bootstrap Confidence Intervals ",
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"text": "We first reaffirm the importance of reporting interval estimates to indicate the range within which an algorithm’s aggregate performance is believed to lie. Concretely, we propose using bootstrap CIs [29] with stratified sampling for aggregate performance, a method that can be applied to small sample sizes and is better justified than reporting sample standard deviations in this context. While prior work has recommended using bootstrap CIs for reporting uncertainty in single task mean scores with $N$ runs [16, 20, 41], this is less useful when $N$ is small (Figure A.18), as bootstrapping assumes that re-sampling from the data approximates sampling from the true distribution. We can do better by aggregating samples across tasks, for a total of $M N$ random samples. ",
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"type": "image",
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"img_path": "images/a8daa5e50e5dfd4123df034240efdc49a14a53a92a2f1c351c7735e62533768c.jpg",
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"image_caption": [
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"Figure 4: Detecting score lifts. Left. $9 5 \\%$ CIs for observed lift with median scores, and Right. $9 5 \\%$ CIs for observed lift with IQM (Section 4.3) when comparing SPR with an algorithm that performs $\\ell \\%$ better. IQM requires fewer runs than median for small uncertainty. "
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"image_caption": [
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"Figure 5: Normalized DER scores with non-standard evaluation protocols. Gains from SUNRISE and CURL over DER can mostly be explained by such protocols. "
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"Figure 6: Validating $95 \\%$ Stratified Bootstrap CIs for a varying number of runs for median and IQM scores for DER. The true coverage $\\%$ is computed by sampling 10,000 sets of K runs without replacement from 200 runs and checking the fraction of $9 5 \\%$ CIs that contains the true estimate approximation based on 200 runs. Note that we evaluate additional 100 runs for DER for an accurate point estimate. Percentile CIs has the best coverage while achieving a small width compared to other methods. Also, CI widths for IQM are much smaller than that of median. We also note that with 3 runs, bootstrap CIs underestimate the true $9 5 \\%$ CIs and might require a larger nominal coverage rate to achieve true $9 5 \\%$ coverage. "
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"text": "To compute the stratified bootstrap CIs, we re-sample runs with replacement independently for each task to construct an empirical bootstrap sample with $N$ runs each for $M$ tasks from which we calculate a statistic and repeat this process many times to approximate the sampling distribution of the statistic. We measure the reliability of this technique in Atari $1 0 0 \\mathrm { k }$ for variable $N$ , by comparing the nominal coverage of $9 5 \\%$ to the “true” coverage from the estimated CIs (Figure 6) for different bootstrap methods (see [30] and Appendix A.5). We find that percentile CIs provide good interval estimates for as few as $N = 1 0$ runs for both median and IQM scores (Section 4.3). ",
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"text": "4.2 Performance Profiles ",
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"text": "Most deep RL benchmarks yield scores that vary widely between tasks and may be heavy-tailed, multimodal, or possess outliers (e.g., Figure A.14). In this regime, both point estimates, such as mean and median scores, and interval estimates of these quantities paint an incomplete picture of an algorithm’s performance [24, Section 3]. Instead, we recommend the use of performance profiles [26], commonly used in benchmarking optimization software. While performance profiles from Dolan and Moré [26] correspond to empirical cumulative distribution functions without any uncertainty estimates, profiles proposed herein visualize the empirical tail distribution function (Section 2) of a random score (higher curve is better), with pointwise confidence bands based on stratified bootstrap. ",
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"text": "By representing the entire set of normalized scores $x _ { 1 : M , 1 : N }$ visually, performance profiles reveal performance variability across tasks much better than interval estimates of aggregate metrics. Although tables containing per-task mean scores and standard deviations can reveal this variability, such tables tend to be overwhelming for more than a few tasks.7 In addition, performance profiles are robust to outlier runs and insensitive to small changes in performance across all tasks [26]. ",
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"text": "In this paper, we propose the use of a performance profile we call run-score distributions or simply score distributions (Figure 7, right), particularly well-suited to the few-run regime. A score distribution shows the fraction of runs above a certain normalized score and is given by ",
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"image_caption": [
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"Figure 7: Performance profiles on Atari 100k based on score distributions (left), which we recommend, and average score distributions (right). Shaded regions show pointwise $9 5 \\%$ confidence bands based on percentile bootstrap with stratified sampling. The profiles on the left are more robust to outliers and have smaller confidence bands. We use 10 runs to show the robustness of profiles with a few runs. For SimPLe [50], we use the 5 runs from their reported results. The $\\tau$ value where the profiles intersect $y = 0 . 5$ shows the median while for a non-negative random variable, area under the performance profile corresponds to the mean. "
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"text": "$$\n\\hat { F } _ { X } ( \\tau ) = \\hat { F } ( \\tau ; x _ { 1 : M , 1 : N } ) = \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\hat { F } _ { m } ( \\tau ) = \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\mathbb { 1 } [ x _ { m , n } > \\tau ] .\n$$",
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"text": "One advantage of the score distribution is that it is an unbiased estimator of the underlying distribution $\\begin{array} { r } { F ( \\tau ) = \\frac { 1 } { N } \\sum _ { m = 1 } ^ { M } F _ { m } ( \\tau ) } \\end{array}$ . Another advantage is that an outlier run with extremely high score can change the output of score distribution for any $\\tau$ by at most a value of $\\frac { 1 } { M N }$ . ",
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"text": "It is useful to contrast score distributions to average-score distributions, originally proposed in the context of the ALE [5] as a generalization of the median score. Average-score distributions correspond to the performance profile of a random variable $\\bar { X }$ , $\\hat { F } _ { \\bar { X } } ( \\tau ) = \\hat { F } ( \\tau ; \\bar { x } _ { 1 : M } )$ , which shows the fraction of tasks on which an algorithm performs better than a certain score. However, such distributions are a biased estimate of the thing they seek to represent. Run-score distributions are more robust than average-score distributions, as they are a step function in $1 / M N$ versus $1 / M$ intervals, and typically has less variance: $\\begin{array} { r } { \\sigma _ { X } ^ { 2 } = \\frac { 1 } { M ^ { 2 } N } \\sum _ { m = 1 } ^ { M } F _ { m } ( \\tau ) ( 1 - F _ { m } ( \\tau ) ) \\quad } \\end{array}$ versus $\\begin{array} { r } { \\sigma _ { \\bar { X } } ^ { 2 } = \\frac { 1 } { M ^ { 2 } } \\sum _ { m = 1 } ^ { M } F _ { \\bar { X } _ { m } } ( \\tau ) ( 1 - F _ { \\bar { X } _ { m } } ( \\tau ) ) } \\end{array}$ . Figure 7 illustrates these differences. ",
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"text": "4.3 Robust and Efficient Aggregate Metrics ",
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"text": "Performance profiles allow us to compare different methods at a glance. If one curve is strictly above another, the better method is said to stochastically dominate8 the other [27, 61]. In RL benchmarks with a large number of tasks, however, stochastic dominance is rarely observed: performance profiles often intersect at multiple points. Finer quantitative comparisons must therefore entail aggregate metrics. ",
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"text": "We can extract a number of aggregate metrics from score distributions, including median (mixing runs and tasks) and mean normalized scores (matching our usual definition). As we already argued that these metrics are deficient, we now consider interesting alternatives also derived from score distributions. ",
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"image_caption": [
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| 688 |
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"Figure 8: Aggregate metrics. For a non-negative random variable $X$ , IQM corresponds to the red shaded region while optimality gap corresponds to the orange shaded region in the performance profile of $X$ . "
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"text": "culates the mean score of the remaining $50 \\%$ runs $\\scriptstyle ( = \\lfloor N M / 2 \\rfloor$ for $N$ runs each on $M$ tasks). IQM interpolates between mean and median across runs, which are $0 \\%$ and almost $5 0 \\%$ trimmed means respectively. Compared to sample median, IQM is a better indicator of overall performance as it is calculated using $50 \\%$ of the combined runs while median only depends on the performance ordering across tasks and not on the magnitude except at most 2 tasks. For example, zero scores on nearly half of the tasks does not affect the median while IQM exhibits a severe degradation. Compared to mean, IQM is robust to outliers, yet has considerably less bias than median (Figure A.17). While median is more robust to outliers than IQM, this robustness comes at the expense of statistical efficiency, which is crucial in the few-run regime: IQM results in much smaller CIs (Figure 2 (right) and 6) and is able to detect a given improvement with far fewer runs (Figures 4 and A.15). ",
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"img_path": "images/560e26e5f432686fc445b55c288373a79afcfc1d603e379fa2269bb41358c228.jpg",
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"image_caption": [
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"Figure 9: Aggregate metrics on Atari 200M with $9 5 \\%$ CIs based on 55 games with sticky actions [69]. Higher mean, median and IQM scores and lower optimality gap are better. The CIs are estimated using the percentile bootstrap with stratified sampling. IQM typically results in smaller CIs than median scores. Large values of mean scores relative to median and IQM indicate being dominated by a few high performing tasks, for example, DreamerV2 and M-IQN obtain normalized scores above 50 on the game JAMESBOND. Optimality gap is less susceptible to outliers compared to mean scores. We compare DQN (Nature) [75], DQN with Adam optimizer, C51 [6], REM [1], Rainbow [42], IQN [22], Munchausen-IQN (M-IQN) [109], and DreamerV2 [38]. All results are based on 5 runs per game except for M-IQN and DreamerV2 which report results with 3 and 11 runs. "
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"text": "As a robust alternative to mean, we recommend using the optimality gap: the amount by which the algorithm fails to meet a minimum score of $\\gamma = 1 . 0$ (orange region in Figure 8). This assumes that a score of 1.0 is a desirable target beyond which improvements are not very important, for example when the aim is to obtain human-level performance [e.g., 3, 23]. Naturally, the threshold $\\gamma$ may be chosen differently, which we discuss further in Appendix A.7. ",
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"text": "If one is interested in knowing how robust an improvement from an algorithm $X$ over an algorithm $Y$ is, another possible metric to consider is the average probability of improvement – this metric shows how likely it is for $X$ to outperform $Y$ on a randomly selected task. Specifically, $P ( X >$ $\\begin{array} { r } { Y ) = \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\overset { \\textstyle } { P } ( X _ { m } > Y _ { m } ) , } \\end{array}$ , where $P ( X _ { m } > Y _ { m } )$ (Equation A.2) is the probability that $X$ is better than $Y$ on task $m$ . Note that, unlike IQM and optimality gap, this metric does not account for the size of improvement. While finding the best aggregate metric is still an open question and is often dependent on underlying normalized score distribution, our proposed alternatives avoid the failure modes of prevalent metrics while being robust and requiring fewer runs to reduce uncertainty. ",
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"type": "text",
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"text": "5 Re-evaluating Evaluation on Deep RL Benchmarks ",
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"text": "Arcade Learning Environment. Training RL agents for 200M frames on the ALE [5, 69] is the most widely recognized benchmark in deep RL. We revisit some popular methods which demonstrated progress on this benchmark and reveal discrepancies in their findings as a consequence of ignoring the uncertainty in their results (Figure 9). For example, DreamerV2 [38] exhibits a large amount of uncertainty in aggregate scores. While M-IQN [109] claimed better performance than Dopamine Rainbow9 [42] in terms of median normalized scores, their interval estimates strikingly overlap. Similarly, while C51 [5] is considered substantially better than DQN [75], the interval estimates as well as performance profiles for DQN (Adam) and C51 overlap significantly. ",
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"type": "text",
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"text": "Figure 9 reveals an interesting limitation of aggregate metrics: depending on the choice of metric, the ordering between algorithms changes (e.g., Median vs. IQM). The inconsistency in ranking across aggregate metrics arises from the fact that such metrics only capture a specific aspect of overall performance across tasks and runs. Additionally, the change of algorithm ranking between optimality gap and IQM/median scores reveal that while recent algorithms typically show performance gains relative to humans on average, their performance seems to be worse on games below human performance. Since performance profiles capture the full picture, they would often illustrate why such inconsistencies exist. For example, optimality gap and IQM can be both read as areas in the profile (Figure 8). The performance profile in Figure 10 (left) illustrates the nuances present when comparing different algorithms. For example, IQN seems to be better than Rainbow for $\\tau \\geq 2$ , but worse for $\\tau < 2$ . Similarly, the profiles of DreamerV2 and M-IQN for $\\tau < 8$ intersect at multiple points. To compare sample efficiency of the agents, we also present their IQM scores as a function of number of frames in Figure 10 (right). ",
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"img_path": "images/830f89aa4c531f81455785c434fee73d1deb1cbf9f3a25f14b3b4134b3919561.jpg",
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"image_caption": [
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| 796 |
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"Figure 10: Atari 200M evaluation. Left. Score distributions using human-normalized scores obtained after training for 200M frames. Right. Sample-efficiency of agents as a function of number of frames measured via IQM human-normalized scores. Shaded regions show pointwise $9 5 \\%$ percentile stratified bootstrap CIs. "
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"image_caption": [
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"Figure 11: DeepMind Control Suite evaluation results, averaged across 6 tasks, on the 100k and $5 0 0 \\mathrm { k }$ benchmark. We compare $\\mathrm { S A C + A E }$ [114], SLAC [58], Dreamer [37], CURL [98], RAD [57], DrQ [53], PISAC [60], SUNRISE [59], and CURL-D2RL [97]. The ordering of the algorithms in the left figure is based on their claimed relative performance – all algorithms except Dreamer claimed improvement over at least one algorithm placed below them. (a) Interval estimates show $9 5 \\%$ stratified bootstrap CIs for methods with individual runs provided by their respective authors and $9 5 \\%$ studentized CIs for CURL, CURL-D2RL, and SUNRISE. Normalized scores are computed by dividing by the maximum score $( = 1 0 0 0 )$ . (b) Score distributions. (c) The $i ^ { t h }$ column in the rank distribution plots show the probability that a given method is assigned rank $_ { i }$ averaged across all tasks. The ranks are estimated using 200,000 stratified bootstrap re-samples. "
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"text": "DeepMind Control Suite. Recent continuous control papers benchmark performance on 6 tasks in DM Control [104] at $1 0 0 \\mathrm { k }$ and $5 0 0 \\mathrm { k }$ steps. Typically, such papers claim improvement based on higher mean scores per task regardless of the variability in those scores. However, we find that when accounting for uncertainty in results, most algorithms do not consistently rank above algorithms they claimed to improve upon (Figure 11c and 11b). Furthermore, there are huge overlaps in $9 5 \\%$ CIs of mean normalized scores for most algorithms (Figure 11a). These findings suggest that a lot of the reported improvements are spurious, resulting from randomness in the experimental protocol. ",
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"type": "text",
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"text": "Procgen benchmark. Procgen [18] is a popular benchmark, consisting of 16 diverse tasks, for evaluating generalization in RL. Recent papers report mean PPO-normalized scores on this benchmark to emphasize the gains relative to PPO [92] as most methods are built on top of it. However, Figure 12 (left) shows that PPO-normalized scores typically have a heavy-tailed distribution making the mean scores highly dependent on performance on a small fraction of tasks. Instead, we recommend using normalization based on the estimated minimum and maximum scores on ProcGen [18] and reporting aggregate metrics based on such scores (Figure A.32). While publications sometimes make binary claims about whether they improve over prior methods, such improvements are inherently probabilistic. To reveal this discrepancy, we investigate the following question: “What is the probability that an algorithm which claimed improvement over a prior algorithm performs better than it?” (Figure 12, right). While this probability does not distinguish between two algorithms which uniformly improve on all tasks by $1 \\%$ and $100 \\%$ , it does highlight how likely an improvement is. For example, there is only a $4 0 - 5 0 \\%$ chance that UCB-DrAC [81] improves upon PLR [48]. We note that a number of improvements reported in the existing literature are only $5 0 - 7 0 \\%$ likely. ",
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"img_path": "images/71d17ad3a8d766f85f7b26fa510b8255bb57edd855d91fe0cb5a61a22b7ca5bd.jpg",
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"image_caption": [
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| 859 |
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"Figure 12: Procgen evaluation results based on easy mode comparisons [80] with 16 tasks. Left. Score distributions which compare PPO [92], MixReg [111], UCB-DrAC [81], PLR [48], PPG [19] and IDAAC [80]. Shaded regions indicate $9 5 \\%$ percentile stratified bootstrap CIs. Right. Each row shows the probability of improvement, with $9 5 \\%$ bootstrap CIs, that the algorithm $X$ on the left outperforms algorithm $Y$ on the right, given that $X$ was claimed to be better than $Y$ . For all algorithms, results are based on 10 runs per task. "
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"type": "text",
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"text": "6 Discussion ",
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"text_level": 1,
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"text": "We saw, both in our case study on the Atari $1 0 0 \\mathrm { k }$ benchmark and with our analysis of other widely-used RL benchmarks, that statistical issues can have a sizeable influence on reported results, in particular when point estimates are used or evaluation protocols are not kept constant within comparisons. Despite earlier calls for more experimental rigor in deep RL [16, 20, 21, 41, 49, 83] (discussed in Appendix A.3), our analysis shows that the field has not yet found sure footing in this regards. ",
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"text": "In part, this is because the issue of reproducibility is a complex one; where our work is concerned with our confidence about and interpretation of reported results (what Goodman et al. [34] calls results reproducibility), others [79] have highlighted that there might be missing information about the experiments themselves (methods reproducibility). We remark that the problem is not solved by fixing random seeds, as has sometimes been proposed [52, 77], since it does not really address the question of whether an algorithm would perform well under similar conditions but with different seeds. Furthermore, fixed seeds might benefit certain algorithms more than others. Nor can the problem be solved by the use of dichotomous statistical significance tests, as discussed in Section 2. ",
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"text": "One way to minimize the risks associated with statistical effects is to report results in a more complete fashion, paying close attention to bias and uncertainty within these estimates. To this end, our recommendations are summarized in Table 1. To further support RL researchers in this endeavour, we released an easy-to-use Python library, rliable along with a Colab notebook for implementing our recommendations, as well as all the individual runs used in our experiments10. Again, we emphasize the importance of published papers providing results for all runs to allow for future statistical analyses. ",
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"text": "A barrier to adoption of evaluation protocols proposed in this work, and more generally, rigorous evaluation, is whether there are clear incentives for researchers to do so, as more rigor generally entails more nuanced and tempered claims. Arguably, doing good and reproducible science is one such incentive. We hope that our findings about erroneous conclusions in published papers would encourage researchers to avoid fooling themselves, even if that requires tempered claims. That said, a more pragmatic incentive would be if conferences and reviewers required more rigorous evaluation for publication, e.g., NeurIPS 2021 checklist asks whether error bars are reported. Moving towards reliable evaluation is an ongoing process and we believe that this paper would greatly benefit it. ",
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"text": "Given the substantial influence of statistical considerations in experiments involving 40-year old Atari 2600 video games and low-DOF robotic simulations, we argue that it is unlikely that an increase in available computation will resolve the problem for the future generation of RL benchmarks. Instead, just as a well-prepared rock-climber can skirt the edge of the steepest precipices, it seems likely that ongoing progress in reinforcement learning will require greater experimental discipline. ",
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"text": "Societal Impacts ",
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"text": "This paper calls for statistical sophistication in deep RL research by accounting for statistical uncertainty in reported results. However, statistical sophistication can introduce new forms of statistical abuses and monitoring the literature for such abuses should be an ongoing priority for the research community. Moving towards reliable evaluation and reproducible research is an ongoing process and this paper only partly addresses it by providing tools for more reliable evaluation. That said, while accounting for uncertainty in results is not a panacea, it provides a strong foundation for trustworthy results on which the community can build upon, with increased confidence. In terms of broader societal impact of this work, we do not see any foreseeable strongly negative impacts. However, this paper could positively impact society by constituting a step forwards in rigorous few-run evaluation regime, which reduces computational burden on researchers and is “greener” than evaluating a large number of runs. ",
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"text": "Acknowledgments ",
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"text": "We thank Xavier Bouthillier, Dumitru Erhan, Marlos C. Machado, David Ha, Fabio Viola, Fernando Diaz, Stephanie Chan, Jacob Buckman, Danijar Hafner and anonymous NeurIPS’ reviewers for providing valuable feedback for an earlier draft of this work. We also acknowledge Matteo Hessel, David Silver, Tom Schaul, Csaba Szepesvári, Hado van Hasselt, Rosanne Liu, Simon Kornblith, Aviral Kumar, George Tucker, Kevin Murphy, Ankit Anand, Aravind Srinivas, Matthew Botvinick, Clare Lyle, Kimin Lee, Misha Laskin, Ankesh Anand, Joelle Pineau and Braham Synder for helpful discussions. We also thank all the authors who provided individual runs for their corresponding publications. We are also grateful for general support from Google Research teams in Montréal and elsewhere. ",
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"text": "References ",
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Improving sample efficiency in model-free reinforcement learning from images. arXiv preprint arXiv:1910.01741, 2019. \n[115] Jinhua Zhu, Yingce Xia, Lijun Wu, Jiajun Deng, Wengang Zhou, Tao Qin, and Houqiang Li. Masked contrastive representation learning for reinforcement learning. arXiv preprint arXiv:2010.07470, 2020. \n[116] Donglin Zhuang, Xingyao Zhang, Shuaiwen Leon Song, and Sara Hooker. Randomness in neural network training: Characterizing the impact of tooling. arXiv preprint arXiv:2106.11872, 2021. ",
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| 1 |
+
# Implicit Bias of SGD for Diagonal Linear Networks: a Provable Benefit of Stochasticity
|
| 2 |
+
|
| 3 |
+
Scott Pesme EPFL scott.pesme@epfl.ch
|
| 4 |
+
|
| 5 |
+
Loucas Pillaud-Vivien EPFL loucas.pillaud-vivien@epfl.ch
|
| 6 |
+
|
| 7 |
+
Nicolas Flammarion EPFL nicolas.flammarion@epfl.ch
|
| 8 |
+
|
| 9 |
+
# Abstract
|
| 10 |
+
|
| 11 |
+
Understanding the implicit bias of training algorithms is of crucial importance in order to explain the success of overparametrised neural networks. In this paper, we study the dynamics of stochastic gradient descent over diagonal linear networks through its continuous time version, namely stochastic gradient flow. We explicitly characterise the solution chosen by the stochastic flow and prove that it always enjoys better generalisation properties than that of gradient flow. Quite surprisingly, we show that the convergence speed of the training loss controls the magnitude of the biasing effect: the slower the convergence, the better the bias. To fully complete our analysis, we provide convergence guarantees for the dynamics. We also give experimental results which support our theoretical claims. Our findings highlight the fact that structured noise can induce better generalisation and they help explain the greater performances of stochastic gradient descent over gradient descent observed in practice.
|
| 12 |
+
|
| 13 |
+
# 1 Introduction
|
| 14 |
+
|
| 15 |
+
Understanding the performance of neural networks is certainly one of the most thrilling challenges for the current machine learning community. From the theoretical point of view, progress has been made in several directions: we have a better functional analysis description of neural networks [3] and we steadily understand the convergence of training algorithms [29, 10] as well as the role of initialisation [20, 12]. Yet there remain many unanswered questions. One of which is why do the currently used training algorithms converge to solutions which generalise well, and this with very little use of explicit regularisation [39].
|
| 16 |
+
|
| 17 |
+
To understand this phenomenon, the concept of implicit bias has emerged: if over-fitting is benign, it must be because the optimisation procedure converges towards some particular global minimum which enjoys good generalisation properties. Though no explicit regularisation is added, the algorithm is implicitly selecting a particular solution: this is referred to as the implicit bias of the training procedure. The implicit regularisation of several algorithms has been studied, the simplest and most emblematic being that of gradient descent and stochastic gradient descent in the least-squares framework: they both converge towards the global solution which has the lowest squared distance from the initialisation. For logistic regression on separable data, Soudry et al. show in the seminal paper [31] that gradient descent selects the max-margin classifier. This type of result has then been extended to neural networks and to other frameworks. Overall, characterising the implicit bias of gradient methods has almost always come down to unveiling mirror-descent like structures which underlie the algorithms.
|
| 18 |
+
|
| 19 |
+
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| 20 |
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Figure 1: Sparse regression with $n = 4 0$ , $d = 1 0 0$ , $\| \beta _ { \ell _ { 0 } } ^ { * } \| _ { 0 } = 5$ , $x _ { i } \sim \mathcal { N } ( 0 , I ) y _ { i } = x _ { i } ^ { \top } \beta _ { \ell _ { 0 } } ^ { * } .$ Left: for initialisation scale $\alpha = 0 . 0 5$ , SGD converges towards a solution which generalises better than GD. Right: for different values of the initialisation scale $\alpha$ , the solution recovered by SGD has better validation loss than that of GD. The sparsifying effect due to their implicit biases differ by more than an order of magnitude. See Section 5.1 for the precise experimental setup.
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+
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While mostly all of the results focus on gradient descent, it must be pointed out that this full batch algorithm is not used in practice for neural networks since it does not lead to solutions which generalise well [23]. Instead, results on stochastic gradient descent, which is widely used and shows impressive results, are still missing or unsatisfactory. This has certainly to do with the fact that grasping the nature of the noise induced by the stochasticity of the algorithm is particularly hard: it mixes properties from the model’s architecture, the data’s distribution and the loss. In our work, by focusing on simplified neural networks, we answer to the following fundamental questions: do SGD’s and GD’s implicit bias differ? What is the role of SGD’s noise over the algorithm’s implicit bias?
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+
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The simplified neural networks which we consider are diagonal linear neural networks; despite their simplicity they have become popular since they already enable to grasp the complexity of more general networks. Indeed, they highlight important aspects of the theoretical concerns of modern machine learning: the neural tangent kernel regime, the roles of over-parametrisation, of the initialisation and of the step size. For a regression problem where we assume the existence of an interpolating solution, we study stochastic gradient descent through its continuous version, namely stochastic gradient flow (SGF). Though the continuous modelling of SGD has not yet led to many fruitful results compared to the well studied gradient flow, we believe it is because capturing the essence of the stochastic noise is particularly difficult. It has generally been done in a non realistic and over simplified manner, such as considering constant and isotropic noise. In our work, we attach peculiar attention to the adequate modelling of the noise. Tools from Itô calculus are then leveraged in order to derive exact formulas, quantitative bounds and interesting interpretations for our problem.
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# 1.1 Main contributions and paper organisation.
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In Section 2, we start by introducing the setup of our problem as well as the continuous modelisation of stochastic gradient descent. Then, in Section 3, we state our main result on the implicit bias of the stochastic gradient flow. We informally formulate it here and illustrate it in Figure 1:
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+
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Theorem 1 (Informal). Stochastic gradient flow over diagonal linear networks converges with high probability to a zero-loss solution which enjoys better generalisation properties than the one obtained by gradient flow. Furthermore, the speed of convergence of the training loss controls the magnitude of the biasing effect: the slower the convergence, the better the bias.
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+
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Unlike previous works [14, 36], in addition to characterising the implicit bias effect of SGF, we also prove the convergence of the iterates towards a zero-loss solution with high-probability. To accomplish this, we leverage in Section 4 the fact that the iterates follow a stochastic continuous mirror descent with a time-varying potential. We support our results experimentally and validate our model in Section 5.
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| 33 |
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| 34 |
+
# 1.2 Related work
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| 35 |
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| 36 |
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As recalled, implicit bias has a recent history that has been initiated by the seminal work [31] on max-margin classification with log-loss for a linear setup and separable data. This work has been extended to other architectures, e.g. multiplicative parametrisations [14], linear networks [22] and more general homogeneous neural networks [27, 11]. In [36] the authors show that the scale of the initialisation leads to an interpolation between the neural tangent kernel regime [20, 12] (which is a linear regression on fixed features) leading to $\ell _ { 2 }$ minimum norm solutions and the rich regimes leading to $\ell _ { 1 }$ minimum norm solutions. Note that these works focus on full batch gradient descent (or flow) and are deeply linked to mirror descent.
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| 37 |
+
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| 38 |
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While the links between SGD’s stochasticity and generalisation have been looked into in numerous works [28, 21, 16, 18, 24], no such explicit characterisation of implicit regularisation have ever been given. It has been empirically observed that SGD often outputs models which generalise better than GD [23, 21, 16]. One suggested explanation is that SGD is prone to pick flatter solutions than GD and that bad generalisation solutions are correlated with sharp minima, i.e., with strong curvature, while good generalisation solutions are correlated with flat minima, i.e., with low curvature [17, 23]. This idea has been further investigated by adopting a random walk on random landscape modelling [18], by suggesting that SGD’s noise is smoothing the loss landscape, thus eliminating the sharp minima [24], by considering a dynamical stability perspective [38] or by interpreting SGD as a diffusion process [16, 21, 8]. Recently, label-noise has been shown to influence the implicit bias of SGD, by biasing the solution towards the origin for quadratically-parameterized models [15] or by implicitly regularising the expected squared norm of the gradient of the model with respect to the weights [5]. Thus, if the notion of implicit bias of GD is fairly well understood both in the cases of regression and classification, it remains unclear for SGD, and its explicit characterisation is missing.
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+
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The linear diagonal neural networks we consider have been studied in the case of gradient descent [33] and stochastic gradient descent with label noise [15]. In both cases the authors show that this model has the ability to implicitly bias the training procedure to help retrieve a sparse predictor. The link between gradient descent and mirror descent for this model has been initiated by [13] and further exploited by the same author in [37, 34] for its sparse inducing property.
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Contrary to the deterministic case, the modelling of stochastic gradient descent as a stochastic differential equation is quite recent, see [28, 21]. However, as highlighted by [1], early attempts often suffer from the drawback that they model the noise using a constant covariance matrix. On the contrary, state dependant noise has now become the legitimate manner for modelling SGD as a stochastic gradient flow and it is shown in [26] that it can be done consistently. Yet, noise modelling still remains the principal issue [35] as it influences largely the behaviour of the dynamics [8, 9].
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# 1.3 Notations
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For input data $( x _ { 1 } , \ldots , x _ { n } ) \in ( \mathbb { R } ^ { d } ) ^ { n }$ and output $( y _ { 1 } , \dots , y _ { n } ) \in \mathbb { R } ^ { n }$ , we denote respectively $X \in$ $\mathbb { R } ^ { n \times d }$ the design matrix whose $i$ -th row is feature $x _ { i } \in \mathbb { R } ^ { d }$ and $y \in \mathbb { R } ^ { n }$ the vector of outputs. $\mathbb { R } _ { + } ^ { * }$ denotes the set of strictly positive real numbers. For $p = 1 , 2$ , the $\ell _ { p }$ -norm of $x \in \mathbb { R } ^ { d }$ is $\| x \| _ { p } ^ { p } =$ $\sum _ { i } ^ { d } | x _ { i } | ^ { p }$ . The operations $\odot$ will stand for coordinate-wise product between vector: $[ u \odot v ] _ { i } = u _ { i } v _ { i }$ and $u ^ { 2 } = u \odot u$ . For $p \in \mathbb { N } ^ { * }$ , we also define $u ^ { p } : = u \odot \ldots \odot u$ , the $p$ times product of $u$ with itself. All inequalities between vectors should be understood value by value. For $f , g \in \mathbb { R }$ , the existence of $C > 0$ such that $f \leq C g$ and $C g \leq f$ will be denoted $f \leq O ( g )$ and $\Omega ( g ) \leq f$ respectively. We shall use the symbole $\widetilde O$ when this is true up to log factors. For a vector $u \in \mathbb { R } ^ { d }$ , $\mathrm { d i a g } ( u )$ denotes the $d \times d$ diagonal matrix which has its diagonal equal to $u$ . For a matrix $M \in \mathbb { R } ^ { d \times d }$ , $\mathrm { d i a g } ( M )$ denotes the vector $( M _ { 1 1 } , \dots , M _ { d d } ) \in \mathbb { R } ^ { d }$ . The indexed vector $\beta ^ { * }$ will stand for any $\beta$ interpolating the data, i.e. any vector in the affine space $\{ \beta \in \mathbb { R } ^ { d } s . t$ , $X \beta = Y \}$ of dimension at least $d - n$ . Out of all these, let $\begin{array} { r l } { \beta _ { \ell _ { 1 } } ^ { * } = } & { { } \arg \operatorname* { m i n } \quad \| \beta \| _ { 1 } } \end{array}$ . For $z$ any vector, $z _ { \infty }$ or $z ^ { \infty }$ will always designate of $\operatorname* { l i m } _ { t \to \infty } z _ { t }$ . ${ \boldsymbol { \beta } } \in { \mathbb { R } } ^ { d }$ $X \beta { = } y$
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+
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| 48 |
+
# 2 Setup and preliminaries
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| 49 |
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# 2.1 Architecture and algorithm.
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+
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Overparametrised noiseless regression. We consider a linear regression problem with outputs $( y _ { 1 } , \dotsc , y _ { n } ) \in \mathbb { R } ^ { n }$ and inputs $( x _ { 1 } , \ldots , x _ { n } ) \in ( \mathbb { R } ^ { d } ) ^ { n }$ . We study an overparametrised setting $( n < d )$
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| 53 |
+
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| 54 |
+
and assume that there exists at least one interpolating parameter $\beta ^ { * } \in \mathbb { R } ^ { d }$ which perfectly fits the training set, i.e. $y _ { i } = \langle \beta ^ { * } , x _ { i } \rangle$ for all $1 \leq i \leq n$ . We parametrise the regression vector $\beta$ as $\beta _ { w }$ with $w \in \mathbb { R } ^ { p }$ . We will see that though in the end our final models $x \mapsto \langle \beta _ { w } , x \rangle$ are classical linear models whatever the parametrisation $w \mapsto \beta _ { w }$ , the choice of this parametrisation has crucial consequences on the solution recovered by the learning algorithms. We study the quadratic loss and the overall loss is written as:
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| 55 |
+
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| 56 |
+
$$
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| 57 |
+
L ( w ) = L ( \beta _ { w } ) : = \frac { 1 } { 4 n } \sum _ { i = 1 } ^ { n } ( \langle \beta _ { w } , x _ { i } \rangle - y _ { i } ) ^ { 2 } = \frac { 1 } { 4 n } \sum _ { i = 1 } ^ { n } \langle \beta _ { w } - \beta ^ { * } , x _ { i } \rangle ^ { 2 } ,
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| 58 |
+
$$
|
| 59 |
+
|
| 60 |
+
where by abuse of notation we use $L ( w ) = L ( \beta _ { w } )$ .
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| 61 |
+
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| 62 |
+
2-layer diagonal linear network. The simplest parametrisation of $\beta _ { w }$ is to consider $\beta _ { w } \ = \ w$ which corresponds to the classical least-squares framework. It is well known that in this case, many first order methods (GD, SGD, with and without momentum) will converge towards the same solution: we say that they have the same implicit bias. This is experimentally not the case for neural networks where SGD has been shown to lead to solutions which have better generalisation properties compared to GD [23]. To theoretically confirm this observation, we study a simple non-linear parametrisation: $\beta _ { w } = w _ { + } ^ { 2 } - w _ { - } ^ { 2 }$ with $w \doteq [ w _ { + } , w _ { - } ] ^ { \intercal } \in \mathbb { R } ^ { 2 d }$ . We point out that it is 2-positive homogeneous and that it is equivalent to the parametrisation $\beta _ { u , v } = u \odot v$ with $u , v \in \mathbb { R } ^ { d }$ . It should be thought of a simplified linear network of depth 2 (see [36, Section 4] for more details). We consider two weight vectors $w _ { + }$ and $w _ { - }$ (and not only $\beta _ { w } = w ^ { 2 }$ ) in order to ensure that our final linear predictor parameter $\beta _ { w }$ can take negative values. For the sake of completeness, the study of diagonal linear networks of arbitrary depth $p \geq 3$ is done in Appendix E.2. Also note that additionally to being a toy neural model, it has received recent attention for its practical ability to induce sparsity [33, 34, 15] or to solve phase retrieval problems [37].
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+
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| 64 |
+
Stochastic Gradient Descent. With this quadratic parametrisation, the loss now rewrites as: $\begin{array} { r } { L ( w ) = \frac { 1 } { 4 n } \sum _ { i = 1 } ^ { n } \langle w _ { + } ^ { 2 } - w _ { - } ^ { 2 } - \beta ^ { * } , x _ { i } \rangle ^ { 2 } } \end{array}$ . Note that despite its simplicity, this loss is non convex and its minimisation is non trivial. The algorithm we shall consider is the well known SGD algorithm, where for a step size $\gamma > 0$ :
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+
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| 66 |
+
$$
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| 67 |
+
\begin{array} { r l } & { w _ { t + 1 , + } = w _ { t , + } - \gamma \langle \beta _ { w } - \beta ^ { * } , x _ { i _ { t } } \rangle x _ { i _ { t } } \odot w _ { t , + } } \\ & { w _ { t + 1 , - } = w _ { t , - } + \gamma \langle \beta _ { w } - \beta ^ { * } , x _ { i _ { t } } \rangle x _ { i _ { t } } \odot w _ { t , - } } \end{array} \qquad \mathrm { w h e r e } i _ { t } \sim \mathrm { U n i f } ( 1 , n ) .
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| 68 |
+
$$
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| 69 |
+
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+
It is convenient to rewrite this recursion as
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| 71 |
+
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| 72 |
+
$$
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| 73 |
+
w _ { t + 1 , \pm } = w _ { t , \pm } - \gamma \nabla _ { w _ { \pm } } L ( w _ { t } ) \pm \gamma \mathrm { d i a g } ( w _ { t , \pm } ) X ^ { \top } \xi _ { i _ { t } } ( \beta _ { t } ) ,
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| 74 |
+
$$
|
| 75 |
+
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+
where $\xi _ { i _ { t } } ( \beta ) = - \big ( \langle \beta - \beta ^ { * } , x _ { i _ { t } } \rangle \mathbf { e } _ { i _ { t } } - \mathbb { E } _ { i _ { t } } \big [ \langle \beta - \beta ^ { * } , x _ { i _ { t } } \rangle \mathbf { e } _ { i _ { t } } \big ] \big ) \in \mathbb { R } ^ { n }$ is a zero-mean multiplicative noise which vanishes at any global optimum $\mathrm { i } \mathbf { e } _ { i }$ denotes the $i ^ { \mathrm { { t h } } }$ element of the canonical basis). We point out that all the results we shall give hold for any initialisation such that $w _ { t = 0 , + } = w _ { t = 0 , - } \in \mathbb { R } ^ { d }$ , under which we have that $\beta _ { w _ { t = 0 } } = 0$ . To understand under what conditions the SGD procedure converges and towards which point it does, we shall consider its continuous counterpart which has the advantage of leading to clean and intuitive calculations. We highlight the fact that we consider a bath-size equal to 1 for clarity, however all our analysis holds for mini-batch SGD (with and without replacement) simply by considering an effective step-size $\gamma _ { \mathrm { e f f } }$ instead of $\gamma$ , this is clearly explained in Appendix A.
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+
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| 78 |
+
# 2.2 Stochastic gradient flow
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+
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Continuous time modelling of sequential processes offer a large set of tools, such as derivation, which come in helpful to understand the dynamics of the processes. This has led to a large part of the recent literature to consider continuous gradient flow in order and understand the behaviour of gradient descent on complicated architectures such as neural nets. However, the continuous time modelling of stochastic gradient descent is more challenging: it requires to add on top of the gradient flow a diffusion term whose covariance matches the one of SGD. Hence, it is fundamental to understand its structure and scale.
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+
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Understanding the noise’s structure. As seen in equation (2), evaluated at $w _ { \pm }$ , the stochastic noise $\gamma \mathrm { d i a g } ( \tilde { w _ { \pm } } ) X ^ { \top } \xi _ { i _ { t } } ( w )$ has two main characteristics which we want to preserve:
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+
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| 84 |
+
$$
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+
\begin{array} { r } { \ast \Sigma _ { \mathrm { s o p } } ( w _ { \pm } ) : = \gamma ^ { 2 } \dim ( w _ { \pm } ) X ^ { \top } \mathbf { C o v } _ { i _ { t } } ( \xi _ { i _ { t } } ( \beta ) ) X \operatorname { d i a g } ( w _ { \pm } ) \in \mathbb { R } ^ { d \times d } } \end{array}
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| 86 |
+
$$
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| 87 |
+
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+
It remains to understand the structure of the covariance of $\xi _ { i _ { t } }$ which has the following closed form: $\begin{array} { r } { \mathrm { C o v } _ { i _ { t } } \big ( \xi _ { i _ { t } } ( \beta ) \big ) = \frac { 1 } { n } \operatorname { d i a g } \big ( \langle \beta - \beta ^ { * } , x _ { i } \rangle ^ { 2 } \big ) _ { 1 \leq i \leq n } - \frac { 1 } { n ^ { 2 } } \big ( \langle \beta - \beta ^ { * } , x _ { i } \rangle \langle \beta - \beta ^ { * } , x _ { j } \rangle \big ) _ { 1 < i , i < n } } \end{array}$ . We identify the two key facts: (i) it is diagonal at the leading $n ^ { - 1 }$ order and (ii) its trace is linked to the loss as $\begin{array} { r } { \operatorname { V a r } _ { i _ { t } } ( \| \dot { \xi } _ { i _ { t } } ( \beta ) \| _ { 2 } ) = \frac { 4 } { n } L ( \beta ) + O ( \frac { 1 } { n ^ { 2 } } ) } \end{array}$ . This leads us in modelling $\xi _ { i _ { t } } ( \beta )$ ’s covariance matrix as $\textstyle { \frac { 4 } { n } } L ( \beta ) I _ { n }$ as it preserves these two characteristics 1. Finally this brings us to consider the following modelling of the overall noise’s structure: $\begin{array} { r } { \Sigma _ { \scriptscriptstyle \mathrm { S G D } } ( w _ { \pm } ) \cong \frac { 4 } { n } \gamma ^ { 2 } L ( w ) [ \mathrm { d i a g } ( w _ { \pm } ) X ^ { \top } ] ^ { \otimes 2 } } \end{array}$ .
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| 89 |
+
|
| 90 |
+
Stochastic differentiable equation modelling. Guided by the previous considerations, we study the following stochastic gradient flow:
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| 91 |
+
|
| 92 |
+
$$
|
| 93 |
+
\begin{array} { r l } & { \mathrm { d } w _ { t , + } = - \nabla _ { w _ { + } } L ( w _ { t } ) \mathrm { d } t + 2 \sqrt { \gamma n ^ { - 1 } L ( w _ { t } ) } w _ { t , + } \odot [ X ^ { \top } \mathrm { d } B _ { t } ] } \\ & { \mathrm { d } w _ { t , - } = - \nabla _ { w _ { - } } L ( w _ { t } ) \mathrm { d } t - 2 \sqrt { \gamma n ^ { - 1 } L ( w _ { t } ) } w _ { t , - } \odot [ X ^ { \top } \mathrm { d } B _ { t } ] , } \end{array}
|
| 94 |
+
$$
|
| 95 |
+
|
| 96 |
+
where $\mathrm { d } B _ { t }$ is a standard $\mathbb { R } ^ { n }$ Brownian motion. The SDE is a perturbed gradient flow with a diffusion term that is defined such that its Euler discretisation with step size $\gamma$ leads to a Markov Chain whose covariance exactly matches SGD’s noise covariance $\Sigma _ { \mathrm { s g D } } ( w _ { \pm } )$ . We refer to [26] or [25] for the technical details regarding consistency of such a procedure in the limit of small step sizes. This stochastic differential equation is the starting point of the analysis.
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+
|
| 98 |
+
# 3 The implicit bias of the stochastic gradient flow
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+
|
| 100 |
+
Implicit bias and hyperbolic entropy. To understand the relevance of the main result and how stochasticity induces a preferable bias, we start by recalling some known results for gradient flow. In [36] it is shown, assuming global convergence, that the solution selected by the gradient flow initialised at $\alpha \in \mathbb { R } ^ { d }$ and denoted $\beta _ { \infty } ^ { \alpha }$ solves a constrained optimisation problem involving the hyperbolic entropy introduced by [13]:
|
| 101 |
+
|
| 102 |
+
$$
|
| 103 |
+
\beta _ { \infty } ^ { \alpha } = \operatorname * { a r g m i n } _ { \beta \in \mathbb { R } ^ { d } } \operatorname* { m i n } _ { s . t . X \beta = y } \phi _ { \alpha } ( \beta ) : = \frac { 1 } { 4 } \big [ \sum _ { i = 1 } ^ { d } \beta _ { i } \mathrm { a r c s i n h } ( \frac { \beta _ { i } } { 2 \alpha _ { i } ^ { 2 } } ) - \sqrt { \beta _ { i } ^ { 2 } + 4 \alpha _ { i } ^ { 4 } } \big ] ,
|
| 104 |
+
$$
|
| 105 |
+
|
| 106 |
+
Though the hyperbolic entropy function has a non-trivial expression, its principal characteristic is $\ell _ { 1 }$ nd thand $\ell _ { 2 }$ $\alpha$ precisely for. We refer to $\begin{array} { r } { \tau \in \mathbb { R } ^ { 2 } \colon \dot { \phi _ { \alpha } } ( \beta ) \underset { \alpha \to 0 } { \sim } \frac { 1 } { 2 } \ln \left( \frac { 1 } { \alpha } \right) \| \bar { \beta } \| _ { 1 } } \end{array}$ $\begin{array} { r } { \bar { \phi _ { \alpha } ( \beta ) } \underset { \alpha \to + \infty } { = } - \frac { 1 } { 2 } \bar { \alpha ^ { 2 } } + \frac { 1 } { 1 6 \alpha ^ { 2 } } \| \beta \| _ { 2 } ^ { 2 } + o ( \alpha ^ { - 2 } ) . } \end{array}$ [36, Theorem 2] for more details on the asymptotic analysis. The implicit optimisation problem (4) therefore highlights the fact that the initialisation scale of the weights controls the shape of the recovered solution. Small initialisations lead to low $\ell _ { 1 }$ -norm solutions which are known to induce good generalisation properties: this is what is often referred to as the rich regime. Large initialisations lead to low $\ell _ { 2 }$ -norm solutions: this is referred to as the kernel regime or lazy regime in which the weights move only very slightly. The dynamics of the gradient flow are then very similar to the one of kernel linear regression with the kernel depending on the initialisation [20, 12]. Overall, to retrieve a sparse solution, one should initialise with the smallest $\alpha$ possible. However, as is clearly explained in [36], it is important to stress out that there is a generalisation $/$ optimisation tradeoff: the point $w = 0$ happens to be a saddle point for the loss and a smaller $\alpha$ will lead to a longer training time.
|
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+
|
| 108 |
+
Main result. In the main theorem we show that, for an initialisation scale $\alpha$ , the stochasticity of SGF biases the flow towards solutions which still minimise the hyperbolic entropy. However, what is remarkable is that it does so with an effective parameter $\alpha _ { \infty }$ which is strictly smaller than $\alpha$ . The recovered solution therefore minimises an optimisation problem which has better sparsity inducing properties than that of gradient flow.
|
| 109 |
+
|
| 110 |
+
Theorem 1. For $\begin{array} { l } { p \ \leq \ \frac { 1 } { 2 } } \end{array}$ and $w _ { 0 , \pm } = \alpha \in ( \mathbb { R } _ { + } ^ { * } ) ^ { d }$ , let $( w _ { t } ) _ { t \geq 0 }$ follow the stochastic gradient flow (3) with step size $\begin{array} { r } { \gamma \leq O \big ( \big [ \ln ( \frac { 4 } { p } ) \lambda _ { \operatorname* { m a x } } \operatorname* { m a x } \{ \| \beta _ { \ell _ { 1 } } ^ { * } \| _ { 1 } \ln \big ( \frac { \| \beta _ { \ell _ { 1 } } ^ { * } \| _ { 1 } } { \operatorname* { m i n } _ { i } \alpha _ { i } ^ { 2 } } \big ) , \| \alpha \| _ { 2 } ^ { 2 } \} \big ] ^ { - 1 } \big ) } \end{array}$ where $\beta _ { \ell _ { 1 } } ^ { * } =$ arg min $\| \beta \| _ { 1 }$ and $\lambda _ { \mathrm { m a x } }$ is the largest eigenvalue of $X ^ { \top } X / n$ . Then, with probability at least $1 - p$ : $\beta \in \mathbb { R } ^ { d } \ s . t . \ X \beta = y$
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| 111 |
+
|
| 112 |
+
• $( \beta _ { t } ) _ { t \geq 0 }$ converges towards a zero-training error solution $\beta _ { \infty } ^ { \alpha }$
|
| 113 |
+
|
| 114 |
+
• the solution $\beta _ { \infty } ^ { \alpha }$ satisfies
|
| 115 |
+
|
| 116 |
+
$$
|
| 117 |
+
\beta _ { \infty } ^ { \alpha } = \underset { \beta \in \mathbb { R } ^ { d } } { \arg \operatorname* { m i n } } \quad \phi _ { \alpha _ { \infty } } ( \beta ) \quad w h e r e \quad \alpha _ { \infty } = \alpha \odot \exp \left( - 2 \gamma \dim \operatorname { g } \left( \frac { X ^ { \top } X } { n } \right) \int _ { 0 } ^ { + \infty } L ( \beta _ { s } ) \mathrm { d } s \right) .
|
| 118 |
+
$$
|
| 119 |
+
|
| 120 |
+
The theorem is three-fold: with high probability and for an explicit choice of constant step size $\gamma$ , (i) the flow $( \beta _ { t } ) _ { t \geq 0 }$ converges, (ii) its limit $\beta _ { \infty } ^ { \alpha }$ is an interpolating solution, i.e. $X \beta _ { \infty } ^ { \alpha } = y$ , (iii) this solution minimises the hyperbolic entropy problem with a parameter that depends on the dynamics. We illustrate these results in Figure 2. Now let us comment further the theorem.
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+
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| 122 |
+
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| 123 |
+
Figure 2: Sparse regression (see Section 5.1 for the detailed experimental setting). Both SGD and GD are initialised at $\alpha = 0 . 1$ . 2 different runs of SGD over the training set are performed, they differ due to the inner stochasticity of the algorithm. Left: GD and SGD both converge towards a global minimum. Middle and right: for two different trajectories of SGD, the higher the value of the loss integral at convergence, the better the validation loss. In both cases SGD converges towards a solution which generalises better than GD. This figure illustrates Theorem 1.
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+
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Beneficial implicit bias through effective initialisation. The most remarkable aspect of the result is that the recovered solution $\beta _ { \infty } ^ { \alpha }$ minimises the same potential as for gradient flow but with an effective parameter $\alpha _ { \infty }$ which is strictly smaller than $\alpha$ . Hence, the hyperbolic entropy is closer to the $\ell _ { 1 }$ norm compared to the deterministic case, proving a systematic benefit of stochasticity. Note that this effective parameter is random and controlled by the loss integral $\begin{array} { r l } { \int _ { 0 } ^ { + \infty } L ( \beta _ { s } ) \mathrm { d } s , } \end{array}$ : the higher the integral, the smaller the effective initialisation scale. In other words and quite surprisingly, the slower the loss converges to 0, the “richer” the implicit bias. However, it must be kept in mind that, as explained in [36], there is a tension between generalisation and optimisation: a longer training time might improve generalisation but comes at the cost of... a longer training time. Yet it is clear experimentally that SGD systematically largely wins the trade-off over GD (see Figure 2). Interestingly, Problem (5) tells us that the implicit bias of SGD initialised at $\alpha$ acts as if we run GD initialised at $\alpha _ { \infty }$ (see Section 5.3). Note that the minimisation problem (5) only makes sense $a$ posteriori since the quantity $\alpha _ { \infty }$ depends on the whole stochastic trajectory. Finally, an interesting question is whether one can quantify the scale of this beneficial phenomenon, i.e. how small $\alpha _ { \infty }$ is compared to $\alpha$ . To answer this, we quantify the scale of the loss integral w.r.t. $\gamma$ and $\alpha$ (see Proposition 3) and show under slightly stronger conditions that the relative scale $\alpha _ { \infty } / \alpha$ decays as power of $\alpha$ (See Eq. (8) of the main text and Proposition 6 of the appendix for a proof).
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Kernel regime. Though it is less our focus, our result still holds as $\alpha + \infty$ which corresponds to the kernel regime. In this regime, we believe that $\begin{array} { r } { \int _ { 0 } ^ { + \infty } L ( \beta _ { s } ) \mathrm { d } s \underset { \alpha \infty } { } 0 } \end{array}$ (not shown in the paper but experimentally observed) and hence SGF and GF converge towards the same solution. This is expected since in the NTK regime, the iterates follow a kernel linear regression for which the bias of SGF and GF are the same.
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Step size. Note that the convergence of the iterates holds for a constant step size. This is not illogical since in the overparametrised setting, the noise vanishes at the optimum (see [32] for a convergence result in the overparametrised least-squares setup). The explicit formula for the $\gamma$ upper bound is $\begin{array} { r } { \gamma \leq \left( 4 0 0 \ln \left( \frac { 4 } { p } \right) \lambda _ { \operatorname* { m a x } } \big ( \frac { X ^ { \top } X } { n } \big ) \operatorname* { m a x } \left\{ \| \beta _ { \ell _ { 1 } } ^ { * } \| _ { 1 } \ln \left( \sqrt { 2 } \frac { \| \beta _ { \ell _ { 1 } } ^ { * } \| _ { 1 } } { \operatorname* { m i n } _ { i } \alpha _ { i } ^ { 2 } } \right) , \| \alpha \| _ { 2 } ^ { 2 } \right\} \right) ^ { - 1 } } \end{array}$ . It has a classical dependence on $\lambda _ { \operatorname* { m a x } } ( X ^ { \top } X / n )$ which can be computed, but also on the unknown value of $\| \beta _ { \ell _ { 1 } } ^ { * } \| _ { 1 }$ . However in practice we choose the highest value of $\gamma$ for which the iterates converge. Note that in practice the weights are often initialised such that $\| \alpha \| _ { 2 } ^ { 2 }$ is roughly equal to 1 and hence it is sensible to consider $\| \alpha \| _ { 2 } ^ { 2 } < \| \beta _ { \ell _ { 1 } } ^ { * } \| _ { 1 }$ . In the explicit bound, there is a $\ln \left( \lVert \boldsymbol { \beta } _ { \ell _ { 1 } } ^ { * } \rVert _ { 1 } / \operatorname* { m i n } _ { i } \alpha _ { i } ^ { 2 } \right) ^ { - 1 }$ factor, we believe that it is an artefact of our analysis and could be removed. It is hence best to think of the upperbound on $\gamma$ to simply be $\begin{array} { r } { \gamma \le O ( \frac { 1 } { \lambda _ { \operatorname* { m a x } } \| \beta _ { \ell _ { 1 } } ^ { * } \| _ { 1 } } ) } \end{array}$ .
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Convergence and proof sketch. Let us put emphasis on the fact that since we deal with a nonconvex problem, neither convergence nor convergence towards a global minimum are obvious. In most of similar works, convergence of the iterates is assumed [36, 14]. In fact, the hardest and most technical part of our result is to show the convergence of the flow with high probability: once the convergence is shown, describing the minimisation problem $\beta _ { \infty } ^ { \alpha }$ verifies is straightforward. In the following section we give several properties which constitute the major keys of the theorem’s proof.
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# 4 Links with mirror descent
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The aim of this section is to show that the sequence $( \beta _ { t } ) _ { t \geq 0 }$ follows a stochastic version of continuous mirror descent with a time dependent mirror. From this crucial property, we show how the convergence and implicit bias characterisation follow. Finally, as it is one of the central objects of our main theorem, we give an estimation of $\int _ { 0 } ^ { \infty } L ( \beta _ { s } ) \mathrm { d } s$ .
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# 4.1 Stochastic continuous mirror descent with time-varying potential
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We start by recalling known results on the link between implicit bias and mirror descent. We recall also convergence guarantees for mirror descent dynamics.
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Mirror descent: convergence and implicit bias. For any $\beta _ { 0 } \in \mathbb { R } ^ { d }$ and convex potential function $\Psi$ , consider the mirror descent flow $( \beta _ { t } ) _ { t }$ which corresponds to $\mathrm { d } \nabla \Psi ( \beta _ { t } ) = - \bar { \nabla L } ( \beta _ { t } ) \mathrm { d } t$ . Though the convergence of the loss to 0 is straightforward, showing the convergence of the iterates requires more work and is shown in [4, Theorem 2] for strongly convex potentials. Yet, once the convergence of the iterates is shown, deriving the implicit minimisation problem is straightforward. We recall the reasoning here (see Section 3 of [2] for more details): integrating the flow yields $\nabla \Psi ( \beta _ { \infty } ) -$ $\begin{array} { r } { \nabla \Psi ( \beta _ { 0 } ) = - \int _ { 0 } ^ { \infty } \nabla L ( \beta _ { s } ) \mathrm { d } s = - 4 X ^ { \top } \int _ { 0 } ^ { \infty } X ( \beta _ { s } - \beta _ { \infty } ) \mathrm { d } s \in \mathrm { s p a n } ( X ) } \end{array}$ . This condition, along with the fact that $X \beta _ { \infty } = y$ exactly corresponds to the KKT conditions of the problem:
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$$
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\beta _ { \infty } = \underset { \beta \in \mathbb { R } ^ { d } \mathrm { ~ s . t . ~ } X \beta = y } { \arg \operatorname* { m i n } } D _ { \Psi } ( \beta , \beta _ { 0 } ) ,
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$$
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where $D _ { \Psi } ( \beta , \beta _ { 0 } ) = \Psi ( \beta ) - \Psi ( \beta _ { 0 } ) - \langle \nabla \Psi ( \beta _ { 0 } ) , \beta - \beta _ { 0 } \rangle$ is the Bregman divergence w.r.t. $\Psi$ .
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Link with our model. It turns out that these general observations on mirror descent apply to our framework when $( w _ { t } ) _ { t }$ follows the gradient flow $\mathrm { d } w _ { t , \pm } = - \nabla _ { w _ { \pm } } L ( w _ { t } ) \mathrm { d } t$ . Indeed it has been shown in [36] that the corresponding iterates $\beta _ { t } = w _ { t , + } ^ { 2 } - w _ { t , - } ^ { 2 }$ follow a mirror descent with potential $\phi _ { \alpha }$ defined in Eq.(4). Therefore we can apply the previous remarks to obtain the convergence towards an interpolator3, as well as the associated implicit minimisation problem which in our case can be rewritten as $\beta _ { \infty } ^ { \alpha } = \arg \operatorname* { m i n } \quad \phi _ { \alpha } ( \beta )$ since $\nabla \phi _ { \alpha } ( \beta _ { 0 } = 0 ) = 0$ .
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$\beta \in \mathbb { R } ^ { d }$ s.t. $X \beta { = } y$
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Stochastic Mirror descent with a time varying potential. To address the problem where $( w _ { t } ) _ { t }$ follows a stochastic gradient flow instead of a gradient flow, it is natural, as in the deterministic framework, to see what type of flow $( \beta _ { t } ) _ { t }$ follows. Because of the noise, we cannot hope to simply recover a classical mirror descent. However interestingly the next property shows that it follows a stochastic mirror-like descent with a geometry that depends on time.
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Proposition 1. Consider the iterates $( w _ { t } ) _ { t \geq 0 }$ issued from the stochastic gradient flow in Eq.(3) with initialisation $w _ { 0 , \pm } = \alpha \in ( \mathbb { R } _ { + } ^ { * } ) ^ { d }$ . Then the corresponding flow $( \beta _ { t } ) _ { t \geq 0 }$ follows a “stochastic continuous mirror descent with time varying potential” defined by:
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$$
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\mathrm { d } \nabla \phi _ { \alpha _ { t } } ( \beta _ { t } ) = - \nabla L ( \beta _ { t } ) \mathrm { d } t + \sqrt { \gamma n ^ { - 1 } L ( \beta _ { t } ) } \boldsymbol { X } ^ { \top } \mathrm { d } \boldsymbol { B } _ { t } ,
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$$
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where $\begin{array} { r } { \alpha _ { t } = \alpha \odot \exp \left( - 2 \gamma \operatorname { d i a g } \left( \frac { X ^ { \top } X } { n } \right) \int _ { 0 } ^ { t } L ( \beta _ { s } ) \mathrm { d } s \right) } \end{array}$ and $\phi _ { \alpha }$ is the hyperbolic entropy defined in (4).
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Under this form we clearly see that the iterates $( \beta _ { t } ) _ { t }$ follow a flow which closely resembles that of mirror descent but with two major differences: (i) the potential $\phi _ { \alpha _ { t } }$ changes over time according to the random quantity $\int _ { 0 } ^ { t } L ( \beta _ { s } ) \mathrm { d } s$ , (ii) the flow is perturbed by noise. We highlight the fact that viewing the dynamics this way has the major advantage of giving a clear roadmap for the proof of Theorem 1: (i) we can adapt classical mirror-descent results to our framework and construct appropriate Lyapunov functions to prove the convergence of the flow with high probability to some interpolator $\beta _ { \infty } ^ { \alpha }$ , (ii) we immediately recover the corresponding minimisation problem as in the deterministic case. Indeed, integrating Eq.(7) still yields $\nabla \phi _ { \alpha _ { \infty } } ( \beta _ { \infty } ^ { \alpha } ) \in \mathrm { s p a n } ( X )$ which, along with $X \beta _ { \infty } ^ { \alpha } = y$ , are the KKT conditions of the implicit minimisation problem (5). We emphasise the fact that the structure of the noise, belonging to $\operatorname { s p a n } ( X )$ , is crucial in order to obtain this minimisation problem. This would for instance clearly not be true if we considered isotropic noise in the SDE modelling. This highlights the fact that not every form of noise improves the implicit bias: the shape of the intrinsic SGD noise is of primal importance [15].
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# 4.2 Convergence and control of $\int _ { 0 } ^ { \infty } L ( \beta _ { s } ) \mathrm { d } s$
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Though it seems easy to derive the implicit minimisation problem (5) from the mirror-like structure of Eq.(7), it is necessary to ensure that the iterates converge towards an interpolator $\beta _ { \infty }$ . This is the purpose of the following proposition.
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Proposition 2 (Convergence of the iterates). Consider the iterates $( w _ { t } ) _ { t \geq 0 }$ issued from the stochastic gradient flow (3), initialised at $w _ { 0 , \pm } = \alpha \in ( \mathbb { R } _ { + } ^ { * } ) ^ { d }$ . For $\begin{array} { r } { p \leq \frac { 1 } { 2 } } \end{array}$ and $\gamma$ such as in Theorem $^ { l }$ , then with probability at least $1 - p ,$ the flow $( \beta _ { t } ) _ { t }$ converges to an interpolating solution $\beta _ { \infty } ^ { \alpha }$ .
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The convergence of the iterates is technical and requires several intermediate results. We start by considering an appropriate Bregman-type stochastic function with a time-varying potential and show that it converges with high probability. Leveraging the fact that we are able to bound the iterates $\beta _ { t }$ , we are able to show that the limit of the function is in fact 0. Owing to the fact that the function we consider also controls the distance of $\beta _ { t }$ to a particular $\beta ^ { * }$ we finally get that the iterates converge.
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However for the objects (such as $\alpha _ { \infty }$ ) and functions we introduce to be well defined, we need to guarantee the convergence of $\int _ { 0 } ^ { \infty } L ( \beta _ { s } ) \mathrm { d } s$ . Besides, it is crucial to grasp the scale of this quantity since it gives the overall scale of $\alpha _ { \infty }$ . This is done in the following proposition where we lower and upper bound its value.
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Proposition 3. Under the same setting as in Proposition 2 with initialisation $w _ { 0 , \pm } = \alpha \mathbf { 1 }$ , we have with probability at least $1 - p$ :
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$$
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\Omega \Big ( \| \beta _ { \ell _ { 1 } } ^ { * } \| _ { 1 } \ln \Big ( \frac { \| \beta _ { \ell _ { 1 } } ^ { * } \| _ { 1 } } { \alpha ^ { 2 } } \Big ) \Big ) \underset { \alpha \to 0 } { \leqslant } \int _ { 0 } ^ { + \infty } L ( \beta _ { s } ) { \mathrm { d } } s \leqslant O \Big ( \operatorname* { m a x } \big \{ \| \beta _ { \ell _ { 1 } } ^ { * } \| _ { 1 } \ln \Big ( \frac { \| \beta _ { \ell _ { 1 } } ^ { * } \| _ { 1 } } { \alpha ^ { 2 } } \Big ) , \alpha ^ { 2 } d \big \} \Big ) .
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$$
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We point out that the lower bound is given for small $\alpha$ ’s for simplicity but we provide in Lemma 7 (Appendix B.5) a lower bound which holds for all $\alpha$ ’s. Note that when $\gamma = 0$ , which corresponds to deterministic gradient flow, we can give the exact value for the integral: $\begin{array} { r } { \int _ { 0 } ^ { + \infty } L ( \beta _ { s } ) \mathrm { d } s = } \end{array}$ $\begin{array} { c l c r } { \frac { 1 } { 2 } D _ { \phi _ { \alpha } } \big ( \beta _ { \infty } ^ { \alpha } , \beta _ { 0 } \big ) } \end{array}$ (see Proposition 7 in Appendix C). This matches the scale of the bounds given in Proposition 3, hence showing the tightness of the result. We focus now on how this translates to the scale of the effective initialisation w.r.t. $\alpha$ when this latter is small enough. In fact, this lower bound on the integral of the loss along with a stronger assumption on the boundedness of the iterates lead to
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$$
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\frac { \alpha _ { \infty } } { \alpha } \underset { \alpha 0 } { \leqslant } ( \frac { \alpha ^ { 2 } } { \| \beta _ { \ell _ { 1 } } ^ { * } \| _ { 1 } } ) ^ { \zeta } ,
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$$
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for some $\zeta > 0$ . Hence the smaller the initialisation scale $\alpha$ and the greater the benefit of SGD over GD in terms of implicit bias (see Appendix B.6 for more details).
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Again, the proof of this proposition is technical and relies on considering appropriate Lyapunov functions which highly resemble to Bregman divergences, but which take into account the fact that the geometry changes over time. These overall decreasing Lyapunov’s enable to bound the iterates as well as lower and upper bound the integral of the loss. The stochastic integrals which naturally appear are controlled with high probability using time-uniform concentration of martingales [19].
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# 5 Experiments
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# 5.1 Experimental setup for sparse regression
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We consider the following sparse regression setup for our experiments. We choose $n = 4 0$ , $d = 1 0 0$ and randomly generate a sparse model $\beta _ { \ell _ { 0 } } ^ { \ast }$ such that $\| \beta _ { \ell _ { 0 } } ^ { * } \| _ { 0 } = 5$ . We generate the features as $x _ { i } \sim \mathcal { N } ( 0 , I )$ and the labels as $y _ { i } = x _ { i } ^ { \top } \beta _ { \ell _ { 0 } } ^ { * }$ . SGD, GD and the SGF are always initialised using the same scale $\alpha > 0$ and it is specified each time. We use the same step size for GD and SGD and choose it to be the biggest as possible why still ensuring convergence. Note that since the true population covariance $\mathbb { E } [ x x ^ { \top } ]$ is equal to identity, the quantity $\lVert \beta _ { t } - \beta _ { \ell _ { 0 } } ^ { * } \rVert _ { 2 } ^ { 2 }$ corresponds to the validation loss.
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# 5.2 Validation of the SDE model
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In this section, we present an experimental validation of the stochastic gradient flow model. In Figure 3, for the same step size, we run: (i) the trajectory of gradient descent, (ii) 5 trajectories of stochastic gradient descent that correspond to different realisations of the uniform sampling over the data, (iii) 5 trajectories of the stochastic gradient flow (its Euler discretisation with $\mathrm { d } t = \gamma / 1 0 $ )) corresponding to different realisations of the Brownian. We clearly see (left) that the loss behaves similarly for SGD and SGF across time. We also see that the validation losses (right) of the iterates of SGD and SGF have very similar behaviours. This tends to validate our continuous modelling from Section 2.2.
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Figure 3: Sparse regression (see Section 5.1 for the detailed experimental setup). Left and right: the training and the validation losses behave very similarly, corroborating the continuous modelling.
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# 5.3 GD and SGD have the same implicit bias, but from different initialisations.
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In order to confirm and illustrate the main Theorem 1, we provide the following experiment which is illustrated Figure 4. We first run GD and SGD with the same step-size and initialise them both at $\alpha \mathbf { 1 }$ with $\alpha ~ = ~ 0 . 0 1$ . As expected, the solution recovered by SGD generalises better. Then, using the iterates $\beta _ { t } ^ { \mathrm { S G D } }$ from the first SGD run, we compute the value $\begin{array} { r } { \alpha _ { \infty } = \alpha \exp ( - 2 \gamma \operatorname { d i a g } ( X ^ { \top } X / n ) \int _ { 0 } ^ { \infty } L ( \beta _ { s } ^ { \mathrm { S G D } } ) \mathrm { d } s ) \in \mathbb { R } ^ { d } } \end{array}$ (the integral is approximated by its discrete time approximation with $\mathrm { d } t = \gamma$ ). We then run gradient descent but this time initialised at $w _ { 0 , \pm } = \alpha _ { \infty }$ . According to our main result from Theorem 1, it should approximately (it would be exact if we ran SGF and GF) converge to the same solution as SGD initialised at $\alpha \mathbf { 1 }$ . This is clearly observed Figure 4 (right). Also note that SGD and GD (initialised at $\alpha _ { \infty }$ ) seem to have overall very similar dynamics, this is not shown by our results and we leave this as future work. However keep in mind that though the validation losses converge at the same iteration rate, in terms of computation time, SGD is $n$ times faster.
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Figure 4: Sparse regression (see Section 5.1 for the detailed experimental setup). Left and right: SGD initialised at $\alpha \mathbf { 1 }$ converges towards the same point as GD initialised at $\begin{array} { r l } { \alpha _ { \infty } } & { { } = } \end{array}$ $\begin{array} { r } { \tilde { \alpha \exp ( - 2 \gamma \operatorname { d i a g } ( X ^ { \top } X / n ) \int _ { 0 } ^ { \infty } { { L } ( \beta _ { s } ^ { \mathrm { { S G } \tilde { D } } } ) \mathrm { { d } } s } ) } } \end{array}$ .
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# 5.4 Doping the implicit bias with label noise
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As largely discussed throughout the paper, the effect of the implicit bias is controlled by the convergence speed of the loss: the slower it converges, the sparser the selected solution will be. Hence the following question: can we leverage this knowledge to dope the implicit bias? We argue in this Section that the answer to this question is affirmative. Indeed, consider a sequence $( \delta _ { t } ) _ { t \in \mathbb { N } } \overline { { \in } } { \mathbb { R } } _ { + } ^ { \mathbb { N } }$ and assume that we artificially inject some label noise $\Delta _ { t }$ at time $t$ , say for example $\Delta _ { t } \sim \mathrm { U n i f } \{ 2 \delta _ { t } , - 2 \delta _ { t } \}$ (independently from $i _ { t }$ ). This injected label noise perturbs the SGD recursion as follows:
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$$
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w _ { t + 1 , \pm } = w _ { t , \pm } \mp \gamma \left( \langle \beta _ { w } - \beta ^ { * } , x _ { i _ { t } } \rangle + \Delta _ { t } \right) x _ { i _ { t } } \odot w _ { t , + } ,
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$$
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As in Section 2.2, we can derive its related stochastic gradient flow (see Appendix D.1 for more details):
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$$
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\mathrm { d } w _ { t , \pm } = - \nabla _ { w _ { \pm } } L ( w _ { t } ) \mathrm { d } t \pm 2 \sqrt { \gamma n ^ { - 1 } ( L ( w _ { t } ) + \delta _ { t } ^ { 2 } ) } w _ { t , + } \odot [ X ^ { \top } \mathrm { d } B _ { t } ] .
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$$
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Assuming that $( \delta _ { t } ) _ { t \geq 0 } \in ( \mathbb { R } _ { + } ) ^ { \mathbb { R } }$ and $\gamma$ are such that the iterates converge, the corresponding implicit regularisation minimisation problem is preserved but with a "slowed down" loss: $\tilde { L } ( \beta _ { t } ) : = L ( \beta _ { t } ) + \delta _ { t } ^ { 2 }$ and the effective initialisation writes: $\begin{array} { r } { \tilde { \alpha } _ { \infty } = \alpha \odot \exp \left( - 2 \gamma \mathrm { d i a g } ( \frac { X ^ { \top } X } { n } ) \int _ { 0 } ^ { + \infty } \tilde { L } ( \beta _ { s } ) \mathrm { d } s \right) } \end{array}$ . The label noise therefore helps recovering a solution which has better sparsity properties. However, it must be kept in mind that adding too much label noise can significantly slow down the convergence of the validation loss or even prevent the iterates from converging. Yet, experimental results showing the impressive effect of label noise are provided Figure 5 in Appendix D.1.
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# 6 Conclusion and Perspectives
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In this paper, we have shown the benefit of using stochastic gradient descent over gradient descent for diagonal linear networks in terms of their implicit bias. Indeed, we prove that stochastic gradient flow acts as gradient flow but initialised at a smaller scale: this induces a sparser finale iterate. This effect is controlled by the speed of convergence of the loss. Moreover, we prove the convergence of the flow and exhibit an interesting link with mirror descent. Fully understanding this novel type of dynamics could help to grasp the implicit biasing properties of stochastic gradient descent in other frameworks. It is also natural to ask whether the integral of the loss also controls the difference of implicit regularisation for more general architectures. It would also be interesting to analyse how this property adapts to log losses known to lead to max-margin solutions in classification.
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Acknowledgements. NF would like to thank Nathan Srebro for introducing him to the question of SGD’s implicit bias as well as for the stimulating discussions they had during his visit at EPFL.
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|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "Implicit Bias of SGD for Diagonal Linear Networks: a Provable Benefit of Stochasticity ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
186,
|
| 8 |
+
122,
|
| 9 |
+
816,
|
| 10 |
+
172
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Scott Pesme EPFL scott.pesme@epfl.ch ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
251,
|
| 19 |
+
233,
|
| 20 |
+
418,
|
| 21 |
+
275
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "Loucas Pillaud-Vivien EPFL loucas.pillaud-vivien@epfl.ch ",
|
| 28 |
+
"bbox": [
|
| 29 |
+
495,
|
| 30 |
+
233,
|
| 31 |
+
745,
|
| 32 |
+
275
|
| 33 |
+
],
|
| 34 |
+
"page_idx": 0
|
| 35 |
+
},
|
| 36 |
+
{
|
| 37 |
+
"type": "text",
|
| 38 |
+
"text": "Nicolas Flammarion EPFL nicolas.flammarion@epfl.ch ",
|
| 39 |
+
"bbox": [
|
| 40 |
+
385,
|
| 41 |
+
290,
|
| 42 |
+
611,
|
| 43 |
+
332
|
| 44 |
+
],
|
| 45 |
+
"page_idx": 0
|
| 46 |
+
},
|
| 47 |
+
{
|
| 48 |
+
"type": "text",
|
| 49 |
+
"text": "Abstract ",
|
| 50 |
+
"text_level": 1,
|
| 51 |
+
"bbox": [
|
| 52 |
+
462,
|
| 53 |
+
367,
|
| 54 |
+
535,
|
| 55 |
+
383
|
| 56 |
+
],
|
| 57 |
+
"page_idx": 0
|
| 58 |
+
},
|
| 59 |
+
{
|
| 60 |
+
"type": "text",
|
| 61 |
+
"text": "Understanding the implicit bias of training algorithms is of crucial importance in order to explain the success of overparametrised neural networks. In this paper, we study the dynamics of stochastic gradient descent over diagonal linear networks through its continuous time version, namely stochastic gradient flow. We explicitly characterise the solution chosen by the stochastic flow and prove that it always enjoys better generalisation properties than that of gradient flow. Quite surprisingly, we show that the convergence speed of the training loss controls the magnitude of the biasing effect: the slower the convergence, the better the bias. To fully complete our analysis, we provide convergence guarantees for the dynamics. We also give experimental results which support our theoretical claims. Our findings highlight the fact that structured noise can induce better generalisation and they help explain the greater performances of stochastic gradient descent over gradient descent observed in practice. ",
|
| 62 |
+
"bbox": [
|
| 63 |
+
233,
|
| 64 |
+
397,
|
| 65 |
+
766,
|
| 66 |
+
578
|
| 67 |
+
],
|
| 68 |
+
"page_idx": 0
|
| 69 |
+
},
|
| 70 |
+
{
|
| 71 |
+
"type": "text",
|
| 72 |
+
"text": "1 Introduction ",
|
| 73 |
+
"text_level": 1,
|
| 74 |
+
"bbox": [
|
| 75 |
+
174,
|
| 76 |
+
602,
|
| 77 |
+
310,
|
| 78 |
+
619
|
| 79 |
+
],
|
| 80 |
+
"page_idx": 0
|
| 81 |
+
},
|
| 82 |
+
{
|
| 83 |
+
"type": "text",
|
| 84 |
+
"text": "Understanding the performance of neural networks is certainly one of the most thrilling challenges for the current machine learning community. From the theoretical point of view, progress has been made in several directions: we have a better functional analysis description of neural networks [3] and we steadily understand the convergence of training algorithms [29, 10] as well as the role of initialisation [20, 12]. Yet there remain many unanswered questions. One of which is why do the currently used training algorithms converge to solutions which generalise well, and this with very little use of explicit regularisation [39]. ",
|
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"text": "To understand this phenomenon, the concept of implicit bias has emerged: if over-fitting is benign, it must be because the optimisation procedure converges towards some particular global minimum which enjoys good generalisation properties. Though no explicit regularisation is added, the algorithm is implicitly selecting a particular solution: this is referred to as the implicit bias of the training procedure. The implicit regularisation of several algorithms has been studied, the simplest and most emblematic being that of gradient descent and stochastic gradient descent in the least-squares framework: they both converge towards the global solution which has the lowest squared distance from the initialisation. For logistic regression on separable data, Soudry et al. show in the seminal paper [31] that gradient descent selects the max-margin classifier. This type of result has then been extended to neural networks and to other frameworks. Overall, characterising the implicit bias of gradient methods has almost always come down to unveiling mirror-descent like structures which underlie the algorithms. ",
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"type": "image",
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"img_path": "images/4845f93997685dd5783933ff3e9d3b42c81826d435a11147baa74f7318c4bd5b.jpg",
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"image_caption": [
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"Figure 1: Sparse regression with $n = 4 0$ , $d = 1 0 0$ , $\\| \\beta _ { \\ell _ { 0 } } ^ { * } \\| _ { 0 } = 5$ , $x _ { i } \\sim \\mathcal { N } ( 0 , I ) y _ { i } = x _ { i } ^ { \\top } \\beta _ { \\ell _ { 0 } } ^ { * } .$ Left: for initialisation scale $\\alpha = 0 . 0 5$ , SGD converges towards a solution which generalises better than GD. Right: for different values of the initialisation scale $\\alpha$ , the solution recovered by SGD has better validation loss than that of GD. The sparsifying effect due to their implicit biases differ by more than an order of magnitude. See Section 5.1 for the precise experimental setup. "
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|
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"text": "While mostly all of the results focus on gradient descent, it must be pointed out that this full batch algorithm is not used in practice for neural networks since it does not lead to solutions which generalise well [23]. Instead, results on stochastic gradient descent, which is widely used and shows impressive results, are still missing or unsatisfactory. This has certainly to do with the fact that grasping the nature of the noise induced by the stochasticity of the algorithm is particularly hard: it mixes properties from the model’s architecture, the data’s distribution and the loss. In our work, by focusing on simplified neural networks, we answer to the following fundamental questions: do SGD’s and GD’s implicit bias differ? What is the role of SGD’s noise over the algorithm’s implicit bias? ",
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"text": "The simplified neural networks which we consider are diagonal linear neural networks; despite their simplicity they have become popular since they already enable to grasp the complexity of more general networks. Indeed, they highlight important aspects of the theoretical concerns of modern machine learning: the neural tangent kernel regime, the roles of over-parametrisation, of the initialisation and of the step size. For a regression problem where we assume the existence of an interpolating solution, we study stochastic gradient descent through its continuous version, namely stochastic gradient flow (SGF). Though the continuous modelling of SGD has not yet led to many fruitful results compared to the well studied gradient flow, we believe it is because capturing the essence of the stochastic noise is particularly difficult. It has generally been done in a non realistic and over simplified manner, such as considering constant and isotropic noise. In our work, we attach peculiar attention to the adequate modelling of the noise. Tools from Itô calculus are then leveraged in order to derive exact formulas, quantitative bounds and interesting interpretations for our problem. ",
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"type": "text",
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"text": "1.1 Main contributions and paper organisation. ",
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"type": "text",
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"text": "In Section 2, we start by introducing the setup of our problem as well as the continuous modelisation of stochastic gradient descent. Then, in Section 3, we state our main result on the implicit bias of the stochastic gradient flow. We informally formulate it here and illustrate it in Figure 1: ",
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"type": "text",
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"text": "Theorem 1 (Informal). Stochastic gradient flow over diagonal linear networks converges with high probability to a zero-loss solution which enjoys better generalisation properties than the one obtained by gradient flow. Furthermore, the speed of convergence of the training loss controls the magnitude of the biasing effect: the slower the convergence, the better the bias. ",
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"text": "Unlike previous works [14, 36], in addition to characterising the implicit bias effect of SGF, we also prove the convergence of the iterates towards a zero-loss solution with high-probability. To accomplish this, we leverage in Section 4 the fact that the iterates follow a stochastic continuous mirror descent with a time-varying potential. We support our results experimentally and validate our model in Section 5. ",
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"type": "text",
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"text": "1.2 Related work ",
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"text": "As recalled, implicit bias has a recent history that has been initiated by the seminal work [31] on max-margin classification with log-loss for a linear setup and separable data. This work has been extended to other architectures, e.g. multiplicative parametrisations [14], linear networks [22] and more general homogeneous neural networks [27, 11]. In [36] the authors show that the scale of the initialisation leads to an interpolation between the neural tangent kernel regime [20, 12] (which is a linear regression on fixed features) leading to $\\ell _ { 2 }$ minimum norm solutions and the rich regimes leading to $\\ell _ { 1 }$ minimum norm solutions. Note that these works focus on full batch gradient descent (or flow) and are deeply linked to mirror descent. ",
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"type": "text",
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"text": "While the links between SGD’s stochasticity and generalisation have been looked into in numerous works [28, 21, 16, 18, 24], no such explicit characterisation of implicit regularisation have ever been given. It has been empirically observed that SGD often outputs models which generalise better than GD [23, 21, 16]. One suggested explanation is that SGD is prone to pick flatter solutions than GD and that bad generalisation solutions are correlated with sharp minima, i.e., with strong curvature, while good generalisation solutions are correlated with flat minima, i.e., with low curvature [17, 23]. This idea has been further investigated by adopting a random walk on random landscape modelling [18], by suggesting that SGD’s noise is smoothing the loss landscape, thus eliminating the sharp minima [24], by considering a dynamical stability perspective [38] or by interpreting SGD as a diffusion process [16, 21, 8]. Recently, label-noise has been shown to influence the implicit bias of SGD, by biasing the solution towards the origin for quadratically-parameterized models [15] or by implicitly regularising the expected squared norm of the gradient of the model with respect to the weights [5]. Thus, if the notion of implicit bias of GD is fairly well understood both in the cases of regression and classification, it remains unclear for SGD, and its explicit characterisation is missing. ",
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| 212 |
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| 218 |
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"page_idx": 2
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"type": "text",
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"text": "The linear diagonal neural networks we consider have been studied in the case of gradient descent [33] and stochastic gradient descent with label noise [15]. In both cases the authors show that this model has the ability to implicitly bias the training procedure to help retrieve a sparse predictor. The link between gradient descent and mirror descent for this model has been initiated by [13] and further exploited by the same author in [37, 34] for its sparse inducing property. ",
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| 223 |
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"text": "Contrary to the deterministic case, the modelling of stochastic gradient descent as a stochastic differential equation is quite recent, see [28, 21]. However, as highlighted by [1], early attempts often suffer from the drawback that they model the noise using a constant covariance matrix. On the contrary, state dependant noise has now become the legitimate manner for modelling SGD as a stochastic gradient flow and it is shown in [26] that it can be done consistently. Yet, noise modelling still remains the principal issue [35] as it influences largely the behaviour of the dynamics [8, 9]. ",
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| 234 |
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"type": "text",
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"text": "1.3 Notations ",
|
| 245 |
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"text_level": 1,
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| 246 |
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},
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{
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"type": "text",
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"text": "For input data $( x _ { 1 } , \\ldots , x _ { n } ) \\in ( \\mathbb { R } ^ { d } ) ^ { n }$ and output $( y _ { 1 } , \\dots , y _ { n } ) \\in \\mathbb { R } ^ { n }$ , we denote respectively $X \\in$ $\\mathbb { R } ^ { n \\times d }$ the design matrix whose $i$ -th row is feature $x _ { i } \\in \\mathbb { R } ^ { d }$ and $y \\in \\mathbb { R } ^ { n }$ the vector of outputs. $\\mathbb { R } _ { + } ^ { * }$ denotes the set of strictly positive real numbers. For $p = 1 , 2$ , the $\\ell _ { p }$ -norm of $x \\in \\mathbb { R } ^ { d }$ is $\\| x \\| _ { p } ^ { p } =$ $\\sum _ { i } ^ { d } | x _ { i } | ^ { p }$ . The operations $\\odot$ will stand for coordinate-wise product between vector: $[ u \\odot v ] _ { i } = u _ { i } v _ { i }$ and $u ^ { 2 } = u \\odot u$ . For $p \\in \\mathbb { N } ^ { * }$ , we also define $u ^ { p } : = u \\odot \\ldots \\odot u$ , the $p$ times product of $u$ with itself. All inequalities between vectors should be understood value by value. For $f , g \\in \\mathbb { R }$ , the existence of $C > 0$ such that $f \\leq C g$ and $C g \\leq f$ will be denoted $f \\leq O ( g )$ and $\\Omega ( g ) \\leq f$ respectively. We shall use the symbole $\\widetilde O$ when this is true up to log factors. For a vector $u \\in \\mathbb { R } ^ { d }$ , $\\mathrm { d i a g } ( u )$ denotes the $d \\times d$ diagonal matrix which has its diagonal equal to $u$ . For a matrix $M \\in \\mathbb { R } ^ { d \\times d }$ , $\\mathrm { d i a g } ( M )$ denotes the vector $( M _ { 1 1 } , \\dots , M _ { d d } ) \\in \\mathbb { R } ^ { d }$ . The indexed vector $\\beta ^ { * }$ will stand for any $\\beta$ interpolating the data, i.e. any vector in the affine space $\\{ \\beta \\in \\mathbb { R } ^ { d } s . t$ , $X \\beta = Y \\}$ of dimension at least $d - n$ . Out of all these, let $\\begin{array} { r l } { \\beta _ { \\ell _ { 1 } } ^ { * } = } & { { } \\arg \\operatorname* { m i n } \\quad \\| \\beta \\| _ { 1 } } \\end{array}$ . For $z$ any vector, $z _ { \\infty }$ or $z ^ { \\infty }$ will always designate of $\\operatorname* { l i m } _ { t \\to \\infty } z _ { t }$ . ${ \\boldsymbol { \\beta } } \\in { \\mathbb { R } } ^ { d }$ $X \\beta { = } y$ ",
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"type": "text",
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"text": "2 Setup and preliminaries ",
|
| 268 |
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"text_level": 1,
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"type": "text",
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"text": "2.1 Architecture and algorithm. ",
|
| 280 |
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"text_level": 1,
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| 281 |
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"text": "Overparametrised noiseless regression. We consider a linear regression problem with outputs $( y _ { 1 } , \\dotsc , y _ { n } ) \\in \\mathbb { R } ^ { n }$ and inputs $( x _ { 1 } , \\ldots , x _ { n } ) \\in ( \\mathbb { R } ^ { d } ) ^ { n }$ . We study an overparametrised setting $( n < d )$ ",
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"type": "text",
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"text": "and assume that there exists at least one interpolating parameter $\\beta ^ { * } \\in \\mathbb { R } ^ { d }$ which perfectly fits the training set, i.e. $y _ { i } = \\langle \\beta ^ { * } , x _ { i } \\rangle$ for all $1 \\leq i \\leq n$ . We parametrise the regression vector $\\beta$ as $\\beta _ { w }$ with $w \\in \\mathbb { R } ^ { p }$ . We will see that though in the end our final models $x \\mapsto \\langle \\beta _ { w } , x \\rangle$ are classical linear models whatever the parametrisation $w \\mapsto \\beta _ { w }$ , the choice of this parametrisation has crucial consequences on the solution recovered by the learning algorithms. We study the quadratic loss and the overall loss is written as: ",
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"type": "equation",
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"img_path": "images/026a34e831f7ad0d265167e31a3d9a29e1f371cb6e6c827fa44925b636f52bd4.jpg",
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"text": "$$\nL ( w ) = L ( \\beta _ { w } ) : = \\frac { 1 } { 4 n } \\sum _ { i = 1 } ^ { n } ( \\langle \\beta _ { w } , x _ { i } \\rangle - y _ { i } ) ^ { 2 } = \\frac { 1 } { 4 n } \\sum _ { i = 1 } ^ { n } \\langle \\beta _ { w } - \\beta ^ { * } , x _ { i } \\rangle ^ { 2 } ,\n$$",
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| 315 |
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"text_format": "latex",
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| 316 |
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{
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"type": "text",
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| 326 |
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"text": "where by abuse of notation we use $L ( w ) = L ( \\beta _ { w } )$ . ",
|
| 327 |
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{
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"type": "text",
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"text": "2-layer diagonal linear network. The simplest parametrisation of $\\beta _ { w }$ is to consider $\\beta _ { w } \\ = \\ w$ which corresponds to the classical least-squares framework. It is well known that in this case, many first order methods (GD, SGD, with and without momentum) will converge towards the same solution: we say that they have the same implicit bias. This is experimentally not the case for neural networks where SGD has been shown to lead to solutions which have better generalisation properties compared to GD [23]. To theoretically confirm this observation, we study a simple non-linear parametrisation: $\\beta _ { w } = w _ { + } ^ { 2 } - w _ { - } ^ { 2 }$ with $w \\doteq [ w _ { + } , w _ { - } ] ^ { \\intercal } \\in \\mathbb { R } ^ { 2 d }$ . We point out that it is 2-positive homogeneous and that it is equivalent to the parametrisation $\\beta _ { u , v } = u \\odot v$ with $u , v \\in \\mathbb { R } ^ { d }$ . It should be thought of a simplified linear network of depth 2 (see [36, Section 4] for more details). We consider two weight vectors $w _ { + }$ and $w _ { - }$ (and not only $\\beta _ { w } = w ^ { 2 }$ ) in order to ensure that our final linear predictor parameter $\\beta _ { w }$ can take negative values. For the sake of completeness, the study of diagonal linear networks of arbitrary depth $p \\geq 3$ is done in Appendix E.2. Also note that additionally to being a toy neural model, it has received recent attention for its practical ability to induce sparsity [33, 34, 15] or to solve phase retrieval problems [37]. ",
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| 338 |
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},
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| 346 |
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{
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| 347 |
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"type": "text",
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| 348 |
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"text": "Stochastic Gradient Descent. With this quadratic parametrisation, the loss now rewrites as: $\\begin{array} { r } { L ( w ) = \\frac { 1 } { 4 n } \\sum _ { i = 1 } ^ { n } \\langle w _ { + } ^ { 2 } - w _ { - } ^ { 2 } - \\beta ^ { * } , x _ { i } \\rangle ^ { 2 } } \\end{array}$ . Note that despite its simplicity, this loss is non convex and its minimisation is non trivial. The algorithm we shall consider is the well known SGD algorithm, where for a step size $\\gamma > 0$ : ",
|
| 349 |
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"type": "equation",
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| 360 |
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"text": "$$\n\\begin{array} { r l } & { w _ { t + 1 , + } = w _ { t , + } - \\gamma \\langle \\beta _ { w } - \\beta ^ { * } , x _ { i _ { t } } \\rangle x _ { i _ { t } } \\odot w _ { t , + } } \\\\ & { w _ { t + 1 , - } = w _ { t , - } + \\gamma \\langle \\beta _ { w } - \\beta ^ { * } , x _ { i _ { t } } \\rangle x _ { i _ { t } } \\odot w _ { t , - } } \\end{array} \\qquad \\mathrm { w h e r e } i _ { t } \\sim \\mathrm { U n i f } ( 1 , n ) .\n$$",
|
| 361 |
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| 372 |
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"text": "It is convenient to rewrite this recursion as ",
|
| 373 |
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"img_path": "images/dac707330c9914d46adb89e0870ada6d826861fb28f152fb1988ac3f4f74fab6.jpg",
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| 384 |
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"text": "$$\nw _ { t + 1 , \\pm } = w _ { t , \\pm } - \\gamma \\nabla _ { w _ { \\pm } } L ( w _ { t } ) \\pm \\gamma \\mathrm { d i a g } ( w _ { t , \\pm } ) X ^ { \\top } \\xi _ { i _ { t } } ( \\beta _ { t } ) ,\n$$",
|
| 385 |
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"text_format": "latex",
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"type": "text",
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| 396 |
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"text": "where $\\xi _ { i _ { t } } ( \\beta ) = - \\big ( \\langle \\beta - \\beta ^ { * } , x _ { i _ { t } } \\rangle \\mathbf { e } _ { i _ { t } } - \\mathbb { E } _ { i _ { t } } \\big [ \\langle \\beta - \\beta ^ { * } , x _ { i _ { t } } \\rangle \\mathbf { e } _ { i _ { t } } \\big ] \\big ) \\in \\mathbb { R } ^ { n }$ is a zero-mean multiplicative noise which vanishes at any global optimum $\\mathrm { i } \\mathbf { e } _ { i }$ denotes the $i ^ { \\mathrm { { t h } } }$ element of the canonical basis). We point out that all the results we shall give hold for any initialisation such that $w _ { t = 0 , + } = w _ { t = 0 , - } \\in \\mathbb { R } ^ { d }$ , under which we have that $\\beta _ { w _ { t = 0 } } = 0$ . To understand under what conditions the SGD procedure converges and towards which point it does, we shall consider its continuous counterpart which has the advantage of leading to clean and intuitive calculations. We highlight the fact that we consider a bath-size equal to 1 for clarity, however all our analysis holds for mini-batch SGD (with and without replacement) simply by considering an effective step-size $\\gamma _ { \\mathrm { e f f } }$ instead of $\\gamma$ , this is clearly explained in Appendix A. ",
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"type": "text",
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"text": "2.2 Stochastic gradient flow ",
|
| 408 |
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"text": "Continuous time modelling of sequential processes offer a large set of tools, such as derivation, which come in helpful to understand the dynamics of the processes. This has led to a large part of the recent literature to consider continuous gradient flow in order and understand the behaviour of gradient descent on complicated architectures such as neural nets. However, the continuous time modelling of stochastic gradient descent is more challenging: it requires to add on top of the gradient flow a diffusion term whose covariance matches the one of SGD. Hence, it is fundamental to understand its structure and scale. ",
|
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"text": "Understanding the noise’s structure. As seen in equation (2), evaluated at $w _ { \\pm }$ , the stochastic noise $\\gamma \\mathrm { d i a g } ( \\tilde { w _ { \\pm } } ) X ^ { \\top } \\xi _ { i _ { t } } ( w )$ has two main characteristics which we want to preserve: ",
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"text": "$$\n\\begin{array} { r } { \\ast \\Sigma _ { \\mathrm { s o p } } ( w _ { \\pm } ) : = \\gamma ^ { 2 } \\dim ( w _ { \\pm } ) X ^ { \\top } \\mathbf { C o v } _ { i _ { t } } ( \\xi _ { i _ { t } } ( \\beta ) ) X \\operatorname { d i a g } ( w _ { \\pm } ) \\in \\mathbb { R } ^ { d \\times d } } \\end{array}\n$$",
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"text": "It remains to understand the structure of the covariance of $\\xi _ { i _ { t } }$ which has the following closed form: $\\begin{array} { r } { \\mathrm { C o v } _ { i _ { t } } \\big ( \\xi _ { i _ { t } } ( \\beta ) \\big ) = \\frac { 1 } { n } \\operatorname { d i a g } \\big ( \\langle \\beta - \\beta ^ { * } , x _ { i } \\rangle ^ { 2 } \\big ) _ { 1 \\leq i \\leq n } - \\frac { 1 } { n ^ { 2 } } \\big ( \\langle \\beta - \\beta ^ { * } , x _ { i } \\rangle \\langle \\beta - \\beta ^ { * } , x _ { j } \\rangle \\big ) _ { 1 < i , i < n } } \\end{array}$ . We identify the two key facts: (i) it is diagonal at the leading $n ^ { - 1 }$ order and (ii) its trace is linked to the loss as $\\begin{array} { r } { \\operatorname { V a r } _ { i _ { t } } ( \\| \\dot { \\xi } _ { i _ { t } } ( \\beta ) \\| _ { 2 } ) = \\frac { 4 } { n } L ( \\beta ) + O ( \\frac { 1 } { n ^ { 2 } } ) } \\end{array}$ . This leads us in modelling $\\xi _ { i _ { t } } ( \\beta )$ ’s covariance matrix as $\\textstyle { \\frac { 4 } { n } } L ( \\beta ) I _ { n }$ as it preserves these two characteristics 1. Finally this brings us to consider the following modelling of the overall noise’s structure: $\\begin{array} { r } { \\Sigma _ { \\scriptscriptstyle \\mathrm { S G D } } ( w _ { \\pm } ) \\cong \\frac { 4 } { n } \\gamma ^ { 2 } L ( w ) [ \\mathrm { d i a g } ( w _ { \\pm } ) X ^ { \\top } ] ^ { \\otimes 2 } } \\end{array}$ . ",
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"text": "Stochastic differentiable equation modelling. Guided by the previous considerations, we study the following stochastic gradient flow: ",
|
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"text": "$$\n\\begin{array} { r l } & { \\mathrm { d } w _ { t , + } = - \\nabla _ { w _ { + } } L ( w _ { t } ) \\mathrm { d } t + 2 \\sqrt { \\gamma n ^ { - 1 } L ( w _ { t } ) } w _ { t , + } \\odot [ X ^ { \\top } \\mathrm { d } B _ { t } ] } \\\\ & { \\mathrm { d } w _ { t , - } = - \\nabla _ { w _ { - } } L ( w _ { t } ) \\mathrm { d } t - 2 \\sqrt { \\gamma n ^ { - 1 } L ( w _ { t } ) } w _ { t , - } \\odot [ X ^ { \\top } \\mathrm { d } B _ { t } ] , } \\end{array}\n$$",
|
| 478 |
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"text_format": "latex",
|
| 479 |
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"bbox": [
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"text": "where $\\mathrm { d } B _ { t }$ is a standard $\\mathbb { R } ^ { n }$ Brownian motion. The SDE is a perturbed gradient flow with a diffusion term that is defined such that its Euler discretisation with step size $\\gamma$ leads to a Markov Chain whose covariance exactly matches SGD’s noise covariance $\\Sigma _ { \\mathrm { s g D } } ( w _ { \\pm } )$ . We refer to [26] or [25] for the technical details regarding consistency of such a procedure in the limit of small step sizes. This stochastic differential equation is the starting point of the analysis. ",
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"text": "3 The implicit bias of the stochastic gradient flow ",
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"text": "Implicit bias and hyperbolic entropy. To understand the relevance of the main result and how stochasticity induces a preferable bias, we start by recalling some known results for gradient flow. In [36] it is shown, assuming global convergence, that the solution selected by the gradient flow initialised at $\\alpha \\in \\mathbb { R } ^ { d }$ and denoted $\\beta _ { \\infty } ^ { \\alpha }$ solves a constrained optimisation problem involving the hyperbolic entropy introduced by [13]: ",
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|
| 524 |
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"text": "$$\n\\beta _ { \\infty } ^ { \\alpha } = \\operatorname * { a r g m i n } _ { \\beta \\in \\mathbb { R } ^ { d } } \\operatorname* { m i n } _ { s . t . X \\beta = y } \\phi _ { \\alpha } ( \\beta ) : = \\frac { 1 } { 4 } \\big [ \\sum _ { i = 1 } ^ { d } \\beta _ { i } \\mathrm { a r c s i n h } ( \\frac { \\beta _ { i } } { 2 \\alpha _ { i } ^ { 2 } } ) - \\sqrt { \\beta _ { i } ^ { 2 } + 4 \\alpha _ { i } ^ { 4 } } \\big ] ,\n$$",
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| 525 |
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"type": "text",
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"text": "Though the hyperbolic entropy function has a non-trivial expression, its principal characteristic is $\\ell _ { 1 }$ nd thand $\\ell _ { 2 }$ $\\alpha$ precisely for. We refer to $\\begin{array} { r } { \\tau \\in \\mathbb { R } ^ { 2 } \\colon \\dot { \\phi _ { \\alpha } } ( \\beta ) \\underset { \\alpha \\to 0 } { \\sim } \\frac { 1 } { 2 } \\ln \\left( \\frac { 1 } { \\alpha } \\right) \\| \\bar { \\beta } \\| _ { 1 } } \\end{array}$ $\\begin{array} { r } { \\bar { \\phi _ { \\alpha } ( \\beta ) } \\underset { \\alpha \\to + \\infty } { = } - \\frac { 1 } { 2 } \\bar { \\alpha ^ { 2 } } + \\frac { 1 } { 1 6 \\alpha ^ { 2 } } \\| \\beta \\| _ { 2 } ^ { 2 } + o ( \\alpha ^ { - 2 } ) . } \\end{array}$ [36, Theorem 2] for more details on the asymptotic analysis. The implicit optimisation problem (4) therefore highlights the fact that the initialisation scale of the weights controls the shape of the recovered solution. Small initialisations lead to low $\\ell _ { 1 }$ -norm solutions which are known to induce good generalisation properties: this is what is often referred to as the rich regime. Large initialisations lead to low $\\ell _ { 2 }$ -norm solutions: this is referred to as the kernel regime or lazy regime in which the weights move only very slightly. The dynamics of the gradient flow are then very similar to the one of kernel linear regression with the kernel depending on the initialisation [20, 12]. Overall, to retrieve a sparse solution, one should initialise with the smallest $\\alpha$ possible. However, as is clearly explained in [36], it is important to stress out that there is a generalisation $/$ optimisation tradeoff: the point $w = 0$ happens to be a saddle point for the loss and a smaller $\\alpha$ will lead to a longer training time. ",
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"text": "Main result. In the main theorem we show that, for an initialisation scale $\\alpha$ , the stochasticity of SGF biases the flow towards solutions which still minimise the hyperbolic entropy. However, what is remarkable is that it does so with an effective parameter $\\alpha _ { \\infty }$ which is strictly smaller than $\\alpha$ . The recovered solution therefore minimises an optimisation problem which has better sparsity inducing properties than that of gradient flow. ",
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"type": "text",
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| 558 |
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"text": "Theorem 1. For $\\begin{array} { l } { p \\ \\leq \\ \\frac { 1 } { 2 } } \\end{array}$ and $w _ { 0 , \\pm } = \\alpha \\in ( \\mathbb { R } _ { + } ^ { * } ) ^ { d }$ , let $( w _ { t } ) _ { t \\geq 0 }$ follow the stochastic gradient flow (3) with step size $\\begin{array} { r } { \\gamma \\leq O \\big ( \\big [ \\ln ( \\frac { 4 } { p } ) \\lambda _ { \\operatorname* { m a x } } \\operatorname* { m a x } \\{ \\| \\beta _ { \\ell _ { 1 } } ^ { * } \\| _ { 1 } \\ln \\big ( \\frac { \\| \\beta _ { \\ell _ { 1 } } ^ { * } \\| _ { 1 } } { \\operatorname* { m i n } _ { i } \\alpha _ { i } ^ { 2 } } \\big ) , \\| \\alpha \\| _ { 2 } ^ { 2 } \\} \\big ] ^ { - 1 } \\big ) } \\end{array}$ where $\\beta _ { \\ell _ { 1 } } ^ { * } =$ arg min $\\| \\beta \\| _ { 1 }$ and $\\lambda _ { \\mathrm { m a x } }$ is the largest eigenvalue of $X ^ { \\top } X / n$ . Then, with probability at least $1 - p$ : $\\beta \\in \\mathbb { R } ^ { d } \\ s . t . \\ X \\beta = y$ ",
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| 559 |
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"type": "text",
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"text": "• $( \\beta _ { t } ) _ { t \\geq 0 }$ converges towards a zero-training error solution $\\beta _ { \\infty } ^ { \\alpha }$ ",
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| 570 |
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"type": "text",
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"text": "• the solution $\\beta _ { \\infty } ^ { \\alpha }$ satisfies ",
|
| 581 |
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"text": "$$\n\\beta _ { \\infty } ^ { \\alpha } = \\underset { \\beta \\in \\mathbb { R } ^ { d } } { \\arg \\operatorname* { m i n } } \\quad \\phi _ { \\alpha _ { \\infty } } ( \\beta ) \\quad w h e r e \\quad \\alpha _ { \\infty } = \\alpha \\odot \\exp \\left( - 2 \\gamma \\dim \\operatorname { g } \\left( \\frac { X ^ { \\top } X } { n } \\right) \\int _ { 0 } ^ { + \\infty } L ( \\beta _ { s } ) \\mathrm { d } s \\right) .\n$$",
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"text": "The theorem is three-fold: with high probability and for an explicit choice of constant step size $\\gamma$ , (i) the flow $( \\beta _ { t } ) _ { t \\geq 0 }$ converges, (ii) its limit $\\beta _ { \\infty } ^ { \\alpha }$ is an interpolating solution, i.e. $X \\beta _ { \\infty } ^ { \\alpha } = y$ , (iii) this solution minimises the hyperbolic entropy problem with a parameter that depends on the dynamics. We illustrate these results in Figure 2. Now let us comment further the theorem. ",
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"img_path": "images/fec063349e8357718e26e29511e1cf5c2fea77e382260222c4aad05d6c7c350b.jpg",
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"image_caption": [
|
| 617 |
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"Figure 2: Sparse regression (see Section 5.1 for the detailed experimental setting). Both SGD and GD are initialised at $\\alpha = 0 . 1$ . 2 different runs of SGD over the training set are performed, they differ due to the inner stochasticity of the algorithm. Left: GD and SGD both converge towards a global minimum. Middle and right: for two different trajectories of SGD, the higher the value of the loss integral at convergence, the better the validation loss. In both cases SGD converges towards a solution which generalises better than GD. This figure illustrates Theorem 1. "
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| 630 |
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"text": "Beneficial implicit bias through effective initialisation. The most remarkable aspect of the result is that the recovered solution $\\beta _ { \\infty } ^ { \\alpha }$ minimises the same potential as for gradient flow but with an effective parameter $\\alpha _ { \\infty }$ which is strictly smaller than $\\alpha$ . Hence, the hyperbolic entropy is closer to the $\\ell _ { 1 }$ norm compared to the deterministic case, proving a systematic benefit of stochasticity. Note that this effective parameter is random and controlled by the loss integral $\\begin{array} { r l } { \\int _ { 0 } ^ { + \\infty } L ( \\beta _ { s } ) \\mathrm { d } s , } \\end{array}$ : the higher the integral, the smaller the effective initialisation scale. In other words and quite surprisingly, the slower the loss converges to 0, the “richer” the implicit bias. However, it must be kept in mind that, as explained in [36], there is a tension between generalisation and optimisation: a longer training time might improve generalisation but comes at the cost of... a longer training time. Yet it is clear experimentally that SGD systematically largely wins the trade-off over GD (see Figure 2). Interestingly, Problem (5) tells us that the implicit bias of SGD initialised at $\\alpha$ acts as if we run GD initialised at $\\alpha _ { \\infty }$ (see Section 5.3). Note that the minimisation problem (5) only makes sense $a$ posteriori since the quantity $\\alpha _ { \\infty }$ depends on the whole stochastic trajectory. Finally, an interesting question is whether one can quantify the scale of this beneficial phenomenon, i.e. how small $\\alpha _ { \\infty }$ is compared to $\\alpha$ . To answer this, we quantify the scale of the loss integral w.r.t. $\\gamma$ and $\\alpha$ (see Proposition 3) and show under slightly stronger conditions that the relative scale $\\alpha _ { \\infty } / \\alpha$ decays as power of $\\alpha$ (See Eq. (8) of the main text and Proposition 6 of the appendix for a proof). ",
|
| 631 |
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| 637 |
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|
| 638 |
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|
| 639 |
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|
| 640 |
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"type": "text",
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| 641 |
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"text": "Kernel regime. Though it is less our focus, our result still holds as $\\alpha + \\infty$ which corresponds to the kernel regime. In this regime, we believe that $\\begin{array} { r } { \\int _ { 0 } ^ { + \\infty } L ( \\beta _ { s } ) \\mathrm { d } s \\underset { \\alpha \\infty } { } 0 } \\end{array}$ (not shown in the paper but experimentally observed) and hence SGF and GF converge towards the same solution. This is expected since in the NTK regime, the iterates follow a kernel linear regression for which the bias of SGF and GF are the same. ",
|
| 642 |
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| 650 |
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| 651 |
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"type": "text",
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| 652 |
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"text": "Step size. Note that the convergence of the iterates holds for a constant step size. This is not illogical since in the overparametrised setting, the noise vanishes at the optimum (see [32] for a convergence result in the overparametrised least-squares setup). The explicit formula for the $\\gamma$ upper bound is $\\begin{array} { r } { \\gamma \\leq \\left( 4 0 0 \\ln \\left( \\frac { 4 } { p } \\right) \\lambda _ { \\operatorname* { m a x } } \\big ( \\frac { X ^ { \\top } X } { n } \\big ) \\operatorname* { m a x } \\left\\{ \\| \\beta _ { \\ell _ { 1 } } ^ { * } \\| _ { 1 } \\ln \\left( \\sqrt { 2 } \\frac { \\| \\beta _ { \\ell _ { 1 } } ^ { * } \\| _ { 1 } } { \\operatorname* { m i n } _ { i } \\alpha _ { i } ^ { 2 } } \\right) , \\| \\alpha \\| _ { 2 } ^ { 2 } \\right\\} \\right) ^ { - 1 } } \\end{array}$ . It has a classical dependence on $\\lambda _ { \\operatorname* { m a x } } ( X ^ { \\top } X / n )$ which can be computed, but also on the unknown value of $\\| \\beta _ { \\ell _ { 1 } } ^ { * } \\| _ { 1 }$ . However in practice we choose the highest value of $\\gamma$ for which the iterates converge. Note that in practice the weights are often initialised such that $\\| \\alpha \\| _ { 2 } ^ { 2 }$ is roughly equal to 1 and hence it is sensible to consider $\\| \\alpha \\| _ { 2 } ^ { 2 } < \\| \\beta _ { \\ell _ { 1 } } ^ { * } \\| _ { 1 }$ . In the explicit bound, there is a $\\ln \\left( \\lVert \\boldsymbol { \\beta } _ { \\ell _ { 1 } } ^ { * } \\rVert _ { 1 } / \\operatorname* { m i n } _ { i } \\alpha _ { i } ^ { 2 } \\right) ^ { - 1 }$ factor, we believe that it is an artefact of our analysis and could be removed. It is hence best to think of the upperbound on $\\gamma$ to simply be $\\begin{array} { r } { \\gamma \\le O ( \\frac { 1 } { \\lambda _ { \\operatorname* { m a x } } \\| \\beta _ { \\ell _ { 1 } } ^ { * } \\| _ { 1 } } ) } \\end{array}$ . ",
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| 653 |
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| 660 |
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| 661 |
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| 662 |
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"type": "text",
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| 663 |
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"text": "Convergence and proof sketch. Let us put emphasis on the fact that since we deal with a nonconvex problem, neither convergence nor convergence towards a global minimum are obvious. In most of similar works, convergence of the iterates is assumed [36, 14]. In fact, the hardest and most technical part of our result is to show the convergence of the flow with high probability: once the convergence is shown, describing the minimisation problem $\\beta _ { \\infty } ^ { \\alpha }$ verifies is straightforward. In the following section we give several properties which constitute the major keys of the theorem’s proof. ",
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"type": "text",
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"text": "4 Links with mirror descent ",
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| 675 |
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"text": "The aim of this section is to show that the sequence $( \\beta _ { t } ) _ { t \\geq 0 }$ follows a stochastic version of continuous mirror descent with a time dependent mirror. From this crucial property, we show how the convergence and implicit bias characterisation follow. Finally, as it is one of the central objects of our main theorem, we give an estimation of $\\int _ { 0 } ^ { \\infty } L ( \\beta _ { s } ) \\mathrm { d } s$ . ",
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| 696 |
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"type": "text",
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| 697 |
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"text": "4.1 Stochastic continuous mirror descent with time-varying potential ",
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| 709 |
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"text": "We start by recalling known results on the link between implicit bias and mirror descent. We recall also convergence guarantees for mirror descent dynamics. ",
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| 720 |
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"text": "Mirror descent: convergence and implicit bias. For any $\\beta _ { 0 } \\in \\mathbb { R } ^ { d }$ and convex potential function $\\Psi$ , consider the mirror descent flow $( \\beta _ { t } ) _ { t }$ which corresponds to $\\mathrm { d } \\nabla \\Psi ( \\beta _ { t } ) = - \\bar { \\nabla L } ( \\beta _ { t } ) \\mathrm { d } t$ . Though the convergence of the loss to 0 is straightforward, showing the convergence of the iterates requires more work and is shown in [4, Theorem 2] for strongly convex potentials. Yet, once the convergence of the iterates is shown, deriving the implicit minimisation problem is straightforward. We recall the reasoning here (see Section 3 of [2] for more details): integrating the flow yields $\\nabla \\Psi ( \\beta _ { \\infty } ) -$ $\\begin{array} { r } { \\nabla \\Psi ( \\beta _ { 0 } ) = - \\int _ { 0 } ^ { \\infty } \\nabla L ( \\beta _ { s } ) \\mathrm { d } s = - 4 X ^ { \\top } \\int _ { 0 } ^ { \\infty } X ( \\beta _ { s } - \\beta _ { \\infty } ) \\mathrm { d } s \\in \\mathrm { s p a n } ( X ) } \\end{array}$ . This condition, along with the fact that $X \\beta _ { \\infty } = y$ exactly corresponds to the KKT conditions of the problem: ",
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| 732 |
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"text": "$$\n\\beta _ { \\infty } = \\underset { \\beta \\in \\mathbb { R } ^ { d } \\mathrm { ~ s . t . ~ } X \\beta = y } { \\arg \\operatorname* { m i n } } D _ { \\Psi } ( \\beta , \\beta _ { 0 } ) ,\n$$",
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| 742 |
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| 743 |
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"type": "text",
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| 744 |
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"text": "where $D _ { \\Psi } ( \\beta , \\beta _ { 0 } ) = \\Psi ( \\beta ) - \\Psi ( \\beta _ { 0 } ) - \\langle \\nabla \\Psi ( \\beta _ { 0 } ) , \\beta - \\beta _ { 0 } \\rangle$ is the Bregman divergence w.r.t. $\\Psi$ . ",
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| 754 |
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"type": "text",
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| 755 |
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"text": "Link with our model. It turns out that these general observations on mirror descent apply to our framework when $( w _ { t } ) _ { t }$ follows the gradient flow $\\mathrm { d } w _ { t , \\pm } = - \\nabla _ { w _ { \\pm } } L ( w _ { t } ) \\mathrm { d } t$ . Indeed it has been shown in [36] that the corresponding iterates $\\beta _ { t } = w _ { t , + } ^ { 2 } - w _ { t , - } ^ { 2 }$ follow a mirror descent with potential $\\phi _ { \\alpha }$ defined in Eq.(4). Therefore we can apply the previous remarks to obtain the convergence towards an interpolator3, as well as the associated implicit minimisation problem which in our case can be rewritten as $\\beta _ { \\infty } ^ { \\alpha } = \\arg \\operatorname* { m i n } \\quad \\phi _ { \\alpha } ( \\beta )$ since $\\nabla \\phi _ { \\alpha } ( \\beta _ { 0 } = 0 ) = 0$ . \n$\\beta \\in \\mathbb { R } ^ { d }$ s.t. $X \\beta { = } y$ ",
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| 763 |
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| 764 |
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| 765 |
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"type": "text",
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| 766 |
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"text": "Stochastic Mirror descent with a time varying potential. To address the problem where $( w _ { t } ) _ { t }$ follows a stochastic gradient flow instead of a gradient flow, it is natural, as in the deterministic framework, to see what type of flow $( \\beta _ { t } ) _ { t }$ follows. Because of the noise, we cannot hope to simply recover a classical mirror descent. However interestingly the next property shows that it follows a stochastic mirror-like descent with a geometry that depends on time. ",
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| 767 |
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"type": "text",
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| 777 |
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"text": "",
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| 778 |
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| 787 |
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"type": "text",
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| 788 |
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"text": "Proposition 1. Consider the iterates $( w _ { t } ) _ { t \\geq 0 }$ issued from the stochastic gradient flow in Eq.(3) with initialisation $w _ { 0 , \\pm } = \\alpha \\in ( \\mathbb { R } _ { + } ^ { * } ) ^ { d }$ . Then the corresponding flow $( \\beta _ { t } ) _ { t \\geq 0 }$ follows a “stochastic continuous mirror descent with time varying potential” defined by: ",
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| 789 |
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"img_path": "images/f534a5258dcd0553c7cda83c0e7d1f0884c89265ffbf8245cf3e29bfcc35f3a8.jpg",
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| 800 |
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"text": "$$\n\\mathrm { d } \\nabla \\phi _ { \\alpha _ { t } } ( \\beta _ { t } ) = - \\nabla L ( \\beta _ { t } ) \\mathrm { d } t + \\sqrt { \\gamma n ^ { - 1 } L ( \\beta _ { t } ) } \\boldsymbol { X } ^ { \\top } \\mathrm { d } \\boldsymbol { B } _ { t } ,\n$$",
|
| 801 |
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"text_format": "latex",
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| 802 |
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"bbox": [
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| 809 |
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| 811 |
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| 812 |
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"text": "where $\\begin{array} { r } { \\alpha _ { t } = \\alpha \\odot \\exp \\left( - 2 \\gamma \\operatorname { d i a g } \\left( \\frac { X ^ { \\top } X } { n } \\right) \\int _ { 0 } ^ { t } L ( \\beta _ { s } ) \\mathrm { d } s \\right) } \\end{array}$ and $\\phi _ { \\alpha }$ is the hyperbolic entropy defined in (4). ",
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| 813 |
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"bbox": [
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| 821 |
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|
| 822 |
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"type": "text",
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| 823 |
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"text": "Under this form we clearly see that the iterates $( \\beta _ { t } ) _ { t }$ follow a flow which closely resembles that of mirror descent but with two major differences: (i) the potential $\\phi _ { \\alpha _ { t } }$ changes over time according to the random quantity $\\int _ { 0 } ^ { t } L ( \\beta _ { s } ) \\mathrm { d } s$ , (ii) the flow is perturbed by noise. We highlight the fact that viewing the dynamics this way has the major advantage of giving a clear roadmap for the proof of Theorem 1: (i) we can adapt classical mirror-descent results to our framework and construct appropriate Lyapunov functions to prove the convergence of the flow with high probability to some interpolator $\\beta _ { \\infty } ^ { \\alpha }$ , (ii) we immediately recover the corresponding minimisation problem as in the deterministic case. Indeed, integrating Eq.(7) still yields $\\nabla \\phi _ { \\alpha _ { \\infty } } ( \\beta _ { \\infty } ^ { \\alpha } ) \\in \\mathrm { s p a n } ( X )$ which, along with $X \\beta _ { \\infty } ^ { \\alpha } = y$ , are the KKT conditions of the implicit minimisation problem (5). We emphasise the fact that the structure of the noise, belonging to $\\operatorname { s p a n } ( X )$ , is crucial in order to obtain this minimisation problem. This would for instance clearly not be true if we considered isotropic noise in the SDE modelling. This highlights the fact that not every form of noise improves the implicit bias: the shape of the intrinsic SGD noise is of primal importance [15]. ",
|
| 824 |
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"bbox": [
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| 831 |
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},
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| 832 |
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{
|
| 833 |
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"type": "text",
|
| 834 |
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"text": "4.2 Convergence and control of $\\int _ { 0 } ^ { \\infty } L ( \\beta _ { s } ) \\mathrm { d } s$ ",
|
| 835 |
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"text_level": 1,
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| 836 |
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| 843 |
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"type": "text",
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| 846 |
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"text": "Though it seems easy to derive the implicit minimisation problem (5) from the mirror-like structure of Eq.(7), it is necessary to ensure that the iterates converge towards an interpolator $\\beta _ { \\infty }$ . This is the purpose of the following proposition. ",
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| 847 |
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| 855 |
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|
| 856 |
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"type": "text",
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| 857 |
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"text": "Proposition 2 (Convergence of the iterates). Consider the iterates $( w _ { t } ) _ { t \\geq 0 }$ issued from the stochastic gradient flow (3), initialised at $w _ { 0 , \\pm } = \\alpha \\in ( \\mathbb { R } _ { + } ^ { * } ) ^ { d }$ . For $\\begin{array} { r } { p \\leq \\frac { 1 } { 2 } } \\end{array}$ and $\\gamma$ such as in Theorem $^ { l }$ , then with probability at least $1 - p ,$ the flow $( \\beta _ { t } ) _ { t }$ converges to an interpolating solution $\\beta _ { \\infty } ^ { \\alpha }$ . ",
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| 858 |
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|
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"type": "text",
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| 868 |
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"text": "The convergence of the iterates is technical and requires several intermediate results. We start by considering an appropriate Bregman-type stochastic function with a time-varying potential and show that it converges with high probability. Leveraging the fact that we are able to bound the iterates $\\beta _ { t }$ , we are able to show that the limit of the function is in fact 0. Owing to the fact that the function we consider also controls the distance of $\\beta _ { t }$ to a particular $\\beta ^ { * }$ we finally get that the iterates converge. ",
|
| 869 |
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"bbox": [
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|
| 878 |
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"type": "text",
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| 879 |
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"text": "However for the objects (such as $\\alpha _ { \\infty }$ ) and functions we introduce to be well defined, we need to guarantee the convergence of $\\int _ { 0 } ^ { \\infty } L ( \\beta _ { s } ) \\mathrm { d } s$ . Besides, it is crucial to grasp the scale of this quantity since it gives the overall scale of $\\alpha _ { \\infty }$ . This is done in the following proposition where we lower and upper bound its value. ",
|
| 880 |
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{
|
| 889 |
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"type": "text",
|
| 890 |
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"text": "Proposition 3. Under the same setting as in Proposition 2 with initialisation $w _ { 0 , \\pm } = \\alpha \\mathbf { 1 }$ , we have with probability at least $1 - p$ : ",
|
| 891 |
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"page_idx": 7
|
| 898 |
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|
| 899 |
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{
|
| 900 |
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"type": "equation",
|
| 901 |
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"img_path": "images/b37d9063dbef10395361243242e9518512fc02421666456c8c9b5a25e1e7bdcc.jpg",
|
| 902 |
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"text": "$$\n\\Omega \\Big ( \\| \\beta _ { \\ell _ { 1 } } ^ { * } \\| _ { 1 } \\ln \\Big ( \\frac { \\| \\beta _ { \\ell _ { 1 } } ^ { * } \\| _ { 1 } } { \\alpha ^ { 2 } } \\Big ) \\Big ) \\underset { \\alpha \\to 0 } { \\leqslant } \\int _ { 0 } ^ { + \\infty } L ( \\beta _ { s } ) { \\mathrm { d } } s \\leqslant O \\Big ( \\operatorname* { m a x } \\big \\{ \\| \\beta _ { \\ell _ { 1 } } ^ { * } \\| _ { 1 } \\ln \\Big ( \\frac { \\| \\beta _ { \\ell _ { 1 } } ^ { * } \\| _ { 1 } } { \\alpha ^ { 2 } } \\Big ) , \\alpha ^ { 2 } d \\big \\} \\Big ) .\n$$",
|
| 903 |
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"text_format": "latex",
|
| 904 |
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"bbox": [
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|
| 909 |
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| 910 |
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"page_idx": 7
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| 911 |
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|
| 912 |
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|
| 913 |
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"type": "text",
|
| 914 |
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"text": "We point out that the lower bound is given for small $\\alpha$ ’s for simplicity but we provide in Lemma 7 (Appendix B.5) a lower bound which holds for all $\\alpha$ ’s. Note that when $\\gamma = 0$ , which corresponds to deterministic gradient flow, we can give the exact value for the integral: $\\begin{array} { r } { \\int _ { 0 } ^ { + \\infty } L ( \\beta _ { s } ) \\mathrm { d } s = } \\end{array}$ $\\begin{array} { c l c r } { \\frac { 1 } { 2 } D _ { \\phi _ { \\alpha } } \\big ( \\beta _ { \\infty } ^ { \\alpha } , \\beta _ { 0 } \\big ) } \\end{array}$ (see Proposition 7 in Appendix C). This matches the scale of the bounds given in Proposition 3, hence showing the tightness of the result. We focus now on how this translates to the scale of the effective initialisation w.r.t. $\\alpha$ when this latter is small enough. In fact, this lower bound on the integral of the loss along with a stronger assumption on the boundedness of the iterates lead to ",
|
| 915 |
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| 922 |
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| 923 |
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| 924 |
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"type": "equation",
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"img_path": "images/42b998f00cc0a581decc3c7e9278118d14d05703f778c317a34d0cd3bc2654e6.jpg",
|
| 926 |
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"text": "$$\n\\frac { \\alpha _ { \\infty } } { \\alpha } \\underset { \\alpha 0 } { \\leqslant } ( \\frac { \\alpha ^ { 2 } } { \\| \\beta _ { \\ell _ { 1 } } ^ { * } \\| _ { 1 } } ) ^ { \\zeta } ,\n$$",
|
| 927 |
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"text_format": "latex",
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| 928 |
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"bbox": [
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| 935 |
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| 936 |
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|
| 937 |
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"type": "text",
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| 938 |
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"text": "for some $\\zeta > 0$ . Hence the smaller the initialisation scale $\\alpha$ and the greater the benefit of SGD over GD in terms of implicit bias (see Appendix B.6 for more details). ",
|
| 939 |
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| 948 |
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"type": "text",
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| 949 |
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"text": "Again, the proof of this proposition is technical and relies on considering appropriate Lyapunov functions which highly resemble to Bregman divergences, but which take into account the fact that the geometry changes over time. These overall decreasing Lyapunov’s enable to bound the iterates as well as lower and upper bound the integral of the loss. The stochastic integrals which naturally appear are controlled with high probability using time-uniform concentration of martingales [19]. ",
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| 959 |
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"type": "text",
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"text": "5 Experiments ",
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"text": "5.1 Experimental setup for sparse regression ",
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| 973 |
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"type": "text",
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"text": "We consider the following sparse regression setup for our experiments. We choose $n = 4 0$ , $d = 1 0 0$ and randomly generate a sparse model $\\beta _ { \\ell _ { 0 } } ^ { \\ast }$ such that $\\| \\beta _ { \\ell _ { 0 } } ^ { * } \\| _ { 0 } = 5$ . We generate the features as $x _ { i } \\sim \\mathcal { N } ( 0 , I )$ and the labels as $y _ { i } = x _ { i } ^ { \\top } \\beta _ { \\ell _ { 0 } } ^ { * }$ . SGD, GD and the SGF are always initialised using the same scale $\\alpha > 0$ and it is specified each time. We use the same step size for GD and SGD and choose it to be the biggest as possible why still ensuring convergence. Note that since the true population covariance $\\mathbb { E } [ x x ^ { \\top } ]$ is equal to identity, the quantity $\\lVert \\beta _ { t } - \\beta _ { \\ell _ { 0 } } ^ { * } \\rVert _ { 2 } ^ { 2 }$ corresponds to the validation loss. ",
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"type": "text",
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| 995 |
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"text": "5.2 Validation of the SDE model ",
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"text_level": 1,
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| 1006 |
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"type": "text",
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| 1007 |
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"text": "In this section, we present an experimental validation of the stochastic gradient flow model. In Figure 3, for the same step size, we run: (i) the trajectory of gradient descent, (ii) 5 trajectories of stochastic gradient descent that correspond to different realisations of the uniform sampling over the data, (iii) 5 trajectories of the stochastic gradient flow (its Euler discretisation with $\\mathrm { d } t = \\gamma / 1 0 $ )) corresponding to different realisations of the Brownian. We clearly see (left) that the loss behaves similarly for SGD and SGF across time. We also see that the validation losses (right) of the iterates of SGD and SGF have very similar behaviours. This tends to validate our continuous modelling from Section 2.2. ",
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},
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| 1016 |
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|
| 1017 |
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"type": "image",
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| 1018 |
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"img_path": "images/c80042837b130c94a7a101d35c6c494c46490def198558b9ee38e05f11479992.jpg",
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| 1019 |
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"image_caption": [
|
| 1020 |
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"Figure 3: Sparse regression (see Section 5.1 for the detailed experimental setup). Left and right: the training and the validation losses behave very similarly, corroborating the continuous modelling. "
|
| 1021 |
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|
| 1022 |
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|
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|
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"type": "text",
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| 1033 |
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"text": "5.3 GD and SGD have the same implicit bias, but from different initialisations. ",
|
| 1034 |
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"type": "text",
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| 1045 |
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"text": "In order to confirm and illustrate the main Theorem 1, we provide the following experiment which is illustrated Figure 4. We first run GD and SGD with the same step-size and initialise them both at $\\alpha \\mathbf { 1 }$ with $\\alpha ~ = ~ 0 . 0 1$ . As expected, the solution recovered by SGD generalises better. Then, using the iterates $\\beta _ { t } ^ { \\mathrm { S G D } }$ from the first SGD run, we compute the value $\\begin{array} { r } { \\alpha _ { \\infty } = \\alpha \\exp ( - 2 \\gamma \\operatorname { d i a g } ( X ^ { \\top } X / n ) \\int _ { 0 } ^ { \\infty } L ( \\beta _ { s } ^ { \\mathrm { S G D } } ) \\mathrm { d } s ) \\in \\mathbb { R } ^ { d } } \\end{array}$ (the integral is approximated by its discrete time approximation with $\\mathrm { d } t = \\gamma$ ). We then run gradient descent but this time initialised at $w _ { 0 , \\pm } = \\alpha _ { \\infty }$ . According to our main result from Theorem 1, it should approximately (it would be exact if we ran SGF and GF) converge to the same solution as SGD initialised at $\\alpha \\mathbf { 1 }$ . This is clearly observed Figure 4 (right). Also note that SGD and GD (initialised at $\\alpha _ { \\infty }$ ) seem to have overall very similar dynamics, this is not shown by our results and we leave this as future work. However keep in mind that though the validation losses converge at the same iteration rate, in terms of computation time, SGD is $n$ times faster. ",
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| 1046 |
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"text": "",
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| 1057 |
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|
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|
| 1067 |
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"img_path": "images/9bf19469529b7bb71b26709898ba73a91666e10c1a9460bfd728fa9d739a7fb5.jpg",
|
| 1068 |
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"image_caption": [
|
| 1069 |
+
"Figure 4: Sparse regression (see Section 5.1 for the detailed experimental setup). Left and right: SGD initialised at $\\alpha \\mathbf { 1 }$ converges towards the same point as GD initialised at $\\begin{array} { r l } { \\alpha _ { \\infty } } & { { } = } \\end{array}$ $\\begin{array} { r } { \\tilde { \\alpha \\exp ( - 2 \\gamma \\operatorname { d i a g } ( X ^ { \\top } X / n ) \\int _ { 0 } ^ { \\infty } { { L } ( \\beta _ { s } ^ { \\mathrm { { S G } \\tilde { D } } } ) \\mathrm { { d } } s } ) } } \\end{array}$ . "
|
| 1070 |
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|
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|
| 1072 |
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| 1081 |
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"text": "5.4 Doping the implicit bias with label noise ",
|
| 1083 |
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"text_level": 1,
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"bbox": [
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| 1094 |
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"text": "As largely discussed throughout the paper, the effect of the implicit bias is controlled by the convergence speed of the loss: the slower it converges, the sparser the selected solution will be. Hence the following question: can we leverage this knowledge to dope the implicit bias? We argue in this Section that the answer to this question is affirmative. Indeed, consider a sequence $( \\delta _ { t } ) _ { t \\in \\mathbb { N } } \\overline { { \\in } } { \\mathbb { R } } _ { + } ^ { \\mathbb { N } }$ and assume that we artificially inject some label noise $\\Delta _ { t }$ at time $t$ , say for example $\\Delta _ { t } \\sim \\mathrm { U n i f } \\{ 2 \\delta _ { t } , - 2 \\delta _ { t } \\}$ (independently from $i _ { t }$ ). This injected label noise perturbs the SGD recursion as follows: ",
|
| 1095 |
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"img_path": "images/d4c1efa4e607861dd2c5b3e8b631d2ba1877fa79b72b206368722a21b2ab69ea.jpg",
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| 1106 |
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"text": "$$\nw _ { t + 1 , \\pm } = w _ { t , \\pm } \\mp \\gamma \\left( \\langle \\beta _ { w } - \\beta ^ { * } , x _ { i _ { t } } \\rangle + \\Delta _ { t } \\right) x _ { i _ { t } } \\odot w _ { t , + } ,\n$$",
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"text": "As in Section 2.2, we can derive its related stochastic gradient flow (see Appendix D.1 for more details): ",
|
| 1119 |
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|
| 1130 |
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"text": "$$\n\\mathrm { d } w _ { t , \\pm } = - \\nabla _ { w _ { \\pm } } L ( w _ { t } ) \\mathrm { d } t \\pm 2 \\sqrt { \\gamma n ^ { - 1 } ( L ( w _ { t } ) + \\delta _ { t } ^ { 2 } ) } w _ { t , + } \\odot [ X ^ { \\top } \\mathrm { d } B _ { t } ] .\n$$",
|
| 1131 |
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"text_format": "latex",
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"bbox": [
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"type": "text",
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| 1142 |
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"text": "Assuming that $( \\delta _ { t } ) _ { t \\geq 0 } \\in ( \\mathbb { R } _ { + } ) ^ { \\mathbb { R } }$ and $\\gamma$ are such that the iterates converge, the corresponding implicit regularisation minimisation problem is preserved but with a \"slowed down\" loss: $\\tilde { L } ( \\beta _ { t } ) : = L ( \\beta _ { t } ) + \\delta _ { t } ^ { 2 }$ and the effective initialisation writes: $\\begin{array} { r } { \\tilde { \\alpha } _ { \\infty } = \\alpha \\odot \\exp \\left( - 2 \\gamma \\mathrm { d i a g } ( \\frac { X ^ { \\top } X } { n } ) \\int _ { 0 } ^ { + \\infty } \\tilde { L } ( \\beta _ { s } ) \\mathrm { d } s \\right) } \\end{array}$ . The label noise therefore helps recovering a solution which has better sparsity properties. However, it must be kept in mind that adding too much label noise can significantly slow down the convergence of the validation loss or even prevent the iterates from converging. Yet, experimental results showing the impressive effect of label noise are provided Figure 5 in Appendix D.1. ",
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| 1143 |
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},
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"type": "text",
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"text": "6 Conclusion and Perspectives ",
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| 1154 |
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| 1165 |
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"text": "In this paper, we have shown the benefit of using stochastic gradient descent over gradient descent for diagonal linear networks in terms of their implicit bias. Indeed, we prove that stochastic gradient flow acts as gradient flow but initialised at a smaller scale: this induces a sparser finale iterate. This effect is controlled by the speed of convergence of the loss. Moreover, we prove the convergence of the flow and exhibit an interesting link with mirror descent. Fully understanding this novel type of dynamics could help to grasp the implicit biasing properties of stochastic gradient descent in other frameworks. It is also natural to ask whether the integral of the loss also controls the difference of implicit regularisation for more general architectures. It would also be interesting to analyse how this property adapts to log losses known to lead to max-margin solutions in classification. ",
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"type": "text",
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| 1176 |
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"text": "Acknowledgements. NF would like to thank Nathan Srebro for introducing him to the question of SGD’s implicit bias as well as for the stimulating discussions they had during his visit at EPFL. ",
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| 1177 |
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